ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS IN ANALYSIS
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NORTH-HOLLAND MATHEMATICS STUDIES
47
Analytical and Numerical Approaches to Asymptotic Problems in Analysis Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems Universityof Nijmegen,The Netherlands, June 9-13,1980
Edited b y
0.AXELSSON L. S. FRANK Mathematicallnstitute The University of Nqmegen TheNetherlands and
A. VAN DER SLUE Mathematical institute The University of Utrecht TheNetherlands
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NEW YORK
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North-Holland Publishing Company, I980
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N O R T H - H O L L A N D PUBL.ISHING C O M P A N Y A M S T E R D A M O N E W YORK * O X F O R D S o l r di\rribiiror\for /tic U . . S . A . r r i r t l ~ ' u i r u d ~ r : ELSEVIER N O R T H - H O L L A N D . INC. 5 2 V A N D E R B I L T A V E N U E . NEW YORK. N . Y . 10017
Library of Congress Cataloging in Publication Data Conference on Analytical and Numerical Approaches to Asymptotic Problems, University of Nijmegen, 1980. Analytical and numerical approaches to asymptotic problems in analysis. (North-Holland mathematics studies ; 47) Differential equations--Asymptotic theory-Congresses. 2. Differential equations--Numerical solutions-Congresses. I. kcelsson, Axel, 1934-11. Frank, Leonid S., 1934- 111. Sluis, Abraham van der. IV. Title. &A370.C63 1980 515.3'5' 80-26580 ISBN 0-444-86131-9 1.
P R I N T E D IN T H E N E T H E R L A N D S
PREFACE
A n International Conference on Analytical and Numerical Approaches to Asymptotic
Problems was held in the Faculty of Science, University of Nijmegen, The Netherlaads from June 9th through June 13th, 1980 The development of the analytical asymptotic theory has been essentially stimulated by needs of applied sciences such as fluid dynamics, diffraction theory, reactiondiffusion processes, elasticity theory, quantum mechanics and so on.
On the other
hand, pure aesthetics have also been a strong attraction for mathematicians and theoretical physicists in the elaboration of this beautiful mathematical tool.
The
introduction of numerical methods during the last decades became a new turning point in the further development of asymptotic analysis. The main motivation in organizing this Conference was the idea to bring together people working in asymptotic theory and to try to survey achievements in this field of applied mathematics, due to the combination and synthesis of analytical and numerical approaches. The attendance at the Conference and discussions during and after its sessions were an explicit indication of a growing interest amongst mathematicians, numerical analysts and theoretical physicists for new aspects and methods in asymptotic analysis. The Proceedings of the Conference contain the full text of 17 invited addresses and
11 contributed papers. We are grateful to the participants, especially to the speakers who were instrumental in making this meeting a fruitful and enjoyable scientific event.
We are also most
indebted to the Administration of the Faculty of Science, University of Nijmegen, whose support was an indispensable contribution to the success of this Conference.
0. AXELSSON
L.S. FRANK A. VAN DER SLUIS Editors V
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ACKNOWLEDGEMENTS
The Conference on ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N ANALYSIS w a s sponsored by t h e U n i v e r s i t y of Nijmegen, t h e Dutch M i n i s t r y o f E d u c a t i o n , t h e Dutch Mathematical S o c i e t y , t h e I n t e r n a t i o n a l B u s i n e s s Machines C o r p o r a t i o n , t h e O f f i c e of t h e Naval Research (London), and t h e U n i t e d S t a t e s Army Research and Development Group ( E u r o p e ) .
We are d e e p l y i n d e b t e d t o a l l s p o n s o r s f o r t h e i r s u p p o r t of t h i s Conference.
THE ORGANIZING COMMITTEE
vii
INTERNATIONAL CONFERENCE ON ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N ANALYSIS
JUNE 9-13,
1980, FACULTY OF SCIENCE, UNIVERSITY OF NIJMEGEN, THE NETHERLANDS
Photograph taken of s o m e of t h e p a r t i c i p a n t s b e f o r e an excursion
CONTRIBUTORS
B. Aulbach, Mathematical Institute, The University of Wiirzburg, Germany 0. Axelsson, Mathematical Institute, The University of Nijmegen, The Netherlands
M. Bertsch, The Mathematical Centre, Amsterdam, The Netherlands D. Caillerie, Laboratoire de mCcanique theorique, Universite Pierre et Marie Curie, Paris, France S . Dobrokhotov, Institute of Machinebuiding, Moscow, USSR J.J. Duistermaat, Mathematical Institute, The University of Utrecht, The Netherlands P.C. Fife, Department of Mathematics, The University of Arizona, Tucson, U.S.A. L.S. Frank, Mathematical Institute, The University of Nijmegen, The Netherlands R. Geel, Ubbo Emmius Institute, Groningen, The Netherlands D.F. Griffiths, Department of Mathematics, University of Dundee, Scotland E.W.C. van Groesen, Mathematical Institute, The University of Nijmegen, The Netherlands I. Gustafsson, Mathematical Institute, The University of Nijmegen, The Netherlands F.C. Hoppensteadt, Department of Mathematics, University of Utah, Salt Lake, U.S.A. D. Huet, U.E.R. de Mathematiques, Universite de Nancy I, France A. Iserlies, Department of Applied Mathematics, University of Cambridge, England E.M. de Jager, Mathematical Institute, The University of Amsterdam, The Netherlands C. Johnson, Chalmers Institute of Technology, GGteborg, Sweden Ya. Kannai, The Weizmann Institute of Science, Rehovot, Israel R.B. Kellogg, Department of Mathematics, The University of Maryland, College Park, U.S.A. J . Lorentz, Mathematical Institute, The University of Konstanz, Germany V.P. Maslov, Institute of Machinebuilding, Moscow, USSR R.M.M. Mattheij, Mathematical Institute, The University of Nijmegen, The Netherlands A. Meiring, Department of Mathematics, University of Dundee, Scotland A.R. Mitchell, Department of Mathematics, University of Dundee, Scotland S. Osher, Department of Mathematics, The University of California, Los Angeles, U.S.A.
J.P. Pauwelussen, The Mathematical Centre, Amsterdam, The Netherlands L . A . Peletier, Mathematical Institute, The University of Leiden, The Netherlands H.4. Reinhardt, Mathematical Institute, The J.W. Goethe University, Frankfurt, Germany M. Schatzman, Universite Pierre et Marie Curie, Paris, France K. Soni, Mathematics Department, The University of Tennessee, Knoxville, U.S.A. R.P. Soni, Mathematics Department, The University of Tennessee, Knoxville, U.S.A. M. Tabata, Kyoto University, Japan and Universit6 Pierre et Marie Curie, Paris, France M.E. Taylor, Department of Mathematics, The Rice University, Houston, U.S.A. R. Temam, Laboratoire d'Analyse Numerique, Universite Paris-Sud, Orsay, France
ix
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TABLE OF CONTENTS
PREFACE
V
vi
ACKNOWLEDGEMENTS
vii
CONTRIBUTORS
xiii
OPENING ADDRESS PART I
:
INVITED LECTURES
Finite gap almost periodic solutions in asymptotical expansions S.Yu. DOBROKHOTOV and V.P. MASLOV Periodic solutions near equilibrium pqints of Hamiltonian systems
1
n . DUISTERMAAT
27
Asymptotics for elementary spherical functions n. DUISTERMAAT
35
On the question of the existence and nature of homogeneous-center target patterns in the Belousov-Zhabotinskii reagent P.C. FIFE
45
Singular perturbations of hyperbolic type E.M. DE JAGER and R. GEEL
57
Computation by extrapolation of solutions of singular perturbation problems F.C. HOPPENSTEADT and W.L. MIRANKER
73
Proper approximation of a normed space and singular perturbations D. HUET
a7
An analysis of some finite element methods for advection-diffusion problems C. JOHNSON and U. NAVERT
99
Short time asymptotic behavior for parabolic equations Y. KA"A1
117
Difference approximation for a singular perturbation problem with turning points R.B. KELLOGG
133
Stability and consistency analysis of difference methods for singular perturbation problems J. LORENZ
141
Finite element Galerkin methods for convection-diffusion and reaction-diffusion A.R. MITCHELL, D.F. GRIFFITHS and A. MEIRING
157
xi
xii
CONTENTS
Numerical solution of singular perturbation problems and hyperbolic systems of conservation laws S. OSHER
179
Nonstationary filtration in partially saturated media M. BERTSCH and L.A. PELETIER
205
A-Posteriori error estimates and adaptive finite element computations for singularly perturbed one space dimensional parabolic equations H.-J. REINHARDT
213
Diffraction of waves by cones and polyhedra M.E. TAYLOR
2 35
Some asymptotic problems in mechanics R. TEMAM
249
PART I1
:
CONTRIBUTED PAPERS
Asymptotic amplitude and phase for isochronic familie..of periodic solutions B. AULBACH
265
Quasioptimal finite element approximations of first order hyperbolic and of convection-dominated convection-diffusion equations 0. AXELSSON and I. GUSTAFSSON
273
Homogenization of the equation of stationary diffusion in cylindrical domains D. CAILLERIE
281
Singular perturbations of an elliptic operation with discontinuous nonlinearity L . S . FRANK and E.W. VAN GROESEN
289
Coercive singular perturbations: asymptotics and reduction to regularly perturbed boundary value problems L.S. FRANK and W.D. WENDT
305
Efficient two-step numerical methods for parabolic differential equations A. ISERLES
319
Estimating the discretization error in three point difference schemes for second order linear singularly perturbed BVP R.M.M. MATTHEIJ
327
Failure of nerve impulse propagation for nonuniform nerve axons J.P. PAUWELUSSEN
339
The penalty method for the vibrating string with an obstacle M. SCHATZMAN
345
On the asymptotic behavior of the solution of a nonlinear Volterra integral equation K. SONI and R.P. SONI
359
Conservative upwind finite element approximation and fts applications M. TRBATA
369
OPENING ADDRESS bY L.S. FRANK
Ladies and Gentlemen, Distinguished Guests and Participants of the Conference, Dear Colleagues and Friends It is a privilege to welcome you at the Opening'of the Conference on "Analytical and numerical approaches to asymptotic problems in analysis", which is being held in the Faculty of Science of the University of Nijmegen. The main motivation in organizing such a conference has been the idea to bring together mathematicans and theoretical physicists working in asymptotic analysis and to try to survey recent achievements in this field. Considerable progress in the treatment of applied problems affected by the presence of small or large parameters has resulted from the combination and synthesis of analytical and numerical approaches in asymptotic analysis. The fundamental rdle played by asymptotics in the applied sciences is not only stressed in mathematical research but is also acknowledged by our colleagues in physics, chemistry, biology and other disciplines of the natural sciences. Perhaps one of the most striking illustrations of the efficient use of asymptotic methods is provided by fluid dynamics, that has been particularly inspiring for the development of other mathematical branches as well. It is well known that the Euler-Lagranye equations for nonviscous fluids gave rise to several paradoxes, which could not be explained in the framework of these equations. One can mention here some of them:
lo. The reversibility paradox connected with the drag and lift acting on a solid in steady translation. 2'.
Paradox of D'Alambert leading to an absurd conclusion that the dray and
the lift are both zero. 3'.
Paradox of Jukowski's ideal flow (the drag is zero and the lift is defi-
ned in an artificial way). 4'.
The development of different kinds of shocks, which contradicts Leibnitz' xiii
xiv
OPENING ADDRESS
numerical approaches to asymptotic problems. This combination turns out to be an efficient mathematical tool to investigate asymptotic phenomena in applied scien-
"Factwn e s t " (it is done), a second small parameter h took its place next to
ces.
the biq brother
E.
The name of Gottfried Silbermann is not widely spread in mathematical circles. Indeed, this gentleman was never involved in any business connected with asymptotics. He is known amongst professional musicians as a creator of the piano, one of the wonders of the musical world. Of course, his claim to the piano paternity was disputed by some other European colleagues of him, a situation which might make mathematicians feel a little more at home when talking about Gottfried Silbermann . Unfortunately, the time-limitation does not allow me to go into detailed analysis of Gottfried Silbermann's criativity. But, the wonderful idea of the piano, as combination and synthesis of a clavichord and an harpsichord can be compared with the idea of putting together analytical and numerical aspects leading to asymptotic problems affected by the presence of two parameters
E
and h.
Ladies and Gentlemen, a very old-fashioned principle in criminology (which one should consider nowadays as a very applied science) says (in French): "Chey-
cher la fermne".
This principle, of course, is not any longer fashionable amongst
criminologists. In asymptotic analysis of different phenomena in natural sciences one of the oldest (but not old-fashioned) principles: "look for a srnaZZ p a r ~ m c t e ~ ~ " turns out to be as vital and strong, as the thirst of human beings for knowledge. Indeed, this small thing, the proverbial parameter
E
provides the mathematical
world with such beauty, that one would be tempted to compare its r6le with that played in men's lives by our charming female companions. Of course, one should keep in mind that the beauty is always in the eye of the beholder. I feel that I should stop here, lest I be accused by the Women's Liberation
movement of discriminating against the better half of human kind and before I start talking on behalf of men's liberation. Anyhow, let me express the hope that the conference on "Analytical and numerical approaches to asymptotic problems in analysis", being held in Nijmegen, will be an enjoyable meeting of people who placed their yearnings for beauty in the wonderful world of the small parameter. One last word. We are, certainly, aware, all of us, of a possible misuse of
our knowledge. Namely, some political systems in the Twentieth century, are still making full use of scientific research for the preparation of war, the grave limitation of freedom and the presecution of dissidents and non-conformists. May we hope that our research will be exploited for the protection of our freedom at least as efficiently, as it has been misused for some evil purposes,
OPENING ADDRESS
xv
famous maxim: "natura non f a c i t saZtus" (the nature does not tolerate jumps). In the book by Garrett Birkhoff: "Hydrodynamics, a study in logic, fact and similitude", one comes across of the following witticism, due to Sir Cyril Hinshelwood and quoted previously by M.J. Lighthill: in the last century " f l u i d
dynamists were divided i n t o hydroZic engineers who observed what could not be explained, and muthernaticians who explained things t h a t could not be observed". This witticism reflects the paradoxial state of fluid dynamics in the past.
A
following comment by Birkhoff, himself, is not very reassuring with respect to the modern times: "It is my impression", he says, "that many survivors of both species are still with us".
Shouldn't one recall in this case Carlyle's famous epigram
(to be found in "Heroes and Hero-Worship"): their s i l e n c e is more eloquent than
words If? The Navier-Poisson equations, fully justified later on by the use of the continuum-approach in the works of San-Venan and Stokes, seemed to be an adequate mathematical model in order to save theoretical fluid dynamics and to remove some of its paradoxes, at least in the cases when relativistic, quantum and some other effects could be neglected. The Navier-Stokes equations, thanks to Reinold's number had become one of the most striking examples of singular perturbations, leading to the idea of boundary layer, due to Prandtl. Prandtl's theory (developed later on in a more rigorous and consistent way) provides a beautiful mathematical tool for the investigation of practical hydrodynamical problems. Still, it is unable to explain some other phenomena, such as, for instance, the appearence of turbulent flows (a well-known boundary layer paradox).
Italian is probably a suitable language to describe this situation: se non
& vero 6 bene t r ovato (if it is not true, still it is a good discovery). Further developnents of the small parameter method led to its efficient use in other fields of applied mathematics. Here, one can mention, for instance:
lo. Von Karman's model of thin elastic plates, that stimulated the development of the elliptic singular perturbation theory. 2O.
Diffraction theory, that gave birth to the Fourier integral operators,
and so on. The problems to be treated by means of asymptotic analysis and analysis itself are becoming progressively, more and more, complicated.
')Ad augusta per angusta"(to
august results by narrow paths) says a latin aphorism used also as a catch-word in Victor Hugo's "Hernani". Or, in other words, one achieves a triumph only by overcoming difficulties.
A new turning point in the development of asymptotic analysis occurred with the introduction of numerical methods and the combination of both analytical and
xvi
OPENING ADDRESS
up t o now. May I r e c a l l h e r e , when t a l k i n g a b o u t t h e v a l u e o f p e r s o n a l freedom f o r human b e i n g s , t h e remarkable poem, t h a t was w r i t t e n by P a u l E l u a r d i n 1943 i n o c c u p i e d P a r i s and t h a t ends w i t h t h e f o l l o w i n g words E t p a r l e p o u v o i r d ' u n mot
J e recommence ma v i e . Je s u i s n6 pour t e c o n n a i t r e ,
Pour t e nommer: Liberte. L a d i e s and Gentlemen, t h a n k you f o r your a t t e n t i o n
ANALYTICAL AND NUMERICAL 4PPRCjACtiES TO ASYMPTOTIC PROBLEMS IN A N A L Y S I S 5 . A x e l s s o n , L . S . F r a n k , A . v a n d e r S l u i s (eds.) @ N o r t h - t i o l l a n d Publishing C o m p a n y , 1 9 6 1
FINITE GAP ALMOST PERIODIC SOLUTIONS IN ASYMPTOTICAL EXPANSIONS S.Yu. Dobrokhotov, V . P .
Maslov
Moscow Institute of Electronic Machinebuilding Moscow USSR
Recently discovered almost periodic solutions of some nonlinear equations are shown to be used in asymptotical expansions in the following way: firstly for generalization of the multiphase nonlinear WKB method (Whitham method) both with real and complex phases, secondly for quantization of these solutions being Lagrangian manifolds in quasiclassical approximation.
1.
INTRODUCTION
An important class of exact solutions which were called the almost periodic finite gap solutions was recently obtained for some nonlinear wave equations with partial derivatives. The methods of these solutions constructing became available after the founding paper by S.P. Novikov ilO3 had appeared and are nowadays being further developed and applied to practically all equations, for which the multisoliton solutions have been found earlier. The finite gap almost periodic solutions form the generalization of cnoidal waves, which are known in the shallow water theory, i.e. of solutions of the following form: U(x,t)
,
y(Ux t Vt)
=
x,t
€
R
2
,
(1.1)
where Y(T) is an elliptic function. Thus the finite gap almost periodic solutions are defined by the formula: U(X,t)
a.
y ( U1 x
=
t
v 1 t ,..., u kx +
(1.2)
Vet)
Here the parameters U . and V . as well as U, V in (1.1) may accept both real and 3
J
k
e ( T l,...,~Q),T
...,
similarly = (Tl, T k to elliptic functions, has the property of being 2k-periodic in k-dimensional com-
complex values, and the function y plex-valued space C
(T)
y
=
a. . Namely there exist
29. complex-valued E-dimensional vectors
(vector periods) a , = (ajl,...,a . 1 and bj = (bjl,...,b . 3
3k
3k
)
j = 1 ,...,k which are
linearly independent of the ring of integers, and such that II Q Q y (T+b.) = y fa ) = y (T) for all j,k = 1, . . . , k . Without loss of generality we 3 k
1
2
S
.Yu . DOBROKHOTOV and
V . P . MASLOV
consider the vectors a . being real-valued and a , = 0, if j # i. Thus for real U . 1 ii 7 and V . solution (1.2) represents an almost periodic function in x and in t. Being 3
a limit (degenerate) case the solutions of the form (1.2) contain also multisoliton solutions: then the lengths of P. periods of function y
a.
(7)
are equal to infi-
nity . Cnoidal waves (1.1) appear to be the nonlinear analogue of plane waves, i.e. of the linear wave equations solutions of the form: v
=
A cos(kx
+
wt)
(1.3)
and they form such a narrow class among all solutions of the equation under consideration as some plane waves among all solutions of wave equations. Moreover, it is well known, that solutions of the form (1.3) exist only for equations with constant coefficients except some rare cases. At the same time various problem of mathematical and theoretical physics lead to equations with a small parameter in the highest derivatives and the approximate solutions of such equations can be constructed by means of asymptotic methods. One of the basic linear mathematical physics methods of such type is the WKB method which represents the solution as a sum of "distorted" plane waves:
or in the exponential form: U = $(x,t) exp(iS(x,t)/h).
Here the phase
S
(1.5)
(x,t) and the amplitude A(x,t) (or @ (x,t)) are smooth real-valued
functions and h is the small parameter. The analogous asymptotic solution in the nonlinear case due to the plane waves correspondence to the cnoidal ones has the form of a "distorted" cnoidal wave; 1 U = y (S (x,t)/h)
(1.6)
where y'(~) is the same elliptic function as in (1.1). First in case of nonlinear ordinary differential equations G.E. Kuzmak (1959) suggested to find asymptotical solutions in the form (1.4) and proposed the method of their construction, then G.B. Whitham (1968) made the same for equations with partial derivatives. In a great number of papers the construction method of solutions in the form (1.6)was further developed and improved, and its application to investigation of some real physical problems was considered. According to 1171 we shall call this method a nonlinear one. Rather full bibliography is available in 16, 8, 19, 20, 23, 24, 91. The solutions of plane waves or cnoidal waves form (1.31, (1.1) are self-similar: i.e. at any time T they can be represented as a single sinusoidal or cnoidal wave,
3
F I N I T E GAP ALMOST P E R I O D I C SOLUTIONS
if at any time t < T they were represented in this form. The analogical statement does not hold for asymptotic WKB-solutions of the form (1.3) even in the linear case: as a rule at some critical time T and in some point (called focal) the cr the phase S(x,t) ceases to be a single-valued function. Then at times t > T cr asymptotic solution is constructed by means of a canonical operator r l 3 l and, generally speaking, has the form distinct from (1.3). In particular, outside the vicinity of branching points of function S(x,t) (focal points) at times t
>
T
cr
the corresponding asymptotical solution is represented as a sum: u = A (x,t) cos
1
s1 (x,t) h
~
...
+
sQ (x,t) + A a (x,t) cos ___ , e r 1
thus it is a multiphase asymptotic solution. Here the phases S
j
(1.7)
and the amplitudes
A j are again smooth functions. Note that the solutions of such type arise not only due to focal points, but they necessary arise in boundary value and mixed problems. Analogical multiphase solutions appear as well in nonlinear equations. The necessaty of their investigation was pointed out already in 1967 by Lighthill, Whitham and others. Whitham e.g. writes in c221: "An extended theory with possibility of more than one principal mode would be necessary. It is possible that such a theory could be developed and the interaction treatment for nearly linear modes would help in this connexion" ( p . 1 8 ) . Each summand in (1.7) is an asymptotic solution, thus (1.7) is the superposition
of "distorted" plane waves. Such a representation as a sum of separate solutions is possible only in the linear case, so the nonlinear generalization of the solution has the form 11,231:
where y
a(
...,T a )
so.(x,t)
s1 (x,t)
~
u = y
-(
,
.
.
.
I
h
)
'
(1.8)
is the same function as in (1.2), which holds for exact almost
T ~ ,
periodic solutions. The construction scheme of multiphase asymptotical solutions of the form (1.8) is practically the same (in any case for the principal term) as in case of the single phase solution (1.6). The lack of methods allowing to deter-
a
mine the function y ( T l , . . . , ~) prevented it from realization till the mentioned
a
above paper by S.P. Novikov was published, Nowadays the methods of constructing exact almost periodic solutions (1.2) are essentially developed, see e.g. the review and bibliography in [lo, 16, 241. The authors of this paper used the finite gap almost periodic solutions in the problem of reflection from the boundary for the generalized Sine-Gordon equation: [4,
5, 61:
S.Yu. DOBROKHOTOV and V.P. MASLOV
4
and in wave train existance problem for Korteveg-de Vries equation which was set by Lighthill and Whitham 1221. Independently, H. Flaschka,
G.
Forest and
D.W. Mc Laughlin C8l investigated the "distorted" almost periodic finite gap s o l u tions of equation (1.9). This paper deals with the results obtained in the wave train superposition problem for Korteveg-de Vries equation. First let us mention the following. So far we dealt only with asymptotical solutions with real-valued phases S.(x,t), but S.(x,t) can also be complex-valued 1
1
functions. In linear case the asymptotical solutions with single complex phase S have the form:
and boundedness of these solutions as h
-f
+O implies the inequality
ImStO
(1.11)
The asymptotic solution (1.10) is both fast oscillating (due to the factor exp iRe S/h) and fast diminishing as h equation
r
= {
the vicinity
(x,t)
r,
:
+
+O outside the set
r
defined by the
In S(x,t) = 01: Thus the function (1.10) is concentrated in
i.e. for each fixed x,t
E
r
m
U(x,t,h) = O(h
).
The solutions of
linear 'equations of form (1.10) concentrated in the vicinity of curves being Hamilton systems trajectories are well known and can be obtained by the model equations method 121 or by the complex canonical operator (complex germ) method 1141: the latter concerns the solutions concentrated in the vicinity of some kdimensional surfaces, k > 1 , or more complicated manifolds (see 113,141. In the nonlinear case solutions concentrated in the vicinily of curves in Rn were obtained in the following papers [3,141. Such asymptotical solutions analogical to (1.10) have the form:
1 u = y (S(x,t)/h)
(1.12)
where the complex phase S is a smooth function satisfying (1.11) and the elliptic 1
function y (T) (see (1.1)) is supposed to be bounded in the whole complex halfplane Im
T
2
0. It turns out the last condition defines among solutions (1.1) the
soliton (degenerate) solutions. Thus the asymptotical solutions of form (1.12) of nonlinear equations with complex phase are expressed in terms of soliton functions. Note that solutions (1.10) and (1.12) as well (1.4) are complex. To obtain the real-valued solutions of initial equation, one should set v = Re U = (of course, if the function
4 (U +
-
iS,/h
U) = 5 $ e
~
Li
p
.-is/h
(1.13)
a, complex conjugate to the solution U, is a solution).
In case of a real-valued phase S the function v again can be represented as a
5
FINITE GAP ALMOST PERIODIC SOLUTIONS single phase function
/$I
cos(S/h + Arg
$)
,
in case of an arbitrary complex
function satisfying (1.11) there does not exist such a representation, and (1.13) is a twe phase solution with phases S
1
= S
iSl/h
v = % ~ $ e
and
s
2
=
-
-S:
iS20h + + T e
(1.14)
Hence in linear case the real-valued asymptotical solution with the complex phase S(x,t) is always a multiphase (at least two phase) solution. The same is true for nonlinear equations: the real-valued asyrnptotical solutions are always multiphase, being expressed in terms of exact multisoliton (i.e. degenerate finite gap) solutions of the form (1.2). The multiphase solution of such form, both real and complex, were obtained for the Sine-Gordon equation by the authors of this paper, and for the equation describing a proper nondegenerate semiconductor as well: (1.15) In particular for the last equation the asyrnptotical solutions of Steklov-type problem, concentrated in the vicinity of stable closed geodesics on 252 are obtained in a convex compact domain R with the smooth boundary. Analogical solutions are obtained in a half-finite straight cylinder, and for equation (1.15) with more general nonlinear terms. In the latter case the soliton functions y
I!
(T,
...,7 9.)
can-
not be expressed in terms of elementary functions and are expressed as Dirichlet series. At last note that an exception among asymptotical solutions with the complex phase
S is the case Re S = 0, where the single phase asyrnptotical solutions can be realvalued. Such asymptotical solutions turning into shock wave or infinitely narrow solitons as h
-f
0 were obtained in 1151.
Another domain of finite gap solutions applications is based on the fact that these solutions form Lagrangian tori (generally speaking, in infinitely dimensional phase space), which can be quantized in quasi-classical approximation by the methods of 113,141. The phase space of periodic Toda lattice is finite dimensional and the quasi-classical quantization can be grounded strictly (see 1 6 1 ) .
2.
WAVE TRAIN INTERACTION PROBLEM IN LINEAR CA SE
Let us consider Korteveg-de Vries equation in the following form: Ut - 6UUx
+
2
h Uxxx = 0
,
x E R
,
(2.1)
were h > 0 is a small parameter depending on a number of physical constants. At first we consider the wave train interaction problem for the linearized Korteveg-
S.YU
6
. DOBROKHOTOV
and V . P. MASLOV
de Vries equation: V
2
t + h Vxxx
,
= 0
(2.2)
f o r which w e s e t t h e Cauchy problem w i t h f i n i t e f a s t o s c i l l a t i n g i n i t i a l v a l u e s : V(t,O
= $ ( x ) cos
kx
h
,
(2.3)
where k > 0 i s t h e wave number and $(x) i s a smooth f u n c t i o n w i t h t h e s u p p o r t belonging t o an i n t e r v a l i n R. The a s y m p t o t i c s o l u t i o n of problem (2.21, (2.3) i s e a s i l y o b t a i n e d by means of WKB-method, i t s p r i n c i p a l term h a s t h e form; V = $(x
+
2
3k t ) cos
3
-(
kx+k t h ) '
(2.4)
and a p p e a r s t o b e a p l a n e wave t r a i n . The sum of any two s o l u t i o n s i n form (2.4) c o r r e s p o n d i n g t o d i f f e r e n t wave numbers k l and k2 i s a l s o a s o l u t i o n of e q u a t i o n
(2.2).
L e t us c o n s i d e r s u c h a s o l u t i o n :
v
=
4
1
(X
+ 3k
2
1
3
t)
k x+klt
2 t $ ( x t 3k t ) cos(-2
)
COS(-
3 k2x+k t 2 h
I f
(2.5)
> k2 > 0 and t h e s u p p o r t s of smooth f u n c t i o n s $ ( x ) , $2 (x) belong t o d i s 1 1 j o i n t i n t e r v a l s i a O , b O ] and [ c d 1 r e s p e c t i v e l y , and d < a o . The s o l u t i o n ( 2 . 5 )
where k
0' 0
0
s a t i s r l e s the i n i t i a l conditions: VltzO
=
(XI
cos
k x
1
~
h
+
$ (x) cos
2
k x 2 h
-.
(2.6)
I t c o n s i s t s o f two d i s t a n t running t r a i n s of p l a n e waves till a c e r t a i n i n s t a n t of
t i m e . A s k l > k2 t h e t r a i n w i t h t h e impulse k l r u n s f a s t e r , t h a n t h a t w i t h t h e impulse k2
,
and c a t c h e s i t . They superposed f o r some t i m e . Then t h e t r a i n w i t h
t h e impulse k l r u n s ahead and t h e s o l u t i o n a g a i n c o n s i s t s of two d i s t a n t t r a i n s . Note t h a t t h e f u n c t i o n (2.6) can be r e p r e s e n t e d i n t h e form of an o s c i l l a t i n g function with a s i n g l e phase:
where $
0
,
S o a r e smooth f u n c t i o n s
0
(one s h o u l d s e t :
$ (x) =
tJ1(X)
+ $,(x)
,
(2.8)
and demand t h a t So = k x f o r x E [ a O , b O 1 and So = k x f o r x E [ c o , d o l ) . Consider 1 2 t h e Caushy problem f o r e q u a t i o n (2.2) i n case, when S o i s an i n c r e a s i n g f u n c t i o n e q u a l t o k x f o r x > a. and t o k x f o r x 4 d o . The p r i n c i p a l t e r m of t h i s problem 1 2 a s y m p t o t i c a l s o l u t i o n i s e x p r e s s e d i n terms of s o l u t i o n s of t h e f o l l o w i n g Hamilton i a n system:
FINITE GAP ALMOST PERIODIC SOLUTIONS
7
The set of these solutions has the form: 0 p = p (a)
,
0 2 q = -3(p (a))t
and for each t defines in the phase space the curve:
1
At
=
0 0 2 (p = p (a) , g = -3(p 1 t + a)
.
= min Before the critical time t the I curve A1 can be projected one-tocr t a 6P Pa one on the axis q and the asymptotical solution of problem (2.2), (2.7) has only 1 one phase. After the time tCr the focal points appear on the curve At, which is
not already projected one-to-one on the axis q , and outside the focal points vicinity the asymptotical solution can be represented in the form: 0
vo
=
z
j
@ ( a .(x,t))
1 1 cosl0 h 1-6p0pa(", (x,t))tl
JI
(S
0
( a .(x,t) 3
1
where C . is the index of the curve point l 1 3 1 with the coordinate u.(x,t) defined 1
3
by the equation q(a,t) = x. In case the function Q0 has the form (2.8) this sum consists only of two summands and (2.9) coincided with (2.5). Thus the wave train interaction problem can be regarded as a simplified variant of asymptotical solution construction problem for time greater than critical.
3.
WAVE TRAIN INTERACTION PROBLEM IN NONLINEAR CASE
Traditionally cnoidal wave (real-valued) solutions of Korteveg-de Vries equation (2.1) are defined in terms of elliptic Jacobi function dn(z,k) and have the form: u = b -
u
=
2 dn2(=,
b
4 2 -L
12 K(k) '
v=
t co,
k),
T =
ux + Vt ___ h '
J . 2 2(6b + i e ( k 2 - 2)) , 12 K(k) 2 3
where a, b, co, k are real-valued parameters, a > 0, 1 > k
2
0, K(k) is the com-
plete elliptic integral of the first kind. The function is completely defined by 1 - E(k)/K(k), the parameters a, b, c0 and k. It means value is equal to C = b - 6 where E(k) is the complete elliptic integral of the second kind. Thus u is a fast oscillating function (with the frequency
*
lih as h
+
0) with the mean value C.
S . Yu. DOBROKHOTOV and V . P .
8
MRSLOV
I t i s more c o n v e n i e n t t o e x p r e s s u i n t e r m s of t h e second d e r i v a t i v e of 0, where
0 i s a Riemann f u n c t i o n 116,241. Namely, l e t H , C , C0 and U b e r e a l p a r a m e t e r s , / H I < 1. Cnoidal wave s o l u t i o n s of Korteveg-de V r i e s e q u a t i o n can b e e x p r e s s e d i n
& f u n c t i o n as f o l l o w i n g :
u = y
1
ux + V t ( 7 B) ,
(3.1)
where B = -i I n H , 0 - f u n c t i o n i s d e f i n e d by means of t h e s e r i e s :
k=-m
k=-m
and V i s a f u n c t i o n of H ( o r B ) , U , C. The c o r r e s p o n d i n g formula f o r V ( H , U , C ) be g i v e n below.
will
(We do n o t g i v e t h e r e l a t i o n s between H , U , C and a , b , k because
we need o n l y ( 3 . 1 ) ) . Note t h a t t h e mean v a l u e o f t h e f u n c t i o n u i s e q u a l t o C and that u
-t
C for H
+
0.
The " d i s t o r t e d " f a s t o s c i l l a t i n g c n o i d a l wave can be r e p r e s e n t e d by formula ( 3 . 1 1 , where U, H , C, C
0
and V are f u n c t i o n s smoothly depending on x , t and, p r o b a b l y , on
the parameter h , h e [ O , l l .
This function i s a sel f - si m i l ar
asymptotical s o l u t i o n
o f Korteveg-de V r i e s e q u a t i o n on small time i n t e r v a l s , i . e . f o r s u f f i c i e n t l y small time t t h e a s y m p t o t i c a l s o l u t i o n of Caushy problem w i t h i n i t i a l c o n d i t i o n i n t h e form of a " d i s t o r t e d " c n o i d a l wave (and s a t i s f y i n g some c o n d i t i o n s on U , H and o t h e r s a c c u r a t e t o O ( h ) ) c a n be r e p r e s e n t e d i n t h e same form, of c o u r s e , U , H and o t h e r s depend on x ( s e e C9,231) i n t h e d i f f e r e n t way. The f u n c t i o n d e f i n e d by e q u a l i t y ( 3 . 1 ) , where
T =
s ( x ) and h
C ( x ) , S ( x ) , H(x) = exp
iB are 2
smooth f u n c t i o n s
w i t h supp H e [ a , b l , we s h a l l c a l l a t r a i n c o n c e n t r a t e d on t h e i n t e r v a l Ca,bl of " d i s t o r t e d " c n o i d a l waves o s c i l l a t i n g on a smooth background. N a t u r a l l y , two d i s t a n t c n o i d a l wave t r a i n s on a smooth background a r e d e f i n e d a n a l o g i c a l l y , w i t h t h e s u p p o r t H(x) b e l o n g i n g t o two d i s j o i n t i n t e r v a l s . The e x a c t s t a t e m e n t of t h e wave t r a i n i n t e r a c t i o n problem i s t h e f o l l o w i n g . One s h o u l d o b t a i n t h e e q u a t i o n ( 2 . 1 ) s o l u t i o n , s a t i s f y i n g f o r t = 0 t h e f o l l o w i n g condition:
where B = - 2 i I n H ( x , h ) and C ( x , t ) , S ( x , t ) , H a r e smooth f u n c t i o n s i n x h
E
c O , l l , w i t h H ( x , h ) b e i n g such t h a t I H ( x , h )
I
E
R and
< 1 'dx E R and s u p p H b e l o n g i n g t o
two d i s j o i n t i n t e r v a l s [ a , b l and [ c , d l . On a s u f f i c i e n t l y s m a l l t i m e i n t e r v a l t h e a s y m p t o t i c a l s o l u t i o n of problem ( 2 . 1 ) , ( 3 . 2 ) i s d e f i n e d ( s e e C231) a c c u r a t e t o O(h) by t h e same formula ( 3 . 2 ) , b u t h w i t h o t h e r (depending on t i m e ) f u n c t i o n s S , H and C , t h e f u n c t i o n s u p p o r t H b e l o n g i n g t o two d i s j o i n t i n t e r v a l s [ a t , b t l
and
9
F I N I T E GAP ALMOST P E R I O D I C SOLUTIONS
depending on t i m e . The a s y m p t o t i c a l s o l u t i o n w i l l c o n s i s t of two d i s t a n t
[ct,dtl
" d i s t o r t e d " c n o i d a l wave t r a i n s on a smooth background, d e f i n e d by t h e f u n c t i o n C ( x , t ) . I f t h e i n i t i a l f u n c t i o n s S,
H and C are s u c h t h a t a s y m p t o t i c a l s o l u t i o n
a t time t t h e d i s t a n c e becr' cr becomes e q u a l t o z e r o , t h e n one of t h e
p r e s e r v e s t h e mentioned above form f o r time t < t
tween t h e i n t e r v a l s [ a t , b t l and [ c d 1 t' t wave t r a i n s c a t c h e s t h e o t h e r and t h e y s u p e r p o s e a s i t i s i n t h e l i n e a r c a s e . I n t h i s case f o r t i m e t > t
t h e asymptotical s o l u t i o n r e p r e s e n t a t i o n ( 3 . 1 ) does not cr h o l d . Namely, w e s h a l l c o n s i d e r i n t h i s s i t u a t i o n t h e problem ( 2 . 1 ) , ( 3 . 2 ) and show, t h a t f o r t > t t h e a s y m p t o t i c a l s o l u t i o n c a n b e o b t a i n e d by means of " d i s cr t o r t e d " f i n i t e gap s o l u t i o n s . The e x a c t c o n d i t i o n s on t h e i n i t i a l f u n c t i o n s S, H and C which d e f i n e t h e problem d i s c u s s e d a r e g i v e n below.
4.
MULTIPHASE ASYMPTOTICAL SOLUTIONS
According t o t h e i n t r o d u c t i o n w e s e a r c h f o r t h e a s y m p t o t i c a l wave t r a i n i n t e r a c t i o n problem s o l u t i o n i n t h e form o f t h e series: S
v =
yk(h, x
k=O
,
t)hk
(4.1)
where y ( ~ , x , t ) ,T = (T ,..., T ~ ) ,S = ( S ( x , t ) ,...,S L ( x , t ) ) a r e new unknown k 1 1 smooth f u n c t i o n s , x E R, t E R , T E Rn; yk b e i n g 2 ~ - p e r i o d i c f u n c t i o n s i n each argument T l,...,~k. (We c o n s i d e r t h e method of c o n s t r u c t i o n t h e r n u l t i p h a s e solut i o n s ( 4 . 1 ) w i t h a n a r b i t r a r y number of p h a s e s , though i n t h e wave t r a i n i n t e r a c t i o n problem i t i s s u f f i c i e n t t o c o n s i d e r t h e case o f two p h a s e s : L: = 2 ) . I n s e r t i n g t h e f u n c t i o n ( 4 . 1 ) i n t o e q u a t i o n ( 2 . 1 ) , d i f f e r e n t i a t i n g and e q u a t i n g t o z e r o t h e c o e f f i c i e n t s of e q u a l powers o f h , w e o b t a i n t h e f o l l o w i n g system of relations: (4.21 (4.3) where A and
Fk
= S
a
+ aT 1
- * *
Skt
+
a 5
t
-
B = s
lx
a
aT1
+
...
+
sa.x
a
%-a. '
a r e p o l y n o m i a l s i n f u n c t i o n s ~ ~ , . . . , y ~S j-t ,~ ,S j x and t h e i r d e r i v a t i v e s .
I f the functions Sl , . . . , S k
and yk s a t i s f y t h e r e l a t i o n s ( 3 . 2 ) and ( 3 . 1 ) up t o t h e N
o r d e r N i n c l u s i v e l y , t h e f u n c t i o n uN = k$o y k ( S ( x , t ) / h , x , t ) h k s a t i s f i e s t h e equaN
t i o n ( 1 . 1 ) a c c u r a t e to O(h ) tions $(x,t,h), (x,t)
I
E
R2
X,t'
.
(By O(hN) we d e n o t e , as u s u a l l y , t h e smooth funch t (O,ll,
f o r which t h e f o l l o w i n g estimate h o l d s :
) . The v a r i a b l e s x , t s e r v e as p a r a xtt ( 4 . 3 ) and t h e e s s e n t i a l i n d e p e n d e n t v a r i a b l e s i n t h e s e r e l a t i o n s
l $ J ( x l t l h ) < c o n s t ( l ( ) h on each compact l( c R meters i n ( 4 . 2 ) ,
10
S.YU. DOBROKHOTOV and V . P . MASLOV
are Tl,...,~p..
All the relations ( 4 . 3 ) are linear uniform equations with partial
derivatives in variables ' r l , vatives in
,
T l,...,~Q
..., T~ ,
(4.2)
is also an equation with partial deri-
but a nonlinear one with constant coefficients of
(The number of unknown functions in the system of first relations ( 4 . 2 ) ,
Tl r * *
tTQ.
(4.3)
is
more than N, hence it can not be solved without any additional conditions. Thus the function yo should be 2n-periodic in should be bounded for each fixed x,t). Consider equation ( 4 . 2 ) .
T
~
..., , T~
and the functions y lf...'YN+l
Though the number of independent variables
T
in this
equation is equal to S, in fact there are only two independent variables
n
and 5
defined by the equalities:
Thus by changing the variables in ( 4 . 2 ) T .
1
= T . ( ~ , Q )= S .
1
n +
It
:
w(6,rl) = y (T ( 5 , r l ) ,..., r p . ( S , r i ) ) 0 1
S . 5 , j = 1,. ..,f,: IX
(4.4)
we again obtain the Korteveg-de Vries equation, but without the small parameter h:
w
n
- ~ W W
5
+ W
155
= O
Due to 2n-periodicity of yo in argument
(4.4')
the solution w is a conditionally pel riodic function for each fixed (x.t). Thus the construction problem of multiphase T.
asymptotical solution of Korteveg-de Vries equation is reduced in the principal term to the already solved problem of obtaining the conditionally periodic solution of the same equation. We give the explicit formulas in 0-functions for these solutions, obtained by A.P. Its and V.B. Matveev (see r 1 6 , 2 4 1 ) , which generalize the solutions of the form (3.1), ( 3 . 1 ' ) .
Consider the parameters which define them
to be functions in x and t. Let E (x,t),...,E2p.(x,t)be real-valued functions such that for any fixed (x,t) 0
holds max(Ezj+z,E2j+l) < min(E2j,E2j-1). Consider a set of hyperelliptic Riemann surfaces
r2&
w 2 = '2Q+1'
of the kind S given by the algebraic curves 2p. = Il (2-E.),which present for each (x,t) two complex planes 'ZP+l j=o 1
[E2p.-2rE2S-3]t*.-r[E2' E 1 1 1 glued on cuts, which are the intervals iE2p.~E2Q-llr and glued on the ray [EO,ml in the usual way. We choose from the set r on each
...,
b such that i) the intersecr Xlt the basis of cycles al,..., aS, bl' a. tion matrix of this basis has the form a 0 a . = hi 0 b . = 0, a. 0 b = Aij: ii) i i 1 1 1 and it surrounds each cycle is situated on the upper sheet of the surface r x,t only one cut which connects the points Ezj, E2j-1; iii) each cycle b . is situated surface
on both sheets of
r
it crosses only the cuts [E
1
,E2j-ll and [EO,ml, transi-
21 x,t' tion from the lower sheet to the upper and backward happens at the intersection-
points of b
1
with the cuts [E
1 and [Eo,m) respectively when moving along
211~21-1
11
FINITE GAP ALMOST PERIODIC SOLUTIONS
the cycle b . . One moves along the cycle a . clockwise, if E 3
3
clockwise otherwise; along the cycles b . clockwise. On each surface dW
j
=
E 2j-l, and counter
<
3
r
x,t Q- 1
C m=O
the coefficients
21
let us define the basis of holomorphic differentials:
C.
im
(E)zm dz/w, j
1
=
,..., 2
; E =
( E o , . ..,E
2e
)
,
of which are defined by the norming condition:
C.
3m
6 dW
ak The matrix of B-periods, B
=
1
2116.
=
3k
B(E) we define by the equalities
Note C16,241 that the matrix B is a symmetric one and Im B
>
0, in case under
consideration we have Re B . = 0 for k # J and Re B . = ~i (J-m), if m inequali3k 11 ties E < E and (j-m) inequalities E > E ,i-j hold. 2i 2i-1 2i 2i-1 By means of the matrix of B-periods we construct the Riemann 0-function of dimension 2 , which is the generalization of (3.1 ' )
i O ( T ~ B )= C exp(2 where
T =
t
( C ~ , . . . , T ~k) ,=
t
:
+
Bk,k >
<
(kl,..., k
e)
i
<
?,k
>)
,
(4.5)
are P,-dimensional vector-columns angular
brackets denote the real-valued scalar product, the summing is with respect to all k with integer components. The series (4.5) converges due to positiveness of According to the definition of the matrix B it is easy to
the matrix Im B[16].
E for any j, then Im B . . + m , the summands containing the I3 2j-1 2j in the sum (4.5) disappear, the !,-dimensional 0-function degenerates
show, that if E variable
T.
J
+
into a (P,-l)-dimensional one independent of surface, defined by the equation w:
=
T.
3
and corresponding to the Riemann
P2,e+l(~,E)/(z-E) ( z - E ~ ~ - ~ ) . 21
(In particular when U;=2 the two-dimensional &function
degenerates into one-dimen-
sional 0-function of the form (3.1").
for all j = 1 , ...,L , then a l l Im B , If E2j-1 + E21 33 (4.5) is identically equal to zero. Consider on every surface
r x,t
+
and the
-function
the following differentials: (4.6)
(4.7)
where the coefficients cn are defined from the conditions
6 dR
a m
n
=
0, n = 0, 1
3
;
m = 1 , ..., e , and denote by U and V the 2-dimensional vec-
12
S
.YU .
DOBROKHOTOV and V . P. MASLOV
t o r s w i t h t h e components:
u
1
=
,
u.(E) = bP Ro 1
,
V j = v . ( E ) = b!
1
j
j
=
1 ,..., k
.
1
The a l m o s t p e r i o d i c &-gap s o l u t i o n of t h e e q u a t i o n ( 4 . 4 ) c o r r e s p o n d i n g t o t h e surface
r
h a s t h e form:
X , t
...,C o e ( x , t ) )
where Co = ( C O 1 ( x , t ) ,
i s an a r b i t r a r y , g e n e r a l l y s p e a k i n g , complex2 9. .9
v a l u e d v e c t o r f o r any x , t , C ( E ) = - . Z
j=O
E
j
+ ,Z
6. zdW. j=l a j j
.
...,
j = 0, 2 k and t h e v e c t o r Co smoothly j' depend on x , t , t h e n t h e f u n c t i o n ( 4 . 7 ) a l s o smoothly depends on x , t . Hence, sub-
Evidently, i f t h e branching p o i n t s E s t i t u t i n g t h e v a r i a b l e s 5 and
n
a g a i n by t h e v a r i a b l e s
T
~
..., , T&,
taking i n t o
account ( 4 . 4 ) and ( 4 . 8 ) , w e o b t a i n e t h e f o l l o w i n g s t a t e m e n t :
...
,Cot ( x , t ) ) b e a smooth r e a l - v a l u e d v e c t o r L e t C ( x , t ) = (C ( x , t ) , 0 01 f u n c t i o n and l e t t h e smooth r e a l - v a l u e d v e c t o r - f u n c t i o n s
LEMMA 1.
S = (S ( X , t )
1
,...,S L ( x , t ) ) , E
= (E ( x , t )
0
,...,E k ( x , t ) ) b e
connected by t h e r e l a -
tions:
then t h e f u n c t i o n :
i s a 2n-periodic with r e s p e c t t o
..., T~
s u l o t i o n of e q u a t i o n ( 4 . 4 ) .
By means o f t h e c o n s t r u c t i o n scheme o f f i n i t e gap s o l u t i o n s a s e t of 2 n - p e r i o d i c s o l u t i o n s of e q u a t i o n ( 4 . 2 ) depending on t h e b r a n c h i n g p o i n t s of t h e s u r f a c e
rx , t
can be o b t a i n e d . We c o n s i d e r a n o t h e r scheme of f i n i t e gap s o l u t i o n s c o n s t r u c t i o n , a l l o w i n g t o e x p r e s s t h e p e r i o d i c a l s o l u t i o n s of e q u a t i o n ( 4 . 2 ) i n terms of t h e m a t r i x B e l e m e n t s (see
1 7 1 ) . ( T h i s scheme
i s c o n v e n i e n t when s o l v i n g t h e e q u a t i o n s
( 4 . 9 ) by t h e method o f p e r t u r b a t i o n t h e o r y ) . Consider t h e s e t of complex p a r a m e t e r s B . . = B . . j , i = 1 , . . . , k 11 11' B,, t h e following i n e q u a l i t y holds: matrix B = 11
11
11
I m B > 0. Denote by O n ( r / B ) t h e f u n c t i o n s :
such t h a t f o r t h e
(4.11)
13
FINITE GAP ALMOST PERIODIC SOLUTIONS
,...,k e ) ,
where k = L(kl
n
=
,...,ne 1
L(nl
are L-dimensional vector-columns, the
summing is carried out with respect to all k with integer components and n . 3
E
R.
(The series (4.12) converge due to (4.11) [7,161). The function On(rlB) with n = (0,. . . , O ) denote as before by O ( r IB) . LEMMA C71.
Let the parameters B . . = Bji
complex parameters of 2 '
u
13
i,j = 1, ...,L and, generally speaking,
;
= (U1,...,U e ) , V = (Vl,...,V ) satisfy the following system 9"
relations:
[e(
a 4
ra
a
a
+ 2 ar ar
+ -G10n(?12B) 2 I r=O ' 1
(4.13)
where n denotes all possible L-dimensional vectors with components 0 or ?1. Then for any values (complex) of parameters Co
=
( Col,...,C
OP.
)
the function:
is equation (4.4) solution. This statement evidently implies LEMMA 1'.
Let Co(x,t) = (C (x,t),...,Co9"(x,t))be a smooth real-valued vector-
01
function and for smooth real-valued functions H
, (x,t) = H . (x,t), XI Ik (x,t), V. (x,t), j = 1,. . ,I, G(x,t), C(x,t) let rela3 1 1 = -2i In Hkj. Then the function: tions (4.9), (4.11) and (4.12) hold, where B kj
k , j = 1,.
. . ,9",
S,
(x,t),
U.
is the 2n-periodic with respect to
T l,...,~E
.
solution of equation (4.2).
Consider equation (4.12). First regard the case .9 = 1. Then there are two equations corresponding to n = 0 and n =
4. Releasing these equations from the
function G(x,t) and solving the obtained relations with respect to V, we have: 3 V = U f(H) + 6CU
(4.15)
and the arguments of Oo and 0% are ( r 12B), iB -
H = e
.
Substituting in this relation V and U by St and
obtain the relation connecting S, H and C
S
respectively, we
:
3 St - S f(H) - 6CSx = 0 .
(4.16)
It is easy to small H to obtain from the formula (3.1) the following equality:
14
DOBROKHOTOV and V.P. MASLOV
S.Yu.
yi
=
c+
4u2H COS(T + Co) + O ( H 2)
(4.17)
2 and one can call the function H (accurate to the factor 4U ) the "nonlinear amplitude" of the fast oscillating on the smooth background C(x,t) solution (4.14). For 2 has the form: small H accurate to the dispersion relation (4.16) accurate to O ( H V = U
3
+
bCU
,
and (4.16) accepts the form of Hamilton-Jacobi equation for the
function S: St -
S3
-
6C(x,t)Sx = 0.
(4.18)
The change of relation (4.16) for relation (4.18) means the linearization of the initial equation (2.1) in the vicinity of the smooth solution C(x,t) C9l. Now let us consider the relation (4.12) for P > 1. Analogically to (4.17) we have for small H . , = exp(i B . ,/2), j = 1 ,..., e : I1 I1
,..., T P )
yO(T1 e
= C
COS(Tl+C01)+...+4U~HPP COS(Tk+CoP) + O(H 2 +... + 4U1Hll 2 11
+ H P2e )
and hence, as it was in one-dimensional case, the functions H . . j = 1 , ...,P Ill have the sense of "nonlinear amplitudes". Nevertheless in this case there exists such a set of values iB . H . = e 1k/2, P 1 k > j 2 1, that define the "interaction' 0 s the components L Ik with different phases of the solution Y ~ ( T ~ ~ . . . , T ~For ) . P = 2 there exist four relations (4.12) by solving which with respect to V following relations:
G =
9 (UIH,C) , H . 0 lk
=
1'
G, H12 one can obtain the
(4.20)
$jk(U,H,C)r. ]=1,2, k=2 '
where f i = (Hll,H12),U = (U ,U ) and $ and 0 are functions. 1 2 jk The relations (4.19) are analogous to (4.15) and it is naturally to call them the multidimensional dispersion relations. Distinct from one-dimensional case ( V . = 1) the relations in functions $ , can be obtained only in the form of series in terms 1 of nonlinear amplitude powers H I under the condition U . # 0, IU1 # IU2 These 1 series coefficients expressions as well of the series, which express H12 and G by
I
1.
U, f / and C, can be obtained by a standart methods using the equations (4.12) in
the following form:
(equation (4.131, n
=
(%,O), n = (O,$))
,
15
FINITE GAP ALMOST PERIODIC SOLUTIONS
-
(U1 H12 =
U2)
(U1 + U 2 )
U4 1
G =
2
R
2 +
2
12
+ U2)
U1U2(U1
4
+ H2 U 2 ) =
(4.221
2
(4.23)
Ro
(equation ( 4 . 1 3 1 , n = ( 0 , o ) Here R . , j = 0 , 1 , 2 , R 1 2 3
-
are polynomials in U , C , V .
I'
1
with coefficients analitically
< 1 are such that the expansion of depending on H . and H . , 0 < l H j k l < m, l H j j ik 33' R . in a series in terms of the powers of ff begins from the third degree, and of
3 R12
-
from the second degree. The corresponding formulas for these series coeffi-
cients and their convergence proof we de not need further, and we do not give them here. Not only, that it follows from ( 4 . 2 1 ) , in particular, that the disper-
1 1 .3 )
sion relations accurate to O ( ff
do not contain the phases interaction: every
dispersion relation coincides accurate to O( I If I
3
)
with the one-dimentional disper-
sion relation, has the form ( 4 . 1 5 ) and contains only the values with the single index j . From this fact it follows immediately a weeker statement that the phases S.(x,t) satisfy the same Hamilton-Jacobi equation ( 4 . 1 0 ) in the (linear) limit 1
H
-f
0. For Q > 2 the investigation of the relation system ( 4 . 1 3 ) becomes essen-
tially complicated, it is overdeterminated in the sense the relations ( 4 . 1 9 ) and ( 4 . 2 0 ) , which express V j , B j k ,
j # k and G by U. and B . . can already be obtained 1 31'
from any ( P . + l ) Q / 2 + 1 relations of ( 4 . 1 3 ) , while the total number of relations ( 4 . 1 3 ) is equal to 2'.
The point which teh relations from ( 4 . 1 3 ) one should take in order
to obtain the equalities ( 4 . 1 9 ) satisfying the whole system ( 4 . 1 3 ) (and existing according to lemma 1 ) is yet open. Nevertheless, when we take the small values of iB,. f o r these relations, it is easily seen from lemma 1 that one can H . ,= e I3
'"'
take the relations with the indexes n = (n
,..., nQ )
such that 0 5 nl
+...+ n2
5
1.
The latter can be expressed in the form ( 4 . 1 9 ) , ( 4 . 2 0 ) by means of series in terms of the powers of H l l , . . . , H L Q
5.
analogically to the case P. = 2 .
Higher approximations
iB = e k 1 / 2 ) and Col ,...,Coe, which (or H kj define accurate to O(h) the asymptotical solution ( 4 . 1 ) , cannot be obtained from
The functions S l ,
...,S Q
the system ( 4 . 9 )
(or ( 4 . 1 3 ) ) , because the number of unknowns is greater than the
and E
,...,E 2 Q
number of relations. One should regard the equations for higher approximations in order to get the full system of relations. As we already know, all of them are linear non-homogeneous equation with 2n-periodic with respect to
..., 'T
T ~ ,
16
S.Yu.
DOBROKHOTOV and V . P . MASLOV
coefficients. The necessary existence condition of 2n-periodic with respect to variables
...,T~
T ~ ,
solution of each of these equations if the orthogonality con-
dition of the kernel right-hand parts F joint to the operator the operators L and
L
of the operator L
(see (4.3)). In the variables
-* L , evidently, have
a a a' L=--~w-+-,L
^*
1
','an
=
A
and
a
-
6y B + B3, ad-
/aE,
'
(see (4.4))
the form
a a' 6w - + -
= - _a the leading role in the investigaa53 an dE at3 tions of these operators properties has the function xK(n,<,E), where E is, geneA *
a5
a0
rally speaking, a complex parameter. The detailed describtion of this function properties is given, e.g. in C241, for our purposes it is sufficient to define this function as a real part (for real
which admitts as 4%
+
-
-iax/ac +
x2
&)
of the solution
x
of Recatti equation:
+ w - E ,
(5.1)
the formal expansion: m
x
%
JE + c Xm/(zJE)m.
(5.2)
m= 1
In (5.1) w is the equation (5.1) solution. Due to (5.1) it is easy to prove, that
xR
=
Re
x
has an expansion in odd powers of 1&,
and moreover, to obtain the re-
current formulas, which allow to express the coefficients
xR2m+1
in the form of
polynomials in the function w and its derivatives (see 1241). In the case, when w is the k-gap almost periodic solution, the function
x
satisfies as well the
equation: (5.3) and can be represented in the form: (5.4) where the polynomial P2k+l was defined in the item 4, and y , ( c , r l ) are functions. Due to ( 5 . 1 ) , * 1 tion < =
(5.3)
1
it is easy to prove by direct differentiation that the func-
/ X R h 6,€)
coefficients are independent of
<*
3
-* *
satisfies the equation L 5
=
0. Since the last equation
&, we immediately obtain that the coefficients
of the function 5 expansion in terms of the powers
1
/&
satisfy the same equation. (Note that this fact is valid for any solutions of 1 R with respect to the powers of / &
equation (4.4'). Comparing the expansion
'/X
with (5.4), we immediately obtain,that all the functions C * ( C , n ) 1
represent linear
17
F I N I T E G A P ALMOST P E R I O D I C SOLUTIONS
c o m b i n a t i o n s o f Y. e l e m e n t a r y s y m m e t r i c p o l y n o m i a l s i n y . ; j = 1 , ..., P and 1 . * 1 S i n c e i; a r e p o l y n o m i a l s i n t h e s o l u t i o n w and i t s d e r i v a t i v e s , i t i s e v i d e n t t h a t * j i.d e p e n d i n g on t h e v a r i a b l e s T 1 , . . . , ~ a r e 2n-periodic functions with respect to 3 c * t h e s e v a r i a b l e s . Thus among t h e s o l u t i o n s 5 . t h e r e a r e t + l l i n e a r l y i n d e p e n d e n t -* * 3 2 n - p e r i o d i c s o l u t i o n s o f t h e e q u a t i o n L i, = 0 , e a c h o f w h i c h c a n b e e x p r e s s e d i n t e r m s o f w and i t s d e r i v a t i v e s by means o f ( 5 . 1 ) , (5.2) a n d (5.5), i t i s n a t u r a l
* *
*
t o choose t h e set o f t h e s e s o l u t i o n s i n t h e f o r o f C 0 , C l , .
..,io. The f i r s t t h r e e
of them h a v e t h e form:
*
d2W
< *2 = - - 2 c
c ; = 1 , c ; = w ,
+ 3w
2
(5.6)
S i n c e one can c o n s t r u c t t h e s o l u t i o n i of t h e e q u a t i o n L any s o l u t i o n
<
-* *
*
of t h e equation L
<
<
= 0 corresponding to
' *
= 0 by means o f f o r m u l a
= Bi,
,
we o b t a i n
i n t h e P.-gap c a s e P l i n e a r l y i n d e p e n d e n t s o l u t i o n s o f t h e f i r s t e q u a t i o n :
^ *
*
,...,
= 1 2 . ( F o r j = 0 Bi; = 0 ) . Note t h a t t h e same s o l u t i o n s c a n b e 3 3 0 o b t a i n e d by a n o t h e r way: e a c h o f them r e p r e s e n t s a l i n e a r c o m b i n a t i o n o f f u n c t i o n s
5 . = BC., j P.
a y 0 / 3 ~ . which are a l s o l i n e a r l y i n d e p e n d e n t s o l u t i o n s of e q u a t i o n Lc = 0 :
I'
...,
where P
, a r e polynomials in Ul,...,Uy, Vl, V and C. The v a l i d i t y o f t h e l a s t mi E. e q u a l i t y can e a s i l y be proved by i n d u c t i o n w i t h r e s p e c t t o m u s i n g t h e e q u a t i o n s
(4.2), ( 5 . l ) , ( 5 . 3 ) and t h e f o r m u l a ( 5 . 4 ) . One more E + l - t h l i n e a r l y i n d e D e n d e n t solution
Lr, = 0 c a n b e o b t a i n e d a s a r e s u l t o f d i f f e r e n t i a t i o n
C0 o f t h e e q u a t i o n
o f t h e f u n c t i o n y e w i t h r e s p e c t t o t h e p a r a m e t e r s E 0,...,E2Y. u n d e r t h e c o n d i t i o n
0
= c o n s t , or w i t h r e s p e c t t o t h e p a r a m e t e r s
Uj(Eo,...,E2e)
H l l , . . . ,Hg"p a n d
C. This
s o l u t i o n h a s t h e form
C0 =
a.
L:
EO ,...,
II
.P
a . "O/aE.
i n c a s e when a s t h e p a r a m e t e r s d e f i n i n g y t h e v a l u e s 3 3 E. a OP. a EZP. are u s e d , and t h e form c0 = ( , Z li __ ~ = 1 j aH + B, % ) y o when a s t h e s e
j=O
parameters serve C, U j ,
Hjj, j = 1 ,..., II.
jj
and 8 . d e f i n e d a c c u r a t e t o a m u l t i p l i c a t i v e c o n s t a n t c a n be 3 3 o b t a i n e d from t h e s y s t e m o f e q u a t i o n s : The c o e f f i c i e n t s
(1,
i n t h e f i r s t c a s e and from
i n t h e s e c o n d . The v a l u e s V ( E 0 , . . . , E 2 E . ) ,
m
U ( Eo , . . . , E 2 n )
(Or
,..., H e P . , C ) )
V (Hll
w e r e i n t r o d u c e d i n i t e m 4 ; t h e s o l v a b i l i t y of t h i s e q u a t i o n s y s t e m f o l l o w s from
ia
S . YU
. DOBROKHOTOV
and V . P . MASLOV
the functional independence of parameters E0,...,E29. or H l l ,...,HQe, U 1 , ...
IUQ
and C. Thus the following statement is proved: Let in the K-th equation (4.3) the right-hand part F , be a 2n-periodic
LEMMA 2.
function with respect to the arguments c l , with respect to
...,T~
T ~ ,
...,T
~ .Then
in order the 21~-periodic
solution yk of the equation (4.3) exists, the
orthogonality condition necessary holds:
oi2r
...
oi2n FkC:
d l...d~ 9. = 0
,
.
j = 0,.. . , k
(5.8)
If such a solution yk exists, then it is defined accurate to a linear combination of the functions
<., I
j = 0,..., i? and has the form: i?
c
Yk(T,Xrt) = (Yk)part +
j=o
C
lk
(5.9)
(Xtt)i j'
is a partial solution (4.3) ( ' k ) part and the way of calculation S.(r,x,t) and C,(~,x,t) was given above.
where C
jk
(x,t) are arbitrary smooth functions,
*
3
J
Consider the orthogonality condition ( 5 . 8 ) when k = 1. The function F
1
has the
form:
F.1
As
a.
yot =
a.
+
then F
1
La. ^2 i? 6YoYox - 3B Yox - 3 ( ,I
t
aYo
ax
aT.
1
o
'
9. +
j=l
^ i ?
)BYO
aYo ~
a?.
and
3
(5.9')
*
3
.9
acoj
--at
a
sjxx
3=1
a a . at (Yo~c =const)
acoj
I--j=l
9.
a.
= - Yot +
9.
'ox
- _a
a .
ax (Y01c =const)
-
0
'
'
can be represented as a sum: F1 = ( F
1
holds under condition
Hence, from ( 5 . 8 ) and 2n-periodicity with respect to
T
~
..., , T~
of the function y
0
by integrating by parts we immediately obtain, that the orthogonality condition
(5.8) for k = 1 represents the relations containing the functions
,...,29.
E.(x,t), Ejx, Ejf, j = 0 3
IBkn/2 (or Hkn(x,t) = e , Hknx, Hknt, k,n = I , . . . , & )
S. it' not c o n t a i n i n g the functions Coj (x,t), j = 1
Besides, these relations are linear with respect to Hkmt). Thus the orthogonality condition ( 5 . 8 ) for k
S
jx'
,...,..P
S. E and JXX' jt = 1 appear to
ficient relations, which together with the relations (4.9), ( 4 . 9 ' ) (4.13))
j = 1 , ...,a.
Sjxx,
and
E
(or H jx kmx' be L + 1 insuf(or (4.9),
form the closed system of equations for the definition of functions S.(x,t), 1
19
F I N I T E GAP ALMOST P E R I O D I C SOLUTIONS
j = 1,
...J and Em, m
...,2 P
(or IIkn, k,n = 1 , ...,1). In order to obtain
= 0,
equations for the functions C . (x,t), one should consider the orthogonality con30
ditions ( 5 . 8 ) of the right-hand part F2 to the functions
*
c*
Analogically
to the previous it is easy to show, that these conditions do not contain the func(x,t) (see (5.9)) and represent a system of V. linear homogeneous equa11 tions for C . with coefficients depending on S St, E . (or Hmk) and on their detions C
30
1
rivatives (see 161). Hence in particular, it follows that the functions C . : 0 10 can serve as the solutions of this system. The orthogonality condition of F 2 to * i gives the equation for determination of the function Cgl(x,t). 0
Consider the equation (4.3) for k
2 analogically. Each of the solutions y
k can be represented in the form (5.9) and the corresponding "constants" of integration C
<:
k j
6.
Fk+l
can be defined from the orthogonality condition (5.8) of and of Fk+2 to the functions
*
to the function
., i l i .
<*
I . .
CONSERVATION LAWS AND FLASCHKA-FOREST-MCLAUGHLIN EQUATION
By differentiating the equalities (4.9), (4..9')with respect to t and x and getting rid of the functions
s . in I
these equalities, we obtain the equations:
Show that the orthogonality conditions (5.8), k = 1 , can be represented jn the analogous form and thus show that EO,...,EZR satisfy a quasi-linear system of equations. From the equation ( 5 . 1 ) and the definition of the functions corresponding to w(q,5) in ( 5 . 1 )
','xR
*
and 5 ,
1
=
*
i .Iwl 3
the following equalities can be proved by induc-
tion with respect to j :
zLw1-i~the differential operator with coefficients depending on the function w and its derivatives as polynomials, LI
,
mi
are some numbers. These equalities valid
for any functions w imply, in particular, a number of polynomial conservation laws, i.e. equalities of the form
a an
a
- Q . C ~ +- I - P rwi 3
at
j-
=
o ,
which hold for any solution w of the equation (4.4'). (See explicit formulas for
20
j'
u
S.YU.
and
P,
1' = w( /h
.
1) By the formal change of variables
e.g. in [
,
DOBROKHOTOV and V . P . MASLOV
rl =
t /h, E = X,'h,
/h) in the equation (4.4') we evidently obtain (2.11, and the equa-
lities (6.2) are transformed into the following equalities:
Here by Q.Lu,hl and so on we denote Q,rwl and so on, where w is changed for u and
'
a
the operator
/ 3 5 for h
0
l
/ax. Then we insert into every relation of (6.3) the E
asymptotical expansion (4.1), considering that the function y o satisfies (4.2). By differentiating and equating the coefficients of equal powers of h for each j in the resulting equality, we obtain the system of equations, which is analogical to (4.2), (4.3):
A
n
Here the operators A, B are defined in the item 4, F has the form (5.9'1, O S O E * E -09, Qj[yol, P.[yol, f [yo], z.ryol are functions Q . P., 5 . and operator z . in which I '1 1 I' 1 I I' E 2 ? * 1 1 a. lx, Sjt and their w = y , 5- = A and 2-= B; ( z j , P . are some polynomials in yo, S , 0 or26 I derivatives. 9.
The equations (6.4) represent conservation laws for function yo (when x,t are fixed) and of no new information for us. Considering the functions y;,y, grate (6.5) with respect to ' r l , [0,2nl
x
...
x
10,2111
. At
2n-periodic with respect to
..., T Y.
T1,
LEMMA 3.
k
=
x
...
x
we inte-
last we obtain:
where the angular brackets denote the averaging with respect to cube [0,2nl
...,T E ,
in the k-dimensional cube
T
~
..., , T~
in the
10,2nl. Then we immediately have the following statement:
Y. a) In order the 211-periodic solutions yo,yl of equations ( 4 . 2 ) ,
(4.3),
1 , exist, the following relations ("conservation laws averaging", compare
[21-23 1 ) necessary hold: (6.6)
21
F I N I T E GAP ALMOST P E R I O D I C SOLUTIONS
o f t h e s e r e l a t i o n s are e q u i v a l e n t t o t h e o r t h o g o n a l i t y c o n d i -
b ) The f i r s t Y.+1
t i o n s ( 5 . 8 ) , k = 1.
o e
O
L
S u b s t i t u t i n g i n Q . [ y 1 and p , c y 1 t h e d e r i v a t i v e s S . and S , f o r U . ( E ) and V . ( E l , 3 0. 3 0 I X It I 3 adding t h e ( Q + l ) - t h e r e l a t i o n t o (6.11, w e o b t a i n f o r E 0 , . . . , E 2 e the required q u a s i - l i n e a r s y s t e m o f e q u a t i o n s . Note t h a t t h e i n t e g r a l s i n ( 6 . 6 ) can be w r i t t e n i n t h e form, which d o e s n o t r e q u i r e t h a t t h e p e r i o d s w i t h r e s p e c t t o t h e a r g u 2 ments ' T ~ , . . . , T ~o f k-gap s o l u t i o n y o s h o u l d be e q u a l t o 2n. Then t h e e q u a t i o n s
...,
(6.6), j = 1 , 2 ,
c a n b e i n t e r p r e t a t e d , g e n e r a l l y s p e a k i n g , a s an i n f i n i t e
system o f q u a s i l i n e a r e q u a t i o n s w i t h r e s p e c t t o E 0 , . . . , E 2 e .
F l a s c h k a , F o r e s t and
McLaughlin showed t h a t , f i r s t l y , o n l y (2V+1) e q u a t i o n s are l i n e a r l y i n d e p e n d e n t i n t h i s system; s e c o n d l y , t h i s system i s e q u a v a l e n t t o t h e system ( 6 . 1 1 , ( 6 . 6 ) , j = 1,
...,P.+l, and
t h i r d l y , t h i s system can be r e p r e s e n t e d i n t h e f o l l o w i n g e l e -
g a n t i n v a r i a n t g e o m e t r i c form: ailo
3R 1 12-=o,
-at-
no
where t h e d i f f e r e n t i a l
(6.7)
2X
and i l l are d e f i n e d by t h e e q u a l i t i e s ( 4 . 6 ) , ( 4 . 7 ) . I n
p a r t i c u l a r , ( 6 . 7 ) can be r e p r e s e n t e d in t h e form:
(6.8) where S and M1
m'
= 6[
M
2y.
Z
j=0
E. 3
- 2Em - 2 d e t M
1
m
2 / d e t P4=1,
2 a r e m a t r i c e s with t h e elements m
Hence, i t f o l l o w s t h a t t h e s y s t e m of e q u a t i o n s f o r E 0 , . . . , E 2 k and t h a t t h e b r a n c h i n g p o i n t s E 0 , . . . , E 2 e
of t h e s u r f a c e
v a r i e n t s C8l.
r
X, t
i s a h y p e r b o l i c one,
a r e t h e Riemann i n -
The s t a t e m e n t of Lemma 3 d o e s n o t depend, of c o u r s e , on t h e p a r a m e t r i z a t i o n
e
method of t h e s o l u t i o n yo of e q u a t i o n ( 4 . 2 ) . Though t h e system o f e q u a t i o n s ( 4 . 9 ) , iBk j (4.13) and ( 6 . 6 ) , j = 1 L + l , f o r t h e f u n c t i o n s S . and H = e has n o t 1 kj s u c h a s i m p l e form, as ( 6 . 8 ) , it i s a l s o c o n v e n i e n t f o r some c a l c u l a t i o n s . O m i t -
,...,
t i n g s i m p l e , b u t t e d i o u s c a l c u l a t i o n s , b a s e d on t h e f o r m u l a s ( 5 . 6 ) and on t h e definition of two-gap c a s e :
P . and 1
Q.
I'
we g i v e t h e a v e r a g e d q u a n t i t i e s
< P . > and f o r t h e 3
3
. . DOBROKHOTOV
22
and V . P
S YU
. MASLOV
Due t o t h e s e formulas and t a k i n g i n t o a c c o u n t ( 4 . 9 ) i t i s easy t o r e p r e s e n t t h e e q u a t i o n s ( 6 . 6 ) i n t h e two-gap c a s e i n t h e form:
ac
-ac d t_
6C--=G,
(6.9)
ax
(6.10)
-
where R . a r e a n a l y t i c f u n c t i o n s i n t h e arguments C , G , U . , H 1 2 , V j , Hjj, j = 1,2 3 1 i n t h e v i c i n i t y of t h e same p o i n t s a s t h e f u n c t i o n s R . and R12 i n t h e i t e m 4 . The
1
, .
expansion o f R . i n t h e s e r i e s i n terms of t h e powers o f H 1 1 , H12 b e g i n s from t h e 3 second d e g r e e . Hence a s F I . . + 0 t h e e q u a t i o n s ( 6 . 6 ) , j = 1 , 2 , 3 , a c c u r a t e t o I3 2 2 0 ( H l 1 + H 2 2 ) c o i n s i d e w i t h t h e e q u a t i o n of t r a n s f e r e n c e
-"-
( 6 c + 3 s2 )
at
x
-a4-
35
ax
s
x~
4 -
6 4 -ac =
ax
0 ,
(6.11)
c o r r e s p o n d i n g t o t h e Hamilton-Jacobi e q u a t i o n ( 4 . 1 8 ) . Thus t h e e q u a t i o n ( 6 . 6 ) one can t r e a t a s t h e e q u a t i o n d e f i n i n g t h e "smooth background" of t h e a s y m p t o t i c sol u t i o n , and t h e e q u a t i o n s ( 6 . 6 ) , j
=
2 , 3 a s t h e n o n l i n e a r a n a l o g u e s of t h e t r a n s -
ference equations f o r the "nonlinear" amplitudes H . , . Analogical c o n s i d e r a t i o n s 33 hold f o r t h e e q u a t i o n ( 6 . 6 ) f o r L > 2 .
7.
FINITE GAP SOLUTIONS I N TIIE WAVE T R A I N PROBLEM
Now r e t u r n t o t h e problem ( 2 . 1 ) , ( 3 . 2 ) on wave t r a i n s u p e r p o s i t i o n . Consider f o r 1 = 2 t h e f o l l o w i n g Cauchy problem f o r t h e e q u a t i o n s
( 4 . 9 ) , (4.131,
(6.6)
j = 1,2,3:
iB
H
11
It=O
= e
11/21
=
H(x) , x c [ a O , b O ~
o
t=O
Assume t h a t f o r t E [ O , t s o l u t i o n of t h e problem
zo
1
, ~ ~ r a o , b;0 Hl 2 2
It=O
iB = e 22/2
I
= t=O
t h e c o n d i t i o n s h o l d : 1) There e x i s t 0 0
- 6 C Cx
=
0, C
I
= C (x);
H(x) , x c r c , d 1 0
'(7.1) , x ~ [ c ~ , d ~ 1
( x , t ) - a smooth
2) The jacobian
t t=O 3 q ( 0 ) / ( a , t ) / i 1 a i s n o t e q u a l t o z e r o o n t h e t r a j e c t o r i e s p ( O ) ( a , t ) , q(O) ( a , t ) of
Hamiltonian system intervals q = a
E
6
= 6C'O) ( q , t )
+ 3p2,
4
= -6p
( q ' t ) / a q which s t a r t from t h e
[ a O , b O 1 , q = a E ~ c o , d o lw i t h t h e impulses p = a s ( a ) / a a ;
3 ) There e x i s t s t h e number 6 > 0 a r b i t r a r y s m a l l , independefit of h , such t h a t
F I N I T E GAP ALMOST P E R I O D I C SOLUTIONS
23
Under the assumptions 1) - 3) for any N 2 0 when t E [O,t 1: i) There 0 N N exist smooth functions S S : , CN, G and HN = exp iB. j,m = 1,2 satisfying 1' im Im'N+l the problem (4.9), (4.13), ( 6 . 6 ) , (7.1) accurate to O(h 1 ; ii) The function
THEOREM.
where yk are the solutions of equations (4.2), satisfies the problem (2.11, (3.2) N+ 1 accurate to O(h 1. Consider the behaviour of the solution (7.2). Assume that in addition to 1) - 3) the condition holds: 4) on the time interval t
<
to all the points moving along
the trajectories, which start from the interval [a ,b 1, are on the left side of 0 0 all the points moving along the trajectories, which start from the interval
-
N
[co,d01. Then till an instant of time tl < to H l l = 0 everywhere outside an interN val cct,dtl, and HZ2 = 0 everywhere outside an interval [c d 1, at > dt, which t' t does not intersect with [c , d 1. Outside these intervals we have, Uo = CN (x,t,h), t t and in each of these intervals the function uo presents a cnoidal wave train on
-
the smooth background. At an instant of time tl the point at reaches the interval [ct,dtl and at the points x from [a ,b 1 n cc ,d 1 in the solution two phases t t t t arise, i.e. the wave train interaction occurs. Then after some time t > tl the
- -
train with the phase SN leaves behind the train with the phase S';', and the inter2 vals [ct,dtl, [at,btl became disjoint, [at,b 1 is situated on the left of [c d 1. t t' t Again is the solution describing two distant wave trains on the smooth background. Conditions l), 2) of THEOREM will hold e.g. in case when
Remark.
S =
k x, for 1
[ao,bol, S = k x, x e [co,dol (i.e. when the wave trains almost coincide with 2 plane cnoidal waves). Assumption 4) then holds automatically, if k l > k2. x
E
The following statement implies the proof of the Theorem: LEMMA 4.
Let conditions l), 2) hold and the function H(x,E) smoothly depend on
E > 0 ( i < E ) such that (HI .: c const. Then for t 6 r0,t 1 and x c R 0 (n) (n) (n) (n), c(n) , n =O O,l,. that there exist such smooth functions S m " m I m I m' -M S(n) for any integer M > 0 the functions S (Xtt)En, n=O
parameter
-M 1 3 =~
M
c
n=1
..
-M (x,t)En, m=1,2, til2 =
M
c
n=O
fz)
n -M (x,t't , G
M =
c ) ;9
n=2
(x,t)En,
satisfy the problem (4.9), (4.13), .P = 2 ( 6 . 6 ) , j = 1,2,3, (7.1) accurate to
. . DOBROKHOTOV
24
S YU
and V . P
M O(E ) . If the expansion of such a solution exists
MASLOV
n terms of the powers of E ,
this expansion is unique. In order to prove this Lemma we insert the expansions of functions S etc. into m the equations mentioned and using a standard method we obtain from (4.21)-(4.23), (6.9)-(6.10) the recurrent system of equations for the expansions coefficients. The solvability of this system which coincides with (4.18), (6.11) in the zero approximation follows from conditions 1 1 , 21. To prove the Theorem we use the results of items 4-G setting
CM, M
BN -M CN = exp i-lk = H , 2 Ik' coincides with (3.2).
=
"/A
1
t
-P.l SN = S . ,
I
1
1 and note that for t = 0 the function (7.2)
In conclusion note that we did not actually regard the sufficient conditions of
equation (4.3) solvability in periodic functions, which demands additional complicated investigations and is based on considering of eigen functions of the ^* -1 operator L which have the form ( x ( v , S , < ) ) exp(i!X(v,5,5)dnl .
REFERENCES
111
Ablowitz M.A., Benny D.Y., The evolution of multiphase modes for nonlinear
121
Babich V.M., Buldyrev V.S., Asymptotic methods in short waves problems
131
Dobrokhotov S.Yu., Maslov V.P., Spectral boundary problem asymptotics for
dispersive waves, Stud. Appl. Math. 49 N3 (19701. ("Nauka", Moscow, 1972) (in Russian). non-linear equation of semiconductor, Dokl. &ad.
Nauk SSSR 243, N4 (1978)
897-900 (in Russian). 141
Dobrokhotov S.Yu., Maslov V.P., Solution asymptotics of mixed problem for 2 non-linear wave equation h O u + ashU = 0 Uspekhi Math. Nauk 34, N3 (1979)
151
Dobrokhotov S.Yu., Maslov V.P., Boundary reflection problem for equation 2 h Ou + ashu = 0 and finite gap conditionally periodic solutions, Funkz.
225-226 (in Russian).
Analis i Primenen. 13, N3 (1973) (in Russian).
C6l
Dabrokhotov S.Yu., Maslov V.P., Finite gap almost periodic solutions in asymptotic axpansions, Modern Probl. of Math. 15 (VINITI, Moscow, 1980) (in Russian).
171
Dubrovin B.A., S.P. Novikov, hypothesis in 0-functions theory and non-linear equations of KdV and KP type, Dokl. &ad.
Nauk SSSR 251, N3 (1980) (in
Russian). C8
I
Flaschka H., Forest G., McLaughlin D.W., Multiphase averaging and the inverse spectral solution of KdV, Preprint LA-UR 79-365, Los Alamos Scintific Laboratory.
F I N I T E GAP ALMOST PERIODIC SOLUTIONS
25
191
Karpman V.I., Non-linear waves in dispersive media, ("Nauka",Moscow, 1973).
1 101
Krichever I.M., Algebraic geometry methods in non-linear equations theory,
Clll
Kuzmak G.E., Asymptotical solutions of nonlinear differential equations of
Uspekhi E4ath. Nauk 32, N6 (1977) 183-208 (in Russian). second order with variable coefficients, Priklad. Math. i Mech. 23, N3 (1959) 515-526 (in Russian). 1121
Luke J.C., A perturbation method for non-linear dispersive wave problems, Proc. ROY. SOC. A292, N 1430 (1966) 403-412.
C131
Maslov V.P., Operational methods ("PTIR", Moscow, 1976).
1141 Maslov V . P . ,
Complex WKB-method in non-linear equations ("Nauka", Moscow,
1977) (in Russian).
El51
Maslov V . P . ,
Tcupin V.A., &-like generalized according to Sobolev solutions
of quasi-linear equations, Uspekhi Math. Nauk 34, N1 (1979) 235-236 (in Russian).
C161
Matveev V.B., Abelian functions and solutions, Inst. Theor. Phys., Univ. Wroclaw, Preprint N 373, 1976.
C171
Miura R.M., Kruskal M.D., Application of non-linear WKB-method to KdV equation, SIAM Appl. Math. 26, N2 (1974) 376-395.
C181
Novikov S.P., Periodical problem for KdV-equation, Funkz. Analis i Premenen. 8, N3 (1974) 54-66 (in Russian).
1191 Scott A.C., Active and non-linear wave propagation in Electronics (Wiley Interscience, New York, 1970). r201
Scott A.C., Chu F.Y., McLaughlin D.W., The soliton: a new concept in applied
C211
Whitham G.B., Non-linear dispersive waves, Proc. Roy. SOC., A283, N 1393
science, Proc. IEEE 1 (1973) 1443. (1965) 283-291. c221
Whitham G.B., Variational methods and applications to water waves, Proc.
C231
Whitham G.B., Linear and non-linear waves (Wiley Interscience, New Yourk,
1241
Zakharov E., Manakov S . V . ,
Roy. SOC., A299, 6 (1967). 1974). ("Nauka",Moscow, 1978).
Novikov S.P., Pitaevsky L.P., Theory of solitons
This Page Intentionally Left Blank
A N A L Y T I C A L AND NLI'IZRTCAL APPHOAClfC.? TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S. Axelsson, L.S. F r a n k , A . v a n d c r S l u i s ( e d s . ) @ N o r t h - H o l l a n d P u b l i s h r n y Company, 1981
PERIODIC SOLUTIONS NEAR EQUILIBRIUM POINTS OF HAi'lILTONIAN SYSTEMS
Hans Duistermaat Mathematisch Instituut Rijksuniversiteit Utrecht Utrecht, The Netherlands This is a report on work in progress in collaboration with Richard Cushman. Consider a Hamiltonian system with n degrees of freedom:
m
where F is a smooth ( = C
or real analytic, everything which follows
holds in both categories) real-valued function of the 2n coordinates z = (q,p). Also F may depend smoothly on a finite number of parameters. The vectorfields (in the z-space) on the right hand side of (1) will also be denoted by v(z), it will be assumed throughout that the origin is an equilibrium point of the system ( I ) , that is
-
v(0) = 0, corresponding to dF(0) = 0. The substitution of variables z = E . Z leads to the equation (2)
dz dt
-
E-l
5
.V(EZ) = A(z)
+
higher order terms in
E,
which is the "scaled version" of the system (1). Here A = Dv(0) is the linear part of v at the equilibrium point, the correspondinq linearized system ((2) for E=O) is eaual to the Hamiltonian system defined by the quadratic part of the Taylor expansion of F at the origin. Its solutions are given by =
etZ(z 0 ) )
r
periodic with period w if and only if E N
=
def
ker (ewA-I) = (kEz
E2Tiik,w'IR.
Here EX denotes the eigenspace of A for the eigenvalue X and the suffix IR means that one has to take the real part of the corresponding complex vector space. Note that the dimension of N is always 37
28
H . DUISTERMAAT
even, because the eigenvalues of A occur in complex conjugate pairs. A classical theorem of Lyapounov [ 5 1 states that if for w = w 0 , v = vo the matrix A = Ao has only one complex conjugate pair of eigenvalues in Z -2.rri/w0(each having algebraic multiplicity equal to 1) then the non-linear Hamiltonian system ( 1 ) with v near vo has a family of periodic solutions with period w (depending on the solution) close to w o and filling up a smooth 2-dimensional manifold P through the origin. At the origin the tangent space to P is equal to N and the orbits in P are concentric around the equilibrium point. Also the family depends smoothly on the parameters in the Hamiltonian function F , reflecting the fact that the theorem can be proved by a direct application of the implicit function theorem (see for instance [31).
However, if some eigenvalues of A = A o on the positive imaginary axis are rationally related, called the case of resonance, then for a suitable choice of w = u o the dimension of N is higher than 2. Nevertheless the periodic solutions of period o close to w o of ( 1 ) in the generic case will fill up a set which looks like a 2-dimensional algebraic variety in a (dim N)-dimensional space, having some complicated singularity at (resp. near) the origin for v = vo (resp. for v near v 0 ) . (The genericity assumption is essential here, because in the linear case the set of periodic solutions even has a higher dimension.). It is our aim to make this statement as precise as possible, but let us first mention some results obtained before. For two degrees of freedom and purely imaginary eigenvalues with a ratio 2:l it had been shown in [ 3 1 that generically one gets 3 families of periodic solutions, one filling up a smooth 2-dimensional manifold through the origin, tangent to the plane of shortestperiodic solutions of the linear system, and the other two having a cone-like singularity at the origin, the tangent cone at the origin not being contained in a 2-dimensional plane. The result itself indicates that in this case the implicit function theorem cannot be applied directly but only after a suitable blowing-up procedure, leading to the dividing out of a factor from the periodicity equation which caused the degeneracy at the origin. A description of the situation for other ratio's k : Q ( k , Q E Z) is given in Henrard [41. The analysis in [ 3 1 , 1 4 1 was incomplete because it did not treat the perturbation to eigenvalue ratio's close to, but different from k:Q. The Lyapounov theorem then predicts only two families of periodic solutions (of nearby period), filling up smooth
29
HAMILTONIAN SYSTEMS
2-dimensional manifolds through the origin with complementary tangent spaces at the origin, and the interesting question is how the bifurcation to the singularity at exactly k:R takes place. This has been analyzed by Schmidt [ 9 1 . His essential tool is the use of the Birkhoff normal form which allows to recognize a factor in the equations which vanishes of order k+R-2 at the origin (also after using a blowing up procedure). After dividing out this factor the implicit function theorem can be used to obtain existence of certain families of periodic solutions which appear to be attached to the Lyapounov families outside the origin if the eigenvalue ratio is different from k:R and drawn into the origin (forming singularities there) if the eigenvalues ratio passes through k:R. In [ 9 1 the qeometry of the attaching of the additional families to the Lyapounov ones has not been described in detail and a still more complicated question is how all the families bifurcate at the origin if the eigenvalue ratio passes through k:R. It is our purpose to answer these questions in the following very precise way. To begin with, we allow an arbitrary number n of degrees of freedom, of which only 2 resonate in the sense that
(5) dim N o
=
4
if
0 0
No = Ker(ew A -I)
and Ao NO has 2 eiqenvalues on the positive imaginary axis, each of (6)
I
algebraic multiplicity equal to 1 and with ratio k:L, k # Q, k,R being integers without a common factor.
Then we have the following Theorem. (I) There is a 4-dimensional smooth manifold N through the origin (depending smoothly on the parameters LI in F) with tangent space at 0 equal to N, such that every periodic solution of (1) close to the origin and with period close to w o is contained in N.
(Note that N is not assumed to be invariant under the flow!).
(11) Under a genericity assumption for the coefficients of the Taylor expansion of F up to the order k+R there is (IIa) A standard Hamiltonian function Gu in 2 degrees of freedom 4 which is a polynomial in w EIR of degree k+L depending linearly on at most 4 real parameters u. (IIb) A smooth mapping p a local diffeomorphism
H
0
0
(defined near p = p , ~ ( )p = 0) and from W4 to N ( u ) depending smoothly on p , u(p)
H. DUISTERMAAT
30
such that for 1-1 close to '1the set of periodic orbits of ( 1 ) near the origin and with period near w o is equal to the Q'-image of the set of periodic orbits of the Hamiltonian system is given a s the set of w E IR4 where:
Gul
u
= U ( U ) ~which
0
(7) grad Gu (w) is a multiple of grad G2(w). Here
G:
denotes the quadratic part of the Taylor expansion of Go at
the origin.
(111) The standard Hamiltonian function Gu can be taken as follows (see (11) I (13), (16) below). By a linear canonical change of coordinates and a suitable time scaling (including time reversal) the quadratic part of F o on N can be brouqht into the form (8)
0
G 2 = ilpl
+
kp2
, k,R
Zwithout common factors, 0
E
il
<
lkl.
Here 2 2 (9) P j = q j + Pj
If k
>
I
0, resp. k
.
3 = 1,2.
0 , write
( 1 0 ) p 3 = Re(ql+ipl)k-(q2-iP2)Qlresp. p 3 = Re(ql+iP1)- k - (q2+ipi)Q
For Ikl + il ( 1 1 ) G'
>
5 we can take u E IR3 and
pl+kp2+apl+2 2 (b+v2)P ~ P ~ +2 C P ~2 + ~ ~ P ~ P ~ + P ~ .
= (il+u,)
Here a,b,c are fixed real number which can be chosen arbitrarily within the components of the set determined by the inequalities: (12) a
ka
=
def
-
Rb # 0, y
Lc
=
def
-
kb # 0, k a
-
Ry
0 , ka
+
Ry # 0 .
under the conditions (14) y
0, 3a
( 1 5 ) 15a2
-
For k = '2, ( 1 6 ) G"
6ay
-
y # 0 , 3a
-
y2
$2
+
y
#
0 and
0 if k = + 3 , no extra condition if k = - 3 . 2
R = 1 we can take u E IR
= (R+ul)p
I
+kp2+ap1+2(b+v2)P1P2+CP2+P3 2 2
31
HAMILTONIAN SYSTEMS
under the conditions (17) y # 0 , 2a
-
y # 0 , 2n
+
y # 0.
The point is that using polar coordinates in the (q ,p.)-planes, j i the set where (7) holds can easily be analyzed and the theorem says that for the original Hamiltonian system the set of periodic orbits is equal to the image of this set for v = v(’) under a smooth embedding @’
depending smoothly on the parameters p , w(p) also
depending smoothly on 11. The proof of the theorem consists of a combination of the following techniques i)
The Weinstein-Moser method [71 for the reduction of the search
for periodic solutions to the problem of finding the pointswheredH 0 is a multiple of dH2. Here Hw is a smooth function on dim N variables, H; is the quadratic part of the Taylor expansion of H 0
W
.
One also gets that HV is invariant under the periodic flow of the 0 0 Hamiltonian system defined by H2, that is HV and H2 Poisson-commute. If dim N = 4 this means that the Hamiltonian system defined by HV is completely integrable. ii)
The Birkhoff normal form theorem which asserts that by a
canonical change of coordinates (depending smoothly on the parame0 ters) F can be made to Poisson-commute with F 2 up to any order one likes. As a consequence, the function Hw in i) can be identified with FV up to any desired order. IN
iii) By the theory of stability of functions of Mather in the equivariant version of Wasserman [lo] one can, by a diffeomorphism (depending smoothly on the parameters, being e uivariant with respect 0 G to a to the flow of the Hamiltonian system G 2 = H$ and mapping :
8
function of G 02 ) , combined with replacing H’ by a suitable function of H’ and G 02 , bring H’ into the standard form G V ( ’ ) . Here it is essential that the condition that the change of coordinates is canonical is dropped, because the Birkhoff normal form contains infinitely many invariants, whereas here we end up with at most 4
parameters in the normal form.
A detailed proof, which needs much more space, will be published elsewhere. Instead, let me close with some additional remarks.
32
H
. DUISTERMAAT
Remark 1. In the analytic category, Brjuno [ 2 1 has the statement that the Birkhoff normal form converges on a set of periodic orbits determined by an equation of the form (7). This can be considered as a paraphrase of the Weinstein-Moser method. However, [2] deals also with quasi-periodic solutions, suggesting that the Weinstein-Moser method should have an extension to quasi-periodic solutions as well. Remark 2 . The techniques i) - ii) can also be used to obtain strong asymptotic results (for long time intervals) about all solutions near the equilibrium point (not only the periodic ones). This is closely connected with the averaging method, see Sanders [ 8 1 . Remark 3 . The techniques i) - iii) also work for more degenerate Hamiltonians, and if more than 2 degrees of freedom are in resonance. However, the algebra of standard forms becomes rapidly more complicated. For a description of the 2:l:l resonance (with positive definite F 2 ) see v.d. Aa and Sanders [l]. Remark 4 . The case of an eigenvalue on the positive imaginary axis with multiplicity 2, but with dim N = 2, is very interesting because it describes the transition from an elliptic to a hyperbolic equilibrium point. In this case ' A has a non-zero nilpotent part on the corresponding 4-dimensional generalized eigenspace, so the theorem of Moser [ 7 1 does not apply directly. However, we are convinced that a suitable variant also works in this case. An example is given by the Lagrange equilibria in the restricted 3-body problem for a special value of the mass-ratio. Van der Meer [ 6 1 has computed the relevant part of its Taylor expansion and checked that it is nondegenerate. Remark 5. If one prescribes the period w of the sought periodic solution then it is easy to show the existence of w-dependent diffeomorphisms which bring the periodic solutions into normal form. Oneof our basic problems was that we wanted an w-independent diffeomorphism on a full neighborhood of the origin which brings the whole family of periodic solutions into normal form at one stroke.
33
HAMILTONIAN SYSTEMS
REFERENCES [ 11 Aa, E. van der and Sanders, J., The 1:2:1 resonance, its
periodic orbits and integrals, in 'Asymptotic Analysis, from Theory to Application', ed. F . Verhulst, Lecture Notes in Plath. 711, Sprinqer-Verlag, Heidelberg 1979.
[ 21 Brjuno, A.D., Integral analytic sets, Dokl. Akad. Nauk SSSR
220
(1975), 1255-1258 - Soviet Math. Dokl. 16 (1975), 224-258. See also his Preprints 97, 98 of the Inst. Appl. Math. Acad. Sci. USSR, Ploscow, 1974.
[ 31 Duistermaat, J . J . , On periodic solutions near equilibrium points
of conservative systems, Arch. Rat. Mech. Anal. 160.
5
(19721, 143-
[ 41 Henrard, J . , Lyapounov's center theorem for resonant equi-
librium, J. Diff. Eq.
14
(1973), 431-441.
[ 51 Liapounoff, A.A., ProblZme gi?nSral de la stabiliti? due mouvement,
Ann.of Math.Studies
11,Princeton
University, 1947. (Original 1892)
[ 61 Pleer, J.C. van der, On the computation of 4th order coefficients
of a normalized Hamiltonian at 1:l resonance, Preprint, Utrecht 1980.
[ 71 Moser, J . , Periodic orbits near an equilibrium and a theorem by
Alan Weinstein, Corn. Pure Appl. Math.
2
(1976), 727-747.
[ 81 Sanders, J . A . ,
On the theory of nonlinear resonance, Thesis, Utrecht 1978. See also: Are higher order resonances really interesting?, Celestial Mechanics 16 (1978), 421-440.
[ 91 Schmidt, D . S . ,
Periodic solutions near a resonant equilibrium of a Hamiltonian system, Celestial Mechanics 2 (1974), 81-103.
[lo] Wasserman, G . , Classification of singularities with compact abelian symmetry, Regensburger Math. Schriften I, 1977.
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A N A L Y T I C A L AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS 1 N A N A L Y S I S S . A x e l s s o n , L . S . Frank, A . van d e r Sluis [ r r i s . ) @ N o r t h - H o l l a n d Pub1 i s h i n q C o m p a n y , 1981
ASYMPTOTICS OF ELEMENTARY SPHERICAL FUNCTIONS by Hans Duistermaat ldathematisch Instituut Rijksuniversiteit Utrecht Utrecht, The Netherlands
This is a report on work in progress together with J.A.C. Kolk and V.S. Varadarajan, in continuation of [ l ] . 1. Symmetric spaces of negative curvature.
The most fruitful approach to the analysis of symmetric spaces is via group theory. S o for us a negatively curved symmetric space S is a space on which a noncompact connected semisimple Lie group G acts transitively and such that the stabilizer of a point of S is a maximal compact subgroup K of G. That is, S may be identified with K\G and the space of functions (resp. distributions) on S with the space of left K-invariant functions (resp. distributions) on G. A continuous linear operator 4 : Cm(S) D' ( S ) which commutes with the (right-)action of G on S is always of the form +
c"(K\G) 3 f
(1.1)
;a *
f E c"(K\G)
where is a uniquely determined compactly supported distribution on G which is both left and right K-invariant. A left and right Kinvariant distribution on G is called spherical and the space of compactly supported spherical distributions on G will be denoted by E'(K\G/K). Note that 4 actually is continuous linear: Cm(S) Cm(S), C:(S) C:(S) and extends to continuous linear mappings: D' ( S ) D ' ( S ) , E' ( S ) E ' ( S ) . Also that E ' (K\G/K) is an algebra with respect to convolution, the convolution corresponding to the composition of the corresponding operators on S. +
+
+
If
+
= TeG, resp.%
OJ
=
TeK denote the Lie algebra's of G, resp. K,
then let 4 be the orthogonal complement of
k
in
bd
with respect to
the Killing form (1.2)
:
K
Because on which
(X,Y)
+
Tr(ad X
0
ad Y ) .
is non-degenerate and k is a maximal linear subspace of
K K
is negative definite,g = 35
k 86 and
K
is positive o n h .
,
36
H. DUISTERMAAT
Furthermore the map (x,X) x.exp X is a diffeomorphism: K x /3 G and w : x.exp XI-+ x.exp -X ( x E K, X E n ) defines an automorphism -1 for any of G called the Cartan involution. Writing x' = v(x) x E G, it follows that +
(x,y)
+
xyx'
:
G
x
expb
+
exp4
defines an action of G on e x p h allowing to identify exp.3 with the symmetric space S, having K as a Riemannian metric for which G acts by isometries. L e t a b e a maximal linear subspace o f 4 which at the same time is a commutative sub Lie algebra of 9. Correspondingly, A = exp M is a maximal flat subspace of S through e, r = dim A is called the rank of the symmetric space S. Because the ad X, X E 4 form a commuting set of symmetric (with respect to K @ - KI ) linear operators-in g, one has a common I6 I* decomposition of into eigenspaces:
7
(1.3)
y
=
Z' aEA
, 9,
91,
non-zero linear subspace of
43
such that (1.4) (ad X) (Y) = [X,Y1 = a(Y).Y
for Y E (aa.
Of course the eigenvalues a(X) are real and depend linearly on X E bt, so the a E A actually are elements o f m * , called the roots. if we write m = k n Also N C @do, more precisely q o = m @3
vo.
The null spaces in otof the a E A,a f 0 are called the root hyperplanes. Thier common complement has connected components which are convex open cones with piecewise flat boundaries, called the Weyl chambers. If k E K normalizes tn, that is Ad k ( W c M, then (because Ad k is a Lie algebra homomorphism) Ad k permutes the root hyperplanes and therefore also the Weyl chambers. Furthermore each orthogonal reflection in a root hyperplane arises in this way. In fact the Weyl group W, defined as the group generated by the orthogonal reflections in the root hyperplanes, is equal to the group (1.5) {Ad k l a ; k
z
E
MI
-
,M
=
normalizer of &in
K. In other words,
W = M/M if (1.6) M = {k E K; Ad k(X) = X for each X E
-
denotes the centralizer of &in K. The Lie algebras of and M are both equal t o m , so M is open in M. On the other hand M is compact 5
31
ELEMENTARY S P H E R I C A L FUNCTIONS
as a closed subgroup of K and therefore W is finite (a fact which puts very severe restrictions on the system of root hyperplanes). It is easy to see that W acts transitively on the set of Weyl chambers, it turns out that this action is simple as well, so in fact#(W) is equal to the number of Weyl chambers. Now choose a Weyl chamber, which will be called the positive + Weyl chamber OL from now on. (Because of the above it does not matter which one we choose.) The non-zero roots do not change sign + , write on (1.7)
A+ = {a E A ;
a(X)
>
0 for some (all) X E
m+)
for the corresponding set of positive roots. Then A = ( - A + ) U ( 0 1 u A + and writing (1.8) Z = {a E A+ ; not a = B+y for some 0,y
E
A+]
for the set of simple roots, it turns out that is a basis of m* and that each positive root is a linear combination of simple roots with non-negative integral coefficients. As,a consequence the walls of &+ are contained in the null spaces of the simple roots. Because [ ga~QdBl C (1.9)
Z
?t =
aEA+
qa+Bl the
sum
%a
is a nilpotent sub Lie algebra of OJ and N sub Lie group of G. The map (k,X,Y)
+
=
exp N is a nilpotent
k.exp X.exp Y
is a diffeomorphism: K x O'L x U + G and as a consequence each x E G can be written as x = k.a.n with uniquely determined k E K , a E A, n E N, depending smoothly on x (Iwasawa decomposition). For any 41
E C:(K\G/K)
(1.10) ( A $ ) (X) = e-p'x)
here (1.11)
p =
5
Z
write
I N
@(exp X.n)ds, X
E
n,
dim g,.a.
U€A+
A is a continuous linear map: C I ( K \ G / K )
,
extending to a E' (a).More importantly A is a continuous linear map: E' ( K \ G / K ) homomorphism with respect to convolution and it maps to W-invariant distributions on oi. The deepest result however, on the proof of which we shall comment later,is that A actually is an isomorphism: +
+
Cz(W)
30
H. DUISTERMAAT
resp. (.8(K\G/K), * ) ( d ( M ) W , * ) . For Schwartz functions the result is due to Harish-Chandra and for compactly supported functions to Helgason ( in some special cases) and Gangolli 1 2 1 . In particular the algebra of G-invariant linear operators on S ( € I (K\G/K), * ) is commutative. For example, the G-invariant differential operators D on S are, since they are local operators on K\G, given by convolution with @ E E'(K\G/K) satisfying supp @ C K. But then supp A @ c {O}, that is A @ = Dgt6 for some uniquely determined differential operator D D&is an isomorphism from with constant coefficients on&, and D the algebra Diff (S)G of G-invariant differential operators on S onto the algebra Dif f (a) "61 of W-invariant differential operators with constant coefficients o n m . NOW, if u E D ' ( S ) is a common eigendistribution of all G-invariant differentialoperators D on S, that is (1.12) Du = A ( D ) . u for all D E Diff(S)G , +
+
A(D) is a homomorphism : Diff(S)G then D identification with Diff (a)"& is given by +
+
C, which via the
for some X EdC;f: which is uniquely determined modulo the action of the Weyl group. The space of common eigendistributions for a given X consists of analytic functions. It contains a spherical one, unique up to a factor, which is given by
Here H(y) is the element of ot defined by (1.15) y E K.exp H(y).N.
In view of the Iwasawa decomposition, H is a smooth mapping: G called the Iwasawa projection. In fact @ A is equal to the image of the exponential function X e(xrx' under the transpose: C"(0t) Cm(K\G/K) of the continuous linear map A : E' (K\G/K) E l (OL) , the integration over K is needed in order to make the functionright K-invariant as well. It may also be remarked that @ A is a common eigenfunction for all G-invariant C m ( S ) and that Fourier continuous linear operators : Cm(S) +
+
+
+
+
ELEMENTARY SPHERICAL FUNCTIONS
39
decomposition in ot shows that every element of E ' (K\G/K) can be expanded in @ p E a*.So the name elementary spherical function ip for the $ A is fully deserved. 2. Harish-Chandra's assptotics. The map kl.exp X.k2
-+
X (kl,k2 E K,X E m+) defines a fibration a
+
from an open dense subset U of G to OL I if f E C"(ot+) then a*f is a spherical function on U. For any differential operator D on G I
+
D(a*f)I + is a differential operator onot , called the radial M part Drad of D. On the other hand one has the integration formula f
-+
(2.1) J f(x)dx = J A(X). I f(kl.exp X.k2)dkldk2 dX G KxK R+ where dim q, (2.2) A(X) = const. II (f(ea (X)-e-a (X)) ) c i a +
and following Gangolli [2] it turns out to be somewhat more convenient to work with
Then D
+
6
is a homomorphism from Diff (S)G to the algebra of
+
differential operators on m with smooth coefficients, the conjugation with the factor A' makes that also D* = ( 6 ) * so in particular symmetric operators get mapped to symmetric operators. = A f $ / + are common eigenfunctions of all 6, Obviously the hot
D E Diff(S)G. If the 6 would have constant coefficients then it would follow that the $ A are sums of W(W) many exponential functions which are easy to analyse. This is not true,but asymptotically when + (away from the walls) then D approaches a constant X m in ot coefficient operator in an exponential fashion. In order to describe this, write -+
Then (2.5)
+
01.
=
[z EIR'
: 0
<
z,
1 for
ci
E
XIl
z a = 0 corresponding to infinity and z a = 1 to a wall of the Weyl + i chamber bl . Then 6 is an operator in OL with coefficients which can
40
H . DUISTERMAAT
L
be written as analytic functions of z E Q: , having only poles if z = 1 for some a E Z. The highest order part of 6 has constant caefficients, and also the constant coefficient part of 6, obtained by taking the values of the coefficients at z=O turns out to be equal to the operator DR E Diff (Ot)wrR mentioned in section 1. The common eigenfunctions of the DOLare linear comb nations of the exponential functions
and because Qw*h 1Y (2.7) Q A ( X )
= Qh
for all w
- c(h).wEW C e
E
W, it is likely that asymptotical-
(w(XI)
for some c(A). From this Harish-Chandra [ 3 1 deduces that if F denotes the Fourier transform mapping functions on ot. to functions on oL*, then tAo F - l is a unitary embedding : L2 (m*/W,B) L 2 ( G ) if onot*/W we take the measure +
(2.8)
6
=
Ic(ip)l-2 dp
(li E
m*)
called the Pencherel measure for spherical functions. This is the main step in the proof of the isomorphism property of A mentioned in the previous section, the remarkable fact being that the Plancherel measure is expressed in terms of the coefficient in the leading term of the asymptotic expansion of the elementary spherical m in &+. functions 4 (x) as x ili The most elegant proof of the Harish-Chandra asymptotics can be obtained by observing (like is done by Kashiwara e.a. [ 5 ] ) that in the za-coordinates the satisfy a "holonomic system with regular singularities at z a = 0 " . Because the only other singularities occur at z a = 1 this approach immediately leads to convergent power series expansions with radius of convergence equal to 1. Also this approach leads to a better understanding of the behaviour if A = ip, p approaching a root hyperplane, in which case the Harish-Chandra asymptotics breaks down. If the symmetric space has rank 1 , that is dim or = 1 , then the 6 are ordinary differential operators and one can apply the classical theory of such operators with regular singularities. However, the point is that if dimor > 1 the system of partial differential equations actually leads to a system of ordinary differential equations with regular singularities along +
+,
41
ELEMENTARY SPHERICAL FUNCTIONS
each curve in the z-space, to which then the same asymptotic theory can be applied.
as p
3. Asymptotics of I$~"(X)
+
m
in in*.
In [ 1 1 we studied the asymptotic behaviour of the common spectrum of the G-invariant operators on S, pushed down to a compact quotient S/T of S in order to make the spectrum discrete. It is not surprising that for more detailed information we need the asymptotic
1x1
(keeping Re A bounded) which is complementary to the asymptotics of Harish-Chandra.
behaviour of I$,(X) as
+
m
Here the approach is to incorporate Re X in p in (1.14) and read (1.14) as an oscillatory integral
i( p,H(xk)) g (H(xk)) dk e K with phase function f : k (p,H(xk)) and amplitude k g(H(xk)) . UfX + In view of the decomposition G = K . e x p 4 .K we may take x = exp X, + X E OL and then H(xkm) = H(xk) for m ' E M = dentralizer of M. in K (see (1.6)) , so (3.1) can actually be seen as an integral over the flag variety K/M rather than over K. (The name is because for the classical groups, M is the stabilizer group for a natural action of K on a set of flags.) The first step is to observe that the asymptotic expansion of (3.1) for p = w.v, w E I R , w m is determined by the behaviour of the amplitude near the stationary point of the phase function fv,X. (3.1)
I
+
+
+
3.1. Theorem. The set of stationary points of f
VfX
(3.2)
Cv,x =
is equal to
K'GK',
which is a smooth submanifold of K, and the rank of the Hessian of is equal to the codimension of C in K. f v f X at each point of C vtx V,X (Clean stationary point set in the sense of Bott.) Obviously f is constant on the connected components of C /PI v,x v,x and because W acts transitively on the set of these connected components, an application of the method of stationary phase immediately leeds to the asymptotic expansion
as p = o.v, w
+
m.
42
H . DUISTERMAAT
An explicit calculation of the Hessian of f at c gave us an v,x v,x explicit formula for the leading coefficient cw,ot if both v and X are regular (not in a root hyperplane), then
Although the proof is quite different, the result is very similar to the asymptotics of Harish Chandra. In fact I believe that in the same way as Harish Chandra's asymptotics implied that tA*°F-l is a unitary embedding: L 2 ( O t * / W , B ) L 2 ( G ) , our asymptotics implies that the adjoint F o A is a unitary embedding:-L2 ( K \ G / K ) --* L2( */W,B). As a consequence A is injective, or equivalently A*OF-' has a dense range, which was the missing bit of information in the proof that A is an isomorphism. With a lot more work we are able also to get uniform asymptotic estimates when 1 ~ --*1 m , allowing u to pass through the root hyperplanes. (There the behaviour is quite' singular because the dimension of C then suddenly increases, leading to a jump in the order of v,x the asymptotic expansion ( 3 . 3 ) . This is an example of the Stokes phenomenon.) Here X is kept in compact subsets: we are also working on the still harder problem to understand what happens if both p and X run to infinity, that is to unify our asymptotics with the asymptotics of Harish-Chandra. I would like to close with some comments on the remarkable properties of the Iwasawa projection +
(3.4)
k
H(xk)
:
K/M
+
OL
which are a consequence of the description of the set of stationary points of fv,x. (fv,x is nothing else as testing the image with the linear function v.) The remarkable fact is that the set of stationary points in K/M does not move continuously with v : K v only depends on the set of roots orthogonal to v and this varies within a finite set, the dimension only going up when v enters the intersection of more root hyperplanes. Assuming for convenience that X is regular + ( = not in a wall of O t ) then if v is regular as well, /M = W all v,x the time, so for instance the maximum value of fv,.j( is equal to ( v, (w-l(X)) for some w E W, all the time. It immediately follows
ELEMENTARY SPHERICAL FUNCTIONS
43
that the image of K/M under the Iwasawa projection is contained in the convex hull of the finite set {w-l(X) ; w E W) and expanding this argument a little further one finds back the 3.2.
Theorem (Kostant [ 6 1 ) . The image of K/M under the Iwasawa projection ( 3 . 4 ) is equal to the convex hull of the Weyl group orbit of X in 4. This result is very remarkable indeed because K/M is a beautiful smooth compact manifold without boundary, the Iwasawa projection is analytic, nevertheless the image is a polyeder, full of faces, edges and corners. Of course for us the surprise came already at an earlier stage, namely that the asymptotics of ( 3 . 1 ) could always be obtained by just applying the methods of stationary phase on a set of stationary points depending in a discrete fashion on the parameter V.
For more convexity results, generalizing Kostant's theorem, see the thesis of Heckman [ 4 ] , defended yesterday in Leiden.
REFERENCES Duistermaat, J.J., Kolk, J.A.C. and Varadarajan, V.S., Spectra of compact locally symmetric manifolds of negative curvature, Inv. Math. 2 ( 1 9 7 9 ) , 2 7 - 9 3 . Gangolli, R . , On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Liegroups, Ann. of Math.
93
( 1 9 7 1 ) , 150-165.
Harish-Chandra, Spherical functions on a semi-simple Liegroup I,II, Amer. J. Math. 80 ( 1 9 5 8 ) , 2 4 1 - 3 1 0 , 5 5 3 - 6 1 3 . Heckman, G.J., Projections of orbits and asymptotic behaviour of multiplicities for compact Lie groups, Thesis, Leiden, 1 9 8 0 . Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T. and Tanaka, M., Eigenfunctions of invariant differential operators on a symmetric space, Annals of Math., 107 ( 1 9 7 8 ) , 1-39.
Kostant, B . , On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ec. Norm. Sup. 6 ( 1 9 7 3 ) , 4 1 3 - 4 5 5 .
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A N A L Y T I C A L AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS IN ANAL.I'SIS S . A x f l s s o n , L . S . F r a n k , A . vdn d c r S l u i s @ North-Ho1lCina Publishing C o r n p i n y , 1 9 8 1
(rds.)
ON THE QUESTION OF THE EXISTENCE AND NATURE OF HOMOGENEOUS-CENTER TARGET PATTERNS IN THE BELOUSOV-ZHABOTINSKII REAGENT Paul C. Fife' Mathematics Department University of Arizona Tucson, Arizona USA
This continues previous work by Tyson and the author on the application of multiple-scaling techniques to the modeling and analysis of the expanding concentric rings of chemical activity seen in the Belousov-Zhabotinskii reagent. The model is based on "Oregonator" kinetics. Previous work with these kinetics concentrated on heterogeneous-center patterns, induced by an external particle or other stimulus; this explores the possibility of homogeneous center structures, more controversial but reportedly observed. 1.
INTRODUCTION
In [ l ] , Tyson and the author presented an analysis, based on multiple-scaling techniques, of the phenomenon of expanding concentric rings of chemical activity As model we took Tyson's seen in the Belousov-Zhabotinskii reagent [ 2 - 5 1 . scaled version of the differential equations of the Oregonator [6], reduced to two equations by a pseudo-steady-state argument, and supplemented by a diffusion term for one of the reacting components (the lack of such a diffusion term for the other component was inessential for our argument). These equations contain a "stoichiometric parameter", which we symbolized by "b". We supposed that a catalyst particle or other externally imposed heterogeneity effected a spatial variation in b in a neighborhood of one point, and we found how such a variation could induce the formation of an expanding circular pattern centered at the point mentioned. We invisaged an initial state in which the reacting chemicals were uniformly distributed in space. Away from the heterogeneity, the value of b was such that a stable rest state was possible, and the initial uniform distribution was taken everywhere to equal this state. Near the center, however, the values of b were such as to make the reaction kinetics oscillatory. It was shown how the tendency to oscillate near the origin induces the periodic formation of abrupt (steep-profiled) expanding chemical wave fronts and backs. These generated and propagating fronts and backs eventually result in a periodic target structure. The existence of such heterogeneous-center patterns has been well corroborated by experiment; but there have also been sparse reports [7-91 suggesting the existence of homogeneous-center ones (target patterns without a catalyst particle or other externally-imposed heterogeneity at the center). The purpose of the present paper is to examine the question of their existence, again within the context of the Oregonator model. The supposition here is that the value of b is related to the concentration of some third chemical, besides the two entering into the basic equations, and that the production rate of this third chemical is affected slightly by the others. 45
46
P.C. FIFE
Specifically, we suppose the fast species ( u = HBr02 in the notation of ( 1 1 ) weakly catalyzes the production of the third, and also that the latter slowly decays into an inert (for our purposes) state. No attempt is made here to interpret these effects in terms of the known ingredients of the reaction. As in [l], we work only with one-dimensional patterns; the spatial coordinate should be interpreted as distance from the center.
(XI
The process we describe begins with an assumed initial nonuniform distribution of b (or third chemical), possibly due to uneven mixing, such that a target pattern will be produced according to the process described in 111. Then as a result of the weak interaction between b and the other components, the distribution of b, hence the characteristics of the patterned structure, will slowly evolve in time. Under certain reasonable assumptions, it is shown that a special target pattern is approached in the limit. This special pattern has a unique period; and is such that b is uniformly distributed (independently of x), the waves propagate at constant speed, and successive fronts are equally spaced. It is not known, at least by the author, whether homogeneous center patterns with these characteristics have been observed. Of course heterogeneous-center patterns with diverse periods are in evidence. other hand, it should be noted that the uniqueness in period of the here constructed is seen only after sufficient time has elapsed; in stages, no set period is predicted.
target On the patterns the first
Another analysis of model homogeneous-center patterns was given by the author in [lo]; that analysis differs from the present one in that both the wave fronts and wave backs are of "trigger" type (in Winfree's terminology), whereas we here follow the more reasonable approach in [l], taking the backs to be phase waves and the fronts to be trigger waves. Another difficulty is that the chemistry supposed in [lo] (at least as regards the moderate speed component) was not necessarily linked to the Oregonator model. In [ll] a model for homogeneous center patterns was also proposed, similar in concept to that in [ l o ] , but a minimal amount of analysis was supplied. Approaches to chemical wave propagation by multiple-scaling techniques with sharp wave fronts have appeared elsewhere in the literature; see, for example, [12-191. Other analytical approaches for a class of models have been given in [20-221, and a numerical study was made in [23]. In Section 2, the model described above, but without diffusion or spatial variation, is formulated as a system of ordinary differential equations with three separate time scales. It is shown that such a system may have both a stable rest state and a stable periodic solution, and the domain of attraction of the latter is estimated. In Section 3, small diffusion is added, and the resulting spatial pattern and its slow evolution are investigated. Conditions are given under which the pattern will evolve to a final state. The period of the final state will be unique, and b will be distributed uniformly. Some discussion remarks are given in the final section. I am grateful to J. Tyson for helpful comments regarding this paper.
BELOUSOV-ZHABOTINSKII REAGENT
2.
47
THE KINETICS
As in [ l l , we begin with a system of kinetic equations derived from the Oregonator model:
d * u = dt d A dt
where Ce+4,
=
u and v can be interpreted respectively, E << 1 , and b
1 E
*
*
- f(u,bv) ,
-
*
'
a s scaled concentrations of HBr02 and = O(1) is the stoichiometric parameter.
For convenience, we change variables, setting u = u , v = bv. In all the d d A following, - b will be small compared to - v, so we may set = bv dt dt (using = dt ) . We obtain -
11.11
,,a,,
h
=
-1
f(u,v)
,
$=bu-v. The function
A
f
has the following type of nullcline:
Figure 1
(1)
(2)
48
P.C. F I F E
shown on the figure are various (dotted) lines along which G = 0, for bo > b > b2. In [l], this nullcurve was approximated by a broken line made up 1 of straight segments. ( A l s o , in the Oregonator kinetic equations there is another small parameter a, measuring the distance from the relative minimum in the figure to the origin. We ignore any effect of this small parameter.) Also
In addition to (1) and (2), we suppose that b is related (linearly, for simplicity) to some third chemical w (different from the w in [l]), which evolves slowly according to the equation
4 = 6(au
- w),
6 << 1, a > 0
.
(3)
If u is periodic in the normal time variable t, though possibly modulated in a slower time scale s = 6t, we can suppose w and u are functions of t and s , and write ( 3 ) as
* at
Averaging over a period in
+
6 L a! = 6(au
as
t, we obtain
_ a w- where Let b
u
as
is the t-average of
depend on
w
- w) .
a ~ - w ,
(4)
U.
linearly, by a relation b = - ,Bw + y ,
.
(B > 0)
(5)
For both stationary and strictly periodic solutions (the period being the t scale), the right side of ( 4 ) will vanish, hence from ( 5 ) b=-B<+Y. We suppose the point (21, ( 5 ) :
(u,,vo)
O(1)
on
(6)
in Figure 1 is a stable stationary state for ( l ) ,
f(u0 ,v0 ) = 0, bOuO = v o ,
bo = -Buo + y
.
(7)
For fixed b in the interval bl > b > bz, the system (l), ( 2 ) undergoes relaxation oscillations, and the average value of u during such oscillations will, of course, be a function of b:
-
u = U(B),
(bl > b > b2)
.
(8)
For b > b l , there are no oscillations, but rather stable rest states. We extend the definition of U in a natural way to this range in b by setting U ( b ) equal to the value of u for the corresponding rest state. Strictly periodic solutions of the complete system ( 1 1 , (21, ( 3 ) are, as we have seen, such that b is approximately constant and ( 6 ) holds approximately. Substituting ( 8 ) into (6), we find such values of b are characterized as zeros of the following function f: F(b) E - B U ( b )
+
y
-b=
0
.
(9)
The stability of such solutions can be analyzed as follows. The relaxation oscillatory solution of (11, (2), for fixed b, is well known to be stable. We only need to examine the stability of the value of b with respect to its slow evolution. This evolution process is obtained from ( 4 ) and.assumes the form
BELOUSOV-ZHABOTINSKII REAGENT
_ db ds
F(b)
49
.
Stable zeros of F are therefore those throuqh which F decreases, and the domain of attraction can be found by looking at the signs of F. We shall now examine the behaviour of F (Fig. 2). For b > b l , the only periodic solution of ( l ) , ( 2 ) is the unique stationary point (u,vl found at the appropriate intersection in Figure 1 ((uO,vo) is a typical example). It follows that U(b) decreases continuously and monotonically in the interval b < b < bO, from u1 to u o . On the other hand, we shall show that U tyiically has a downward jump discontinuity at h l , as shown in Fiqure 2 (a), so that lim U(b) > ul. This follows from the fact, shown below, that a point, bfbl initially on the branch L, in Figure 1, moving under the dynamics ( 1 1 , ( 2 ) , will reach ( u l , v ) in finite time. Hence for b slightly less than bl, the time it takes the relaxation oscillator to transverse L 1 is bounded independently of bl - b. It follows that the average of u during such an oscillation will typically be bounded away from u 1 = U(bl). Actually, these statements are only approximately correct, and are also somewhat conditional. They are based on the assumption that ( E being small) the trajectory tracks the curve L1 exactly. The small derivation from L 1 , however, becomes more pronounced (but still o ( 1 ) as E + 0 ) as the bottom of the nullcurve is rounded. There will be a slight overshoot, possibly causing the transition to the oscillatory regime near b = bl to be premature. But these considerations do not affect the qualitative result. Another worry is that the jump in U at bl may be downward rather than upward. The argument is that after transition to oscillation, the trajectory suddenly spends a finite part of its effectively suddenly increasing the average value of U. This time on L j , increase may not come about if a great preponderance of time is spent on L 1 or if the rounded base is very broad. We simply assume that the numbers are such that the jump is upward. It will be so if the curvature near the bottom of the nullcurve is large enough. Although interval
U is discontinuous at b < b < bl. 2
bl, it will clearly be continuous in the
Our main assumptions now are (see Figure 2 ) : ( 1 ) bo bl is so small that F(bl 0) < 0, and (2) B > 0 is so small that
-
-
F'(b) < 0, b
2
< b < b l , and F(b2 +
0)
>
0
.
Condition ( 1 ) is feasible. In fact, by assumption F(bo) = 0 (see ( 7 ) ) ,and so (F(bl)) is small for bo - bl small, and the finite upward jump makes Condition ( 2 ) is also clearly feasible, from the form of the F(bl - 0 ) < 0. dependence of F on B. The main result in this section is:
-
Under the above assumptions, there exists, besides the stable rest state ( 7 ) a unique (up to translation in time) stable periodic solution of ( l ) , (21, ( 3 ) If I for which b lies in the interval (b2,bl) and is approximately constant. initially, b(0) lies in this same interval, then the solution of ( 1 ) - ( 3 ) evolves into this periodic solution as t +
-.
P.C. FIFE
I I
1
I
Figure 2 Schematic representation of
U(b)
and
F(b).
To show this conclusion, we simply note that our assumptions imply that F(b) decreases through 0 at some unique point b* in the interval (b2,bl), as desired. *This value produces a stable *limit cycle. In fact, F(b) > 0 for b e (b2,b 1 and F(b) < 0 for b f (b ,bl), so that if b(0) E (b2,bl), b ( s ) + b* as s + m . In this sense, the oscillatory solution evolves into the unique periodic solution. For future reference, we denote the latter's period by T*. All that remains is to show that a trajectory on L,, for b = bl, reaches (ul,v) in finite time (under the approximation that the trajectory tracks L 1 exactly). Near the bottom point on L,, the curve L 1 is approximated by the function 2
k (v Hence Prom (2) with
-
b = bl, v
= (u
-
- ul)2 ,
small,
k > 0
.
51
BELOUSOV-ZHABOTINSKII REAGENT
Any solution of this with 3.
u
1
-
-k(v
b v 1-
-
k(v
- v) 'I2.
- 1)'I2
v(0) > 1 reaches the value
in finite time.
THE PATTERNS
As basic model, we take ( l ) , ( 2 1 , ( 3 ) , ( 5 ) , with the diffusion term adjoined to (l), and (3) rewritten in terms of b:
u
t
= EU
xx
1 + f(u,v) ,
-
b + y)
xx (11)
E
vt=bu-v, bt = 6(-Bu
EU
(12)
.
(13)
We retain all the assumptions of Section 2 , so that the main conclusion there still holds. As before, b evolves on a slow time scale: b = b(x,s), averaging over a t-cycle, we approximate ( 1 3 ) by
s =
6t, and by
-
bs=-Bu-b+y.
(14)
As mentioned in the introduction, we assume an initial nonuniform distribution b(x,O). We suppose it attains a minimum at x = 0 , such that
b(O.0) < bl;
lim b(x,O) = b 0 1x1-
'
We assume, for simplicity, that the initial distributions of u and v are constant and equal to the rest state corresponding to b = bo: u(x,O) z u 0' v(x,O) :v 0'
On the moderate time scale of order 1, b remains essentially constant in t We assume and a target pattern develops, as detailed by Tyson and Fife in [ll. 6 is so small that this pattern has time to become established as an approximately periodic structure before b changes appreciably. Its period To will be that of the relaxation oscillator with fixed b = b(0,O). x > 0, thz u-v traiectory of the oscillation will be such that the range of v is v (x) < v < v, where in general v+(x) > 1. This results in a typical period loop as shown below: On the other hand, it was shown in [ l ] that for each fixed value of
52
P.C. FIFE
V
1
Figure 3 The approximate time it takes to traverse this loop can be calculated from a knowledge of b and v+, using ( 1 2 ) and the fact that most of the time is spent on L 1 and L3, where u and v are related by f(u,v) = 0. Thus for some function g, we have T = g(b,v+)
.
(15)
This relation, together with the conditions T = To, b = b(x,O), yields v+ a function of x (in fact, see ( 1 7 ) below). The trajectories of the wave fronts can then be found from the differential equation
as
c is the characteristic speed of a front at the value v = v+ (see [ l I The trajectories of-the wave backs are determined from the conditions that they occur where v = V.
where
[lo]).
,
But our main concern is with the subsequent (slow) evolution of this pattern. The basic evolution equation on this slow time scale is ( 1 4 ) . We are assuming it is slow enough that at each value of s, the pattern is an established structure of the type described above and in [ l l , periodic in the faster time variable t. Thus, we are looking for slow modulations of periodic solutions. Therefore (15) holds for each (x,s). In addition, as previously indicated, T will be a function of b(O,s),_ namely the period of a relaxation oscillator with that value of b. Setting b(s) = b(O,s), we have T = h(g) for some function h.
(16)
53
BELOUSOV-ZHABOTINSKII REAGENT
It is pretty easy to see that the function g variable, so (15) may be solved for v+: v+ = V(b,T) Finally, the average
in (15) is monotone in its second
.
(17)
over a cycle will depend on
b
and :'v
This function is related to the function of a single variable in ( 8 ) by U(b) = U(b,v). From (14), (18), and (171, we have
_ ab as
Specializing this to we find
-BU(b,V(b,T))
x = 0
-
b + y E G(b,T)
and using the fact that
. 1 at
V+ =
(191 Y,
= 0
always,
Our problem has now been reduced to
(i)
solving ( 2 0 1 , with the given initial value
(ii) finding T(s)
u(b,v+) > uo
b(0) = b(O,O);
from (16); and
(iii) solving (19) with T = T(s), Our main result is the following:
If
. d
for all
using the prescribed initial data
b(x,O).
(b,v+) admitting oscillations,
b < b(0,O) = g(0) < b l , b(x,O) = bo for large 1x1, and 8 is small 2 enough, then the solution b(x,s) & (19), (20) is such that for all x, lim b(x,s) = b*,
lim T("bs)) = T*
S+m
S+m
.
(21)
Before showing this result, a comment is in order regarding the assumption. For b > b l or €or b < b2, there is a rest point €or ( 1 1 , (2) on branch L1 or Lg in Fig. 1. This means a periodic solution of the type shown in Fig. 3 is impossible unless v+ is larger than the value of v at that rest point. This is the meaning of (b,v+) "admitting oscillations", and of course U(b,v+) is only defined for such (b,v+). If b2 < b < b l , the problem does not arise. For a solution of thfs type, time is spent on both L 1 and Lj, so the condition u = U(b,v ) > uo is apparently not very restrictive. The demonstration of (21) proceeds on the basis of the following lemma, whose proof is straightforward and will not be given: Lemma:Let where F (i) F I
y(s)
satisfy
has the following properties for some interval is differentiable in x
y
and continuous in
I = (y',~"):
(y,s), uniformly in
54
P.C. FIFE
(ii) lim F(y,s) = F(ylm) exists for all S+-
(iii) F
Y
<
-U
<
0
(iv) F ( y ' , s ) > 0
in and
I
X
y
I;
E
(O,m);
F(y",s) < 0
for all
Then if y(0) E I, y(s), approaches a limit unique solution of F(y , w ) = 0.
>
s
0
y*
s +
m,
* where y is the
-
To apply this lemma, we first note that since b(0) $ bl, the global stability result for (10) given in Section 2 implies b(s) + b , hence T(b(s)) + T In the lemma, we interpret y(s) = b(s), y" = bo, y' = b + w for some small W, and F(b,s) :G(b,T(b(s))). Clearly G < 0 for 8 2small enough, and b G
+
-BU(b,V(b,T
*
))
-b+y
.
=
G(b,T
*
)
as s + m. Now U(b,v+) and V(b,T) are uniformly continuous functions, which makes G(b,T(b(s))) uniformly continuous in b and s near s = m. It is clear, again, that for B small enough, G(b,T) > 0 for b = b + W. In fact, ( 7 ) shows y to be near bo than, so G(h,T) will be near bo2- b > 0. The only remaining condition to be verified is that G(bo,T) < 0 €or all S . But G(bO,T) = -BU(bo,V(bo,T)) = -BU(bO)
- bo + Y
- bo + Y
+ 8[-U(bo,V(bo,T)) + U(boll
= B[-U(bo,V(bo,T)) + U(bo)l =
B[-U(bO,V(bO,T)l + u,]
<
0
,
by assumption. (Note that the construction in [ l ] guarantees that (bo,V(bo,T)) admits oscillations, for every T(s) encountered.) The limit relations (21) have now been shown to hold. They imply that the period of the final state is unique, that b approaches a constant independently of x, and that the final state has v+(x) II. 4.
DISCUSSION
It was shown that by allowing the stoichiometric parameter to depend on a third chemical besides those in the Oregonator system, and by assuming that the new chemical's rate of production is af€ected weakly by u and that it decays slowly, we can explain the evolution of homogeneous-center patterns from a state in which the principal reacting components are uniformly distributed in space. It was assumed th t only the component u diffuses, its diffusivity being O ( E ) , .where E-' is the reaction rate of u (the fast variable). In our framework, v and w do not diffuse; however if they do, and their diffusion rates are of the same order of magnitude, then the qualitative picture is unlikely to change. The reason is that v's diffusivity would serve only to slightly round off any sharp corners in v's spatial distribution, and w's diffusivity would in fact only enhance the process of convergence to a uniform distribution. If E < 0(6), this uniformization tendency due to diffusion would have speed comparable to, or less than, that due to the process described in Section 3 , and so would not alter the results. The end result of the evolution process here is essentially a two-component oscillating system, since b (hence w ) is constant. The question may therefore be asked, why do we need a three-component system to study the
BELOUSOV-ZHABOTINSKII REAGENT
phenomenon? The answer lies in the ability of the three-component system to explain the transformation of a uniform quiescent state into an oscillating patterned state; this is not possible with a two-variable model, unless b is prescribed, and fixed, to be nonuniform. FOOTNOTE Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and the National Science Foundation under Grant No. MCS79-04443. REFERENCES Tyson, J. and Fife, P. C., Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys. to appear. Zaikin, A. N. and Zhabotinskii, A. M., Concentration wave propagation in two-dimensional liquid-phase self-oscillating system, Nature 225 (1970) 535-537.
Smoes, M.-L., Chemical waves in the oscillatory Zhabotinskii system: a transition from temporal to spatio-temporal organization, pp. 80-96 in Dynamics of Synergetic Systems, H. Haken, ed., Springer-Verlag, Berlin (1980).
Winfree, A. T., Stably rotating patterns of reaction and diffusion, Theor. Chem. 4 (Academic Pres, New York, 1978) 1-51. Tyson, J. J., The Belousov-Zhabotinskii Reaction, Lect. Notes in Biomathematics 10 (Springer-Verlag, Berlin, 1976). Tyson, J.J., Oscillations, bistability, and echo waves in models of the Belousov-Zhabotinskii reaction, Ann. N. Y. Acad. Sci. 36 (1979) 279-293. Zhabotinsky, A. N. and Zaikin, A. M., Autowave processes in a distributed chemical system, J. Theor. Biol. 40 (1973) 45-61. Marek, M. and Juda, J., Controlled generation of reaction-diffusion waves, Sci. Papers of Prague Inst. of Chem. Technol. Ser. K, to appear. Raschman, P., Kubicek, M., and Marek, M., Waves in distributed chemical systems-experiments and computations, to appear in Proceedings of Conference on Nonlinear Dynamics, Dec, 1979. Fife, P. C., Wave fronts and target patterns, to appear in Applications of Nonlinear Analysis in the Physical Sciences, Pitman Publishing, London. Zaikin, A. N. and Kawczynskii, A. L., Spatial effects in oscillating chemical systems. I. Model of leading center, J. Nonequilibrium Thermodynamics 2 (1977) 39-48. Ostrovskii, L. A. and Yakhno, V. G., The formation of pulses in an excitable medium, Biofizika 20 (1975) 489-493 (Biophysics 20, 498-503). Yakhno, V. G., On a model for leading centers, Biofizika 20 (1975) 669-674 (Biophysics 20, 679-6881. Fife, P. C., Pattern formation in reacting and diffusing systems, J. Chem. Phys. 64 (1976) 854-864.
55
56
P.C. FIFE
[15] Fife, P. C., Singular perturbation and wave front techniques in reactiondiffusion problems, pp. 23-49 in SIAM-AMS Proceedings, Symposium on Asymptotic Methods and Singular Perturbations, New York (1976). [16] Keener, J., Waves in excitable media, to appear. [17] Ortoleva, P. and Ross, J., Theory of propagation of discontinuities in kinetic systems with multiple time scales: fronts, front multiplicity, and pulses, J. Chem. Phys. 63 (1975) 3398-3408. [la] Feinn, D. and Ortoleva, P., Catastrophe and propagation in chemical reactions, J. Chem. PHys. 67 (1977) 2119. [19] Fife, P. and Tyson, J., Propagating waves and target patterns in chemical systems, Mathematics Research Center Technical Summary Report #2074 (1980). See also pp. 99-106 in Dynamics of Synergetic Systems, H. Haken, ed., Springer-Verlag, Berlin (1980). [20] Kopell, N. and Howard, L. N., Target patterns and horseshoes from a perturbed central force problem: some temporally periodic solutions to reaction diffusion equations, to appear. [211
Greenberg, J. M., Axisymmetric time-periodic solutions of reactiondiffusion equations, SIAM J.App1. Math. 34 (1974) 391-397.
[221
Ortoleva, P. and Ross, J., On a variety of wave phenomena in chemical and biochemical oscillations, J. Chem. Phys. 60 (1974) 5090-5107.
1231
Smoes, M.-L. and Dreitlein, J., Dissipative structures in chemical oscillations with concentrations-dependent frequency, J. Chem. Phys. 59 (1973) 6277-6285.
A N A L Y T I C A L AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S . A x e l s s o n , L . S . F r a n k , A . van d e r Sluis ( e d s . ) @ N o r t h - H o l l a n d Publishing C o m p a n y , 1 9 8 1
SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE E.M. de Jager University of Amsterdam R. Geel Ubbo Emmius Institute Groningen
A singular percurbacion of hyperbolic cype of firsc order linear and quasi linear differencia1 equacions is considered. The solucions of initial value problems as well as cheir derivacives have been escimaced explicicly.
1.
INTRODUCTION
Lec
u(x,c) ,
KO
be defined for
t 2
0 and
<
-w
x
<
, be a solucion of che
+m
following Cauchy problem:
wich
The posicive paramecer of che solucion
u
E
is very small
(0 <
E
<< 1 )
and
so
che decerminacion
conscicuces a singular percurbacion problem of hyperbolic
cype, which will be che subjecc of chis leccure.
In concrasc
KO
cype are noe
so
singular percurbacion problems of ellipcic rype chose of hyperbolic frequently creaced in che liceracure. As
and b independenc of
u
and
d(x,t,u)
d(x,c)u
- f(x,c))
LO
the linear case (a we mention che work
by J.M. Blondel Ell, J . Cole C21, M.G. Dzavadov C31, R. Geel [4,5j, R. Geel
-
E.M. de Jager i61, J.L. Lions C101,E.M. de Jager 1 9 3 , D.R. Smich 1121, D.R. Smich
- J.T. Palmer [ I 3 1 and :4.B. Weinsrein - D.R. Smich 1141. For che quasi linear case we refer M. M a d a m e
KO
R. Geel 141, J . Genec - M. Madaui-e C71 and
c 1 I]. 57
58
DE JAGER and R . GEEL
E.M.
J. Genet and M. Madaune have treated singular perturbation problems of t h e follo-
wing kind E L ~ ~ u+ ILllul + FLU] with
L2 =
12 7 -A,A
=
f(t,x) , t
0 , x c l 2 ccIRn
denoting the Laplace operator, and
at
LI =
a
a(c,x)
and where only of
U.
+
n Z
k= I
b (t,x) k
a -
+ c(t,x)
axk
F contains the non linearity in
I n contrast to Genet
-
u,
being a non linear function
Madaune our work is concerned with non linearities also
in the first order part of rhe differential operator but on the other hand we restrict ourselves to only one space variable. A l s o complications due to the boundedness of
11
(region of the space variables) are not considered here.
'Treating singular percurbation problems one has to consider also the unperturbed problem
(E
a(x,t,w)
aw + ax
=
0)
which in our case reads as
b(x t,w)
* at
+ d(x,t,w)
0
=
,
-m
<
x
<
+m
,
t >
0
(1.4)
with w(x,O)
=
0
,
-m
<
x
<
(1.5)
+m
The principal question i s now t o investigate whether the solution of ( 1 . 4 ) is a good approximation to the solution of rhe perturbed problem ( 1 . 1 )
-
(1.5)
- (1.2).
It is clear chat this is only possible if the subcharacteristic through a point with slope 5 lies within the characteristic triangle b xo + (t-t ) S x i xo - (t-ro) , 0 5 c 5 ro , because otherwise r h e value of 0 u(xo,to) for E small would be derermined approximately by an initial value (xo,tO)
in a point outside the base of the characteristic triangle, which is n o t in agreement with the theory of characteristics for hyperbolic differential equations. (see also figure I )
Subcharacteristic timelike
Subcharacteristic spacelike Figure 1
59
S I N G U L A R PERTURBATIONS OF HYPERBOLIC T Y P E
Therefore we suppose in che following chat the subcharacceristic is cime like,
i%i
i.e.
<
I
t,x ad u
for all values of
ic will appear thac we also need are made for che coefficients
i)
b
> la/
b
a,b
>
with
0 and
and
and all values of
c
x,e
and
u
troubled by cedious bookkeeping we cake
a,b
and
d
and
for all
d u
and of class
C2(u)
Besides this condieion
so ehe following assumpcions
are sufficiencly smoorh in
ii) a,b
.
d :
x , all non negative
for all
c '. 0
for all
. In
u
.
order noe
of class
be
LO
C"(x,c)
(x,t).
We remark chac condieion i) may also be inrerpreted as a physical condicion for exponential decay of jumps in che firsc derivaeives along the characreriscics, see a l s o 121. 2.
THE LINEAR PROBLEM
2. I
The formal npproxirnatioii
-
In the linear case we wriee ( 1 . 1 )
(1.2) as
-m
<
x
<
+ m ,
-m
<
x
<
+m
c > 0
wich che inicial condieions u(x,O)
=
au T-(X,O) dL
=
0
,
(2.2)
A formal approximacion of che solucion of ( 2 . 1 ) - (2.2) is readily be given by
(2.3) wich
w
sacisfying rhe reduced problem
aw
a(x,t) - + b(x,c) ax
aw
+ d(x,t)w
w(x,O) and with
=
f(x,c),
= 0
:
(E=O)
-m
<
x
+m
, K > o
, - m < x < + m
v , a correccion cerm co accounc for che second
satisfying che boundary value problem
inicial condition,
60
E.M.
DE JAGER and R. GEEL
Hence
and ic follows chac v and moreover chac v
, uniformly in any bounded set in the region
O(F)
=
c
2
has boundary layer characcer.
Puccing U(X,K)
-
:= U(X,K)
, . ,
+ R(x,K) := W(X,C) + )',X("
we obcain for che remainder cerm R(x,c)
- V(X,o)
the inicial value problem:
LELRj = O ( E ) , uniformly in any bounded domain wich R(x,O)
=
R (x,O)
= 0
,
< x <
-m
t > 0
c
2
0
lim
che unknown solucion
KO
, one has ac leasc R(x,K)
= 0
u(x,c)
-U(X,K)
in some domain
i s indeed a
D
wich
prove chac
KO
, V(x,c)
(2.7)
+m
I n order co show chac che socalled "formal" approximation
good approximacion
(2.6)
+ R(x,K)
F
D
EO '
Therefore we derive in che nexc Section an a priori escimace for che solution of the initial value problem ( 2 . 7 ) . 2.2
A Prior; Estimate
I n order
wich
KO
c 2 0
escimace the funccion R(x,K)
(see ( 2 . 6 ) )
in a bounded domain
, we consider again che initial value problem
(2.1) - (2.2)
D
which
is now wriccen in che form
$(x,O)
=
$c(x,o)
= 0
,
Multiplying ( 2 . 9 ) wich
-m
< x <
(2.10)
+m
2(aOx + b $ c )
we obcain afcer a simple calculacion
(2.11)
with
0
61
SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE
with
Enclosing the domain
D by a crapezium R
(ABCD) with
AC
and
BD characteriscic
(see fig. 2) and integrating (2.11) over ABCD , we obtain by means of Green's cheorem
(2.12)
t
8
A
Figure 2
Assuming only for che moment
C D
definite along respectively
.
B D
d > 0 and remembering
(Q, - Q,)
and
Due to the assumed regularicy of
ABCD
X
conscants
m
and
and a,b
and
(Q, + Q,) d
M , dependent on A B
b > / a / we have non negacive along
Q
positive
1
A C
chere exist for any fixed domain and
T , b u c independent of
E
,
such that
This inequality wich fixed m
ABC'D'
with
C'D'
below
C D
and
M
is a forciori also valid for any crapezium
(see fig. 2 ) ,
i.e.
62
E.M.
DE JAGER and R . G E E L
and hence according co Gronwall's lemma:
(2.13)
where
denotes henceforth a generic constant depending only on AB
C(s2)
but not
on
T ,
and
E
Substitution of (2.13) into (2.12) yields by means of che posicivicy of the non-negativity of
( Q , i Q2)
Q,
and
the complementary results
J
A and I
ID B
(2. 3) , (2.14) and (2.15) present L2- estimates for $ and its derivatives and it is now not difficulc t o derive from these formulae pointwise estimates for
+ , $t and $x , (see lit141, pp. 59-64). The result is
SUP R
Finally we have still to eliminate the restriction d > 0
a
achieved by adding a term
$
. This
is easily
to both sides of equation (2.9) with
a
>
max Id/.
After replacing in ( 2 . 1 2 ) the coefficient d by d + a and the function J1 2 J1 + a$ , majoracing (J1 + a$) by 2J12 + 2a2$2 and swopping 2a2$2 to Q, we arrive again at (2.13) - (2.15), where the generic conscant C(Q) changed independencly of 2.3
is only
E
The Approximation o f t h e SoZution
Applicacion of (2.16) to ( 2 . 7 ) yields immediately for che remainder cerm che a priori estimate
by
SINGULAR PERTURBATIONS OF HYPERBOLIC T Y P E
u n i f o r m l y i n a n y bounded domain
on
D
buc i n d e p e n d e n t o f
,
c 2 0
wich
where
C
63
i s a conscant depending
.
E
I c f o l l o w s now from ( 2 . 5 ) and ( 2 . 6 ) c h a t = w(x,c)
u(x,t)
+
O(E
314)
ux(x,t) = wx(x,t) + O(1) u (x,t) = w (x,t) +
E
(2.17)
V,(X,r-,
+ O(1)
u n i f o r m l y i n a n y bounded domain i n t h e r e g i o n n
-n
u (x,t) =
.
c
E1
i=O
Wi(X,C)
,
i n t o (2.1) - (2.2) powers i n
+
n
c
E
i=O
i+I
:=
0
.
(2.18)
e x p a n s i o n i n t o powers of
E
and e q u a t i n g t e r m s o f e q u a l
g i v e s y s t e m s o f e q u a t i o n s which c o g e t h e r w i t h p r o p e r i n i t i a l c o n d i -
E
P (x,t) "
E
i
w.
and
v.
.
w.(x,C)
+
n c
c
i=O
i+1
we o b c a i n f o r che r e m a i n d e r cerm
V.(X,~) E 1
c
n+ I
v n ( x , o ) + Rn(x,c)
=
3 / 4 1R;l
+
E
3 / 4 IR:
Rn
< C E
s a t i s f i e s the estimate
n + 314
,
u n i f o r m l y i n a n y bounded domain i n c h e r e g i o n (2.20) wich u(x,c)
,
n = I
= w(x,c)
+
O(E) O(E)
(X,t) =
W
(2.20) t t 0
. Using
f i n a l l y (2.18) -
t h e e s t i m a t e ( 2 . 1 7 ) may b e s h a r p e n e d a s f o l l o w s
ux(x,c) = w ( x , t ) + U
t > 0
0
A p p l y i n g a g a i n ( 2 . 1 6 ) we f i n d c h a t
E
(2.19)
che i n i t i a l v a l u e problem
Rn
L E L R n ] = O ( s n + l ) , u n i f o r m l y i n a n y bounded domain w i c h
Rn(x,O) = R:(x,O)
Setcing
+ X"(X,t)
i=(j
IRn( +
S u b s t i t u t i o n of
vi(x,;)
cions y i e l d consecucively t h e funccions u ( x , t ) :=
t 2
(X,C) + EVc(X,:)
+
O(E)
,
u n i f o r m l y i n a n y bounded domain i n t h e r e g i o n F o r d e r a i l s s e e l i c 1 4 1 , p p . 64-71,
o r lit-151.
c 1 0
.
64
DE JAGER and R . GEEL
E.M.
3
THE QUASI LINEAR PROBLEM
3.1
The Formal Approximation
We consider the initial value problem
with au u(x,O) = (x,o) = at
o ,
where the coefficients they are of class (x,t) tive
wich
Cm
0
t 2
< x <
+m
a,b and
d
-m
in
are now functions of
for all u
(x,t)
. We assume again
x,t
and of class
u ;
and
u
in
C2
for all
x , a l l nonnega-
b > la/ for all values of
u , A formal approximation of the solution of the
and all values of
t
,
initial value problem is given by
-u(x,c)
= w(x,c) +
w(x,c)
E
"(X,T)
.
(3.2)
denotes again the solution of the reduced problem
aw aw a(x,t,w) - + b(x,c,w) - + d(x,t,w) ax at w(x,O) E
=
0 ,
v(x,L)
-m
< x <
0
=
, --m
< x <
+m
,
(E
=
0)
,
i.e.
t > 0
(3.3)
t m
is a boundary layer term which corrects w(x,t)
che second initial condition. v(x,~)
=
V(X,T)
in order t o satisfy
is a solution of the boundary
value problem 2
a v 7 +
b(x,O,w(x,O))
av
=
0
,
T
> 0
aT E
av (x,O) = av (x,O) = at
aT
aw - -ac
(x,O)
,
lim
V(X,T)
= 0
T+=
and hence
(3.4)
Under the assumpcion thac w Y
u
is sufficiently regular
sacisfies the initial value problem (3.1) upto order
(w
E
3
C )
L)(E)
,
,
the funccion
SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE
65
In fact we have
-
, u(x,O)
L E G = I)(€)
=
v(x,O)
E
D with
uniformly in compact domains PutLing
-
-
u ( x , t ) = u(x,t)
-
, u (x,O) t 2
= 0
0
,
(3.5)
, where
w
E
C
3
.
E
+
= ;(X,t)
(3.6)
we obtain according t o ( 3 . 5 ) and ( 3 . 6 ) for the remainder term
R
the initial
value problem
2) a 2R + a(;
+ R) aR + b(;
+ {a(a + R)
-
a(u)]
+ {d(ii + R)
-
d(;)}
+ R) aR
at
ax
ax
a;
+ {b(u + R) - b(u)}
au at
(3.7)
, uniformly in 'D ,
= I)(€)
with the initial conditions R(x,O) = R (x,O)
0
=
I n order to simplify the notation we have omitted in the coefficients
d
the arguments x
and
I:
.
a,b and
I n the same way as in the linear case one may construct a better formal approxiu
mation of
by putting
"
U(X,t) =
i
1
E
n
W.(X,t) +
i=O -n
:= u
i=O
(x,t) + Rn(x,t)
where the terms
E
i
wi(x,t)
E
i+ I
t
v.(x,-) 1
E
-
E
n+l
vn(x,o) + Rn(x,t) (3.9)
, and
E
i+ I
v.(x,l) 1
E
are determined consecutively
by differential equations obtained from ( 3 . 1 ) by means of substitution of ( 3 . 9 ) into ( 3 . 1 ) , expansion into powers of of
E
E
and equating terms with equal powers
; for details see licC41, pp. 85-91.
In this way we obtain for the remainder term Rn
the initial value problem
66
E.M.
DE JAGER and R . GEEL
uniformly valid in any compact domain 0 , where
wo
... n)
and so w.(i=1,2,
are sufficiently regular,and with the initial conditions Rn(x,O)
=
Rn(x,O)
(3.8*)
= 0
-
u
In order to prove or to disprove whether the formal approximation
(or more
generally ;")is really an approximation of the unkwown solution u similarly as in the linear case, to estimate R generally Rn However
R
-
from (3.7*)
(or
from (3.7)
-
(3.8)
we have, (or more
(3.8*)).
Rn) satisfies a non linear equation and we need now a more
sophisticated reasoning than in the linear case. 3.2
A Fixed Toini; l'heorem
Van Harten 181 has considered elliptic singular perturbations of quasi-linear operators of the first order and in order to prove the justification of a formal approximation he applied a fixed point theorem for estimating the remainder rerm. I n this seccion we introduce a modification of this cheorem which will serve our
purpose, namely the estimation of the remainder term
.
R
The theorem reads as follows: Fixed Point Theorem. Let and
B
N
be a normed linear space with norm
11 11
a Banach space with norm
N
mapping
+
B with
, z
. Let
B
E
I .I
,
y
E
N
b e a non linear
F
F ( 0 ) = 0 and with (3.10)
where
L is the linearization of F at y
The following conditions are imposed on i)
The mapping
L
from
N
to
B
L
=
0
. I
and
is bijective and
L
-1
is continuous, i.e. (3.11)
R
where ii)
Let
(jN
is some number independent of
There exists IIy ( Y , )
where
Iy c N , IyI
( p ) :=
-
o
y(y2)11
m(p)
5
z
.
PI
such that 5 m(p)
I y,
-
is decreasing for
y2(
p
, v yi
6
nN(p)
decreasing with
, (i
= 1,2),
lim m(p) P-fO
o
= 0
5 p
.
5
P ,
(3. 12)
SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE
67
Finally define (3.13)
Assereion: If
f
B
E
chen there eXiSKS
g
11 f 11
wich E
N
5
? LP o ,
(3.14)
2
wich
Proof: The relation F(y)
=
L(y) + Y(y)
=
f
is equivalent wich che
relacion z =
T(Z) := f z
where
=
L(y)
o L-l(z) ,
Y
.
Consider now the ball
RB(p)
Whenever
and
11 f 11
< fkp
Iz
=
0
,
B
E
5 p 5 po
flB(Lp)
and is strictly contractive in
So the exiscence of a unique fixed poinc
11
.
5 .pi
then
.
z
=
T maps z*
QB(lp)
into
RB(P.p)
is quaranteed and hence
chere exists a solution of che equacion F(y) = f , namely y := g = L- 1 (z* ) , -1 lying in L [R,(P.ep)l c blN(p) Taking P = 2 Q-' 11 we obtain the result
1/f
(3.15). For details of the proof see 1 4 1 , pp. 2 0 - 2 4 , i8l
.
i 6 1 , pp. 30-32 , or
3.3 The Approximation of the SoZution I n order to estimate the remainder term (1 c
Rn
of secEion 3.1 in a trapezium region
D , we cake
q - Qco(n)
N = ( y ; y c C ( R1 ) ,
2
2
ar:
ax
€
, y(x,O)
wich
I I y
=
3 at
(x.0)
=
0)
(3.16) =
max
Iy1
;z
E
11
+
E3'4
max R
I
y,
/
+
=
max
E3'4
max R
lytl
and B = Iz
Co(R)l
wich
/Iz 1 1
12
1z I
(3.17)
68
E.M.
The map
F
L ( R ~ =)
E(7
from
N
i s given by ( 3 . 7 )
B
to
2 n
a2Rn
a R2 ) + - -
at
DE JAGER and R . G E E L
ax
a R" + K
a(?)
or
(3.7*)
and hence
a R" a t
b(?)
(3.18) and Y(Rn) =
If
a,b
{a(;"
+
Rn) - a(?)}
+ {a(;"
+
R")
+ {b(?
+ Rn)
a R" + ax
-
a(;")
-
-
b(?)
-
+ {d(? + Rn) - d(;")
-
and
d
che o p e r a t o r s
are
L
and
Cm Y
in
{b(;
aa -n au (u ) ab
a Rn K
+ Rn) - b(?)I
n R I
n (in) R }
a;" ax
a;" (3.19)
( i n ) Rn} (x,t)
for a l l
u
s a t i s f y the conditions
and
c2
i)
and
in
u
ii)
p o i n t theorem of che p r e c e d i n g s u b s e c t i o n . For t h e q u a n t i t y
9.
-1
for all
(x,t)
of t h e f i x e d we o b t a i n
according (2.16) k
-1
=
C(n)
E
-114
(3.20)
It may be remarked h e r e t h a t t h e norm (3.16) i n t h e s p a c e
N
has been chosen
i n accordance w i t h t h e r e s u l t (2.16) The non l i n e a r p a r t (3.19) of t h e o p e r a t o r calculation
the factor
C(O)
The parameter
po
E -
21C(n)12
E
-
,u
n
yields after a l i t t l e
i s a g e n e r i c c o n s c a n t , dependent on
satisfies
and hence Po =
F
and
il
but independent of
E
69
SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE
Therefore in order c o escimace che remainder cerm we need according to (3.15)
,
a formal approximacion of ac lease order 2
n
i.e.
=
, (compare
I
(3.7*)).
Application of che fixed poinc cheorem yields finally IR
1
I=
max I R
1
I
n
= O(2e-I
+ s 3 I 4 max
J R ~ +/
n
E2)
=
O(E
I
I R ~ I
max
E3'4
n
714)
uniformly in any crapezium region
lying in
D , where wo is sufficiencly
regular. Finally ic follows from (3.9)
I
-
u - w O - E W I - E V O
I
V]
E2
= O(E7I4)
or /u
au
1- ax au
-
wo(
= O(E)
aw
0
-(=
(3.21)
O(E)
ax
- _ ac
-Iae
E
=
O(E)
.
uniformly in che crapezium n
We summarize chis result in the following theorem Theorem Suppose the coefficients a, b
and
of che differential equation ( 3 . 1 )
d
sacis-
fy che following condicions:
i)
a, b
ii)
b > la( for
d
and
C2(u)
are of class Cm(x,c) -m
+
< x <
When che SOluKiOn wo a(x,c,wo)
a wO + ax
-
u(x,O)
K 2
c 2 0
0 and
+ a(x,t,u)
uc(x.O)
.
< u < +-
-m
.
,
d(x,t,wo) -m
= 0
< x < +
,
L
> 0
,
-m
< x < +
cU
in some compacc domain D , chen che initial value problem
2 2) ax =
,
aw
0 ar: +
b(x,c,w o )
is of class c"(x,c) 2
m
and of class
of che initial value problem
wo(x,o) = 0
E(%a t
(x,t) wich
for all values of
u
for all values of
=
0
au ax + b(x,t,u) ,
-a
< x <
+m
au ar: + d(x,c,u) = 0 , c > 0
,
-
< x <
+m
70
DE JAGER and R . GEEL
E.M.
has a solucion in any crapezium region fies che escimaces (3.21), where
$1
,
lying in
D , and chis solucion sacis-
is given by (3.4).
vo(x,5)
Ic is clear chac simplificacions are possible in case a
. For application of che above cheory
on u
che reader
licc41
LO
,
b
do nor depend
some specific examples we refer
pp. 106-120.
Y
Remark
KO
and
Up Kill now singular percurbaeions of ellipcic and hyperbolic cype have been considered in the lieeracure in fair decail. It mighc be interesting
X
to
invescigaee also singular percur-
bation problems of mixed cype, such as Figure 3
a 2u
~ ( 7 y + ax
2
a u 7) + a(x,y,u)
aY
au - + aY
with boundary condieions along
r+
b(x,y,u)
and
r-
aY
+
d(x,y,u)
. Problems of
=
0
,
chis kind are scill
open for research. LITERATURE
ill
Blondel, J.M., Percurbation singulisre pour une Gquaeion aux dGrivi5es parcielles du second ordre, lingaire, du cype hyperbolique normal, C.R. Acad S c . Paris
257
(1963),pp. 353-355.
i 2 1 Cole, J.D., Percurbacion mechods in applied machemacics, Blaindell, Walcham, Mass., 1968. 13 1 Dzavadov, M.G., The Cauchy problem for a hyperbolic equation with a small parameter multiplying the highest derivatives, Izv. Akad. Nauk Ser. Fiz-Tek. Mat. Nauk
5
Azerb. SSR,
( 1 9 6 3 ) , pp. 3-9.
c 4 1 Geel, R., Singular percurbacions of hyperbolic cype, Machemacical Centre Traces, Machemacical Cenere, Amscerdam, 1979. I 5 1 Geel, R., Singular hyperbolic percurbacions of linear equations, co appear in che Proceedings of che Royal Sociecy of Edinburgh, seccion A (Mach), 1980.
C 61 Geel, R., de Jager, E.M., Inicial value problems for singularly percurbed non linear ordinary differencial equacions. Machemacical Chronicle, New Zealand,
8
(1979), pp. 25-38.
71
SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE
C71
Genec, J., Madaune, M., Percurbacions singulisres pour une classe de problgmes hyperboliques non lingaires; Proceedings of che conference in singular percurbacions and boundary layer cheory, Lyon, 1976, Lecture Noces in Machs.
594,
Springer, Berlin, 1977, pp. 201-230.
181 van Harcen, A., Singularly percurbed non-linear 2nd order ellipcic boundary value problems, chesis, Universicy of Ucrechc, 1975. 191
de Jager, E.M., Singular percurbaeions of hyperbolic cype, Nieuw Archief Wisk. ( 3 )
3
(1975),
pp. 145-172.
[ I 0 1 Lions, J.L., Percurbacions singulisres dans les problPmes aux limices ec en concrsle opcimal, Lecture Notes in Machs. 323, Springer, Berlin, 1973. [ I l l Madaune, M., Percurbacions singuliSres de cype hyperbolique - hyperbolique
non lingaire, Publ. Math. de Pau, 1977.
1 1 2 1 Smith, D.R., An aymptotic analysis of a certain hyperbolic Cauchy problem, SIAM J. Mach. Anal. 2 (1971) pp. 375-392. 1 1 3 1 Smich, D.R. and Palmer, J.T., On che behavior of che solucion of che cele-
graphiscs equacion for large absorpcion, Arch. Rae. Mech. Anal. 39 ( 1 9 7 0 ) pp. 146-157. 1 1 4 1 Weinscein, M.B., Smich, D.R., Comparison cechniques for cercain overdamped 6 hyperbolic parcial differencia1 equacions, Rocky Mountain J . Mach. -
pp. 731-742.
(1976)
This Page Intentionally Left Blank
A N A L Y T I C A L A N D N l J M E R I C A L APPROACltES TO A S I ’ N P T O T I C PROBLEMS I N ANA1,I‘SIS S . A x c l s s o n , L . S . F r d n k , A . v a n dci- S l u i s 0 North-Holland P u b l i s h i i i q C o m p ~ n i ] , 1981
(cds.)
COMPUTATION BY EXTRAPOLATION OF SOLUTIONS OF SINGULAR PERTURBATION PROBLEMS
F. C. Hoppensteadt Department of Mathematics University of Utah Salt Lake City, Utah 84112 and W. L. Miranker IBM T. J. Watson Research Center Yorktown Heights, New York 10598
We show how the asymptotic form of the solution of singular perturbation problems can be used to generate associated unperturbed or relaxed equations. Solutions of these relaxed equations are easily calculated and appropriate combinations of them furnish numerical approximations to the original problem. Variations of this method are applied to two classes of initial problems; those with rapidly equilibrating solutions and those with highly oscillatory solutions. INTRODUCTION Perturbation methods are used to derive numerical schemes for solving stiff ordinary differential equations, i s . , equations whose solutions change rapidly over short integration steps. Such schemes actually improve as the stiffness of the system increases. However, numerical implementation of the perturbation methods can involve costly evaluations of integrals which arise in matching and averaging conditions, and some preliminary analysis of the problem is usually required by this approach. Two methods are presented and compared here. These are described in detail for initial value problems to which the method of matched asymptotic expansions and the method of averaging can be applied. In each case, the perturbation solution is first described along with its direct numerical evaluation. This frequently entails lengthy computations. Next, it is shown how the perturbation solution’s form can be used to calculate the stiff solution by combining solutions of associated non-stiff or relaxed, equation. In Section 1 we study initial value problems having matched asymptotic expansion (MAE) solutions. The matching method is reviewed, and then numerical schemes based on direct calculations of the MAE approximation and on a n extrapolation method are described and sample calculations are presented. Problems having rapidly oscillating solutions are considered in Section 2. The averaging procedure is reviewed, and then numerical methods based on it are presented. First, we consider direct evaluation of averages, and then an alternative method based o n the almost periodic structure of the problem is described. This accelerates computation of averages by finding appropriate translation numbers of the function being averaged. Last, an extrapolation 73
74
F.C.
HOPPENSTEADT and W.L. MIRANKER
method based on the already determined translation numbers, along with some illustrative computations are presented. Additional details and computations may be found in [l]. The novelty of this work lies in the introduction of extrapolation formulas (7) in the MAE case; (14) in the averaging case. They are distinct from straightforward Richardson extrapolation formulas which are based on Taylor's formula and which are not usually appropriate for stiff problems. We stress that the extrapolation formulas (7) and (14) break through the typical limitation of the customary numerical evaluation of one or more terms in the asymptotic expansions supplied by the singular perturbation methodology.
1. PROBLEMS CONTAINlNG MATCHED ASYMPTOTIC EXPANSION SOLUTIONS Consider the initial value problem
where the real parameter e is near zero. problem,
Here x, f,[ e R m and y, g, qeR". The reduced
is assumed to have a solution, x = x o ( f ) , y = y o ( t ) , on some interval 0 5 r 5 T. The data f, 5 and 7 are assumed to be smooth functions of their arguments, and the Jacobian matrix
g,
gy(t9x&),
voW70 )
is assumed to be stable. Under these conditions it is known [2] that the solution of ( 1 ) has the form
(3) where the boundary layer terms X, Y have the following bounds. (4)
llX(f/e, E) II
+
II Y(f/e, e l II I Ke-"''.
8 is usually of the order of the smallest eigenvalue of gy, and K depends on other data in the problem. These estimates hold uniformly for 0 5 f I T and for all small positive E. We now show how this form of the perturbation solution can be used to calculate the stiff solution of (1) by combining solutions of aukiliary non-stiff or relaxed equations. This method which is called the extrapolation method begins by identifying a value of e, say e', which is substantially larger than E in magnitude, but for which the solution of ( 1 . 1 ) with e replaced by e' can be used to approximate x(h, E), y ( h , e), where h is some mesh increment. Thus, (1.1) is solved for larger values of E , and so it can be solved more accurately with less effort. The number of operations used in these computations is proportional to 1 / e and l/e', respectively.
75
COMPUTATION BY EXTRAPOLATION
Therefore the ratio E ‘ / E provides a measure of the relative number of operations of direct solution (by some conventional numerical method) compared to the extrapolation method. 1.1 DERIVATION OF THE RELAXED EQUATIONS Referring to (4), a value T is determined so that Ke-8T = O(hP+’).
(5)
Clearly T is only determinable as an approximate value. Next, a value
E’
= h / T is defined, and the system (1) is solved twice by a conventional
(order p ) integration method: first for x ( h , ~ ’ / 2 )y (, h , d / 2 ) and then for x ( h , follows that
= h ( h , E’/2)-X(h,
E’)
+ O(hP+’) + o((E’)2)+ O ( E ) ,
Y ( h , E ) = 2Y(h, E’/2)-Y(h,
E’)
+ O(hP+’) + 0 ( ( E ’ l 2 )
x(h,
E)
E’),
y(h,
E‘).
It
+ W E ) .
This relationship can be derived in the following way. From (3), 2 x ( h , 2 / 2 1 =2xo(h) + X l ( h ) E ’ + 2 X ( 2 T , E‘/2) + 0 ( ( & ‘ ) 2 ) , x(h,
E’)
= x o ( h ) + x , ( h ) ~ ’ + X ( TE, ’ ) + Q ( ( E ’ ) ~ ) ,
and so by subtracting,
(6)
b ( h , E’/2)-X(h,
E’)
= xo(h)
+ o(hP+’) + O((E’)~).
Here the term O(hP+’) comes from estimating the boundary layer term, X in (4) by utilizing ( 5 ) . On the other hand, by utilizing (3), (4) and (S), we find X(h, E )
=
Xo(h)
+ o(E)
O((E’l2),
and similarly, for y ( h , E ) . Here O ( E ) is an estimate for the outer solution while O ( ( E ’ ) ~is,) as in (6), an estimate for the boundary layer. The final result, which constitutes the extrapolation method is that = 2x(h, € ’ / 2 ) - X ( h ,
E’)
+ O(hP+’)+ O ( E ) + o( (+”).
y ( h , E ) = 2 y ( h , E’/2)-Y(h.
E’)
+ O(hP+’) + O(E) + o( + 2 ) .
x(h,
E)
(71
We make the following observations concerning this extrapolatory approach. The expressions h ( h , ~ ’ / 2 ) - x ( h E, ’ ) and 2 y ( h , ~ ’ / 2 ) - y ( h E, ’ ) in (7) which are used as approximate replacements for x(h, E ) and y(h, E ) , respectively, are in fact also approximate replacements for x,(h) and y o ( h ) , respectively. Thus while calculating x(h, E ’ ) and y ( h , E ’ ) from (1) is much easier than calculating x(h, E ) and y ( h , E ) , since E’ >> E , why do we not just calculate xo(h) and y o ( h ) from (5.1.40)? We give several reasons. Remark:
a) The exploitation of the methods of singular perturbation theory for the development of numerical techniques usually proceeds with the numerical determination of values of one or more terms in the asymptotic expansion supplied by that theory [3]. The extrapolation formulas (5.3.6) break through this limitation of approach.
76
F .C.
HOPPENSTEADT and W . L . MIRANKER
b ) Equation ( 1 ) with E = E' is not stiff and may be easily and reliably solved by simple explicit numerical methods. While (2) is also not stiff, employing it for the determination of x o ( h ) and y o ( h ) requires the solution of the nonlinear system g = 0 at each mesh point. This is usually a costly computation. c) Unlike the extrapolatory method, the conventional approach requires that the initial value problem be placed in the form ( I ) , and this may require considerable work.
I .2 COMPUTATIONAL EXPERIMENTS The following two computational experiments compare the extrapolation method (7) and the asymptotic expression itself.
i)
A linear sysrern
We consider first the following linear example
d x- - y - x , dt
x(0) = 5,
d y = - Y + l , y(O)=TJ. dt
E
The exact solution is given by the formulas x ( r ) =e-'[
y(r)= &
+ ( l - e - ' ) E - ( ~1)- (& T J - & ) ( e - " F - e - ' ) ,
+ e-"yTJ-E),
and the leading terms of the matched asymptotic expansion solutions are X(t)=e-'5+€[(1)-1)e-'+ y(r)= E
I ] + ... ,
+ ... .
For any value of the leading terms of the matched asymptotic expansion ( x o ( h ) , y o ( h ) )and the exact solution ( x ( h , E ) , y ( h , E ) ) agree to about four figures. Thus while in Table 1-1, E = lo-' is employed in the column labeled E ' / E , the results in that table are otherwise valid for any ~ < 1 0 - ' . The results using the extrapolation method and evaluation of the matched asymptotic expansion ( t o leading order) are presented in Table 1-1. Since K = 1 + I T J I and 8 = 1 in this case, we take T = - e n [ h P + ' / ( l + 17 I ) ] . In spite of the involved form of this formula for T , the values of the latter should be taken only approximately. The calculated values of E' = h / T are E' = .0082 and E' = ,00304corresponding to h = .1 and .01, respectively. Since these values of E' a r e to be taken only as approximate, calculations for nearby values of E ' , are also presented in the table. Note that p = 4 in the computation displayed in Table 1-1 as well as in Table 1-2 to follow. The relaxed equations are in fact solved on the interval (0,h ) by means of a fourth order Runge-Kutta method for which we employ a submesh with submesh increment k = he'. This choice of submesh increment assures high accuracy and stability of the Runge-Kutta computation.
COMPUTATION BY EXTRAPOLATION
.$=q=l
p=4 h=.l
EXTRAPOLATION M E T H O D
MATCHED SOLUTION
x(h) ,8632 .R648 ,8657 ,8707
-
,9049
E’
h = .01
MATCHED SOLUTION
E’
.005 ,0082 .0 1 .02
EXTRAPOLATION M E T H O D
77
E’
,003 ,00304 .004
x(h) ,9866 .9866 ,9871
-
,9901
E’
= ,0082
y(h) ,0025 ,004 ,005 .o 10
E’IE
500 819 1000 2000
0.0 = ,00304
y(h) ,00155 ,00157 ,00255
E‘/E
3 00 304 400
0.0
Table 1 - 1 Notice that the extrapolation method gives a 4 % answer for h = . I , but it gives better than a 1% answer h = .01.
We compare this extrapolation method with the frequently used package of C. W. Gear for for the second integrating stiff differential equations [4]. We use this package with E’ = case in Table 1 - 1 with the following result. The package reaches h = 0.01 by employing a variable submesh of points which it determines adaptively. To produce a 1Yo answer, the package requires an average submesh size of 1 . 9 6 ~ The extrapolation method with E‘ = .00304 produces its 1% answer with a fixed stepaize of 9 . 7 5 ~ nearly an order of magnitude difference. Of course as E decreases the latter remains invariant, but the average stepsize employed by the package will decrease even further. ii) A
model enzyme reaction
A simple enzyme reaction involves an enzyme E , substrate S , complex C and product P. Schematically, the reaction is
E
+ S+C,
C e E
+ P.
After some preliminary scaling, this reaction can be described by a system of differential equations for the substrate concentration (x) and the complex concentration b) as
dt
= --x
+ ( x + k)y,
x(0) = 1,
78
F.C. HOPPENSTEADT and W.L. MIRANKER
where E measures a typical ratio of enzyme to substrate (O(IO-')), and k and k ' ( k denote ratios of rate constants suitably normalized (O(1)).
< k')
(although as The Table 1-2 summarizes the result of these numerical calculations for E = h = .I noted in (i) above, except for the column E ' / E , the results are valid for any and 0.1, k = 1 , k' = 2. In this case, K = 1 , 6 = k', so we take T = -(&)en h. The calculated values for E' = h/T are E' = .04 and E' = .0009, respectively. Calcu?ations are also presented for some nearby values of E ' . x(0) = 1 y(0) = 0 p = 4 h = .I
EXTRAPOLATION M E T H O D
= .04
.05 .I .I5 .2
x(h) .9530 .9596 .9617 ,9726 .9882 ,9937
y(h) ,3229 .3247 ,3253 ,3285 .3350 .3406
-
.9888
.3308
E'
.01 .04
MATCHED SOLUTION
E'
h = .01 E
I
EXTRAPOLATION M E T H O D ,0004 .0008 .0009 .001 ,0016 MATCHED SOLUTION -
E'
E'/E
1000 4000 5000 10000 15000 20000
= .0009
x(h) ,995 1 .9952 .9952 ,9952 ,9954
y(h) ,3322 .3323 ,3323 .3323 .3323
.99 17
.3315
E'/E
40 80 90 100 160
Table 1-2 The extrapolation method gives a 3% answer for h = . l , but it gives better than a 1% answer for h = .01. As in the case of Table 1-1, a comparison here produces an average step size of 2 . 4 ~ for a 1 % answer for Gear's package as opposed to a fixed stepsize of 2.7 x for a 1 % answer for the extrapolation method with E' = .0009.
2. AVERAGING PROCEDURES Problems to which the Bogoliubov averaging method and various multitime schemes are applied frequently reduce to problems of the following form.
where x, f, SeR" and where f(7, e ) is an almost periodic function of
7.
Multi-time perturbation
COMPUTATION BY EXTRAPOLATION
where x o is determined from the initial value problem dx,
(10) Here
7 is the average o f f ,
-
= f ( x 0 ) . xo(0) =
6,
defined by
The coefficient x1 is determined from the formula (11)
u
~ ~ ( I/&) 1 , = xl(r)
+
i"'[fC~.
xo)-f(xo)]dr.
(For details see [ S ] . ) Thus,
This approximation suggests several numerical schemes for determining x ( h , E ) . In Section 2.1, we consider the computation of the average f; first by the customary method and then by a second difference method which accelerates the computation of 7 in some cases. Then in Section 2.2, we describe an extrapolation method for approximating x ( h , E ) . As in the extrapolation method introduced i n Section 1 , certain larger values E' of E are introduced, and (2.1) is used with this larger value of E' to furnish approximations to x ( h , E ) itself. In Section 2.3, the results of computational experiments which compare the methods are presented. Finally a discussion of these various computational procedures is given .in Section 2.4. 2.1 ACCELERATED COMPUTATION OF 7 We propose two methods for calculating 7. i) Direct evaluation of lim
T-m
SO fdr.
-!-
T
A convergence criterion is first set, and then the integral L T f d r is calculated for increasing values of T until the criterion is met: Given a tolerance 6,txere is a value T ( 6 , x) such that
and
79
80
F .C.
HOPPENSTEADT and W . L . MIRANKER
for all T , , T , 2 T ( 8 , x). Thus, we can write
and proceed t o solve (10). Unfortunately, there is n o certain way of finding T ( 8 , x).
In order to find a candidate for T ( 6 , x), we calculate Y ( T , x) =
L T f ( T , X)dT
for 0 5 T 5 2 P , and keep increasing P until the condition sup
0<7‘
is met.
1 Y ( T * ,x)-I1 7
T * + T‘
Y ( T * + T’, x) 11 5 6
Then we take T ( 6 , x) = 2 T * . Usually 2P is of order 1/6. T , then f(x) = 0 and Y ( T , x) = cos T- 1. Thus
For example, if
f ( ~x), = sin
0 5 II
Y ( T , x ) I1 -5 2 / T ,
the maximum being attained in each interval of length 2n. ii) Second difference method
In most applications, the integral of the almost periodic function f has the form
U T ,x)
=f(x)T
+ p ( T , x).
where p is an almost periodic function of its first argument and with mean zero. Thus, given a tolerance 6, there is a 6-translation number B(8,x) such that IIdT
+ B(6,x), x ) - p ( T ,
x) II
<6
for all T 2 0; in particular, since p ( 0 , x) = 0, then l p ( 8 ( 6 , x), x) I < 6. T h e determination of such translation numbers is difficult and we consider a method of differencing for finding candidate values of 8. Note that
Y ( 2 T , x ) - 2 Y ( T , x) = p ( 2 T . x)-22p(T, x).
In particular for T = 8 ( 6 , x), (12)
Y ( 2 8 , x) -2 Y ( Y , x) = p ( 2 3 ’ , x ) - p ( 8 , x) - p W , x) + p ( 0 , x) = O(6).
Thus, any &translation number of p makes (12) of order 6. Unfortunately, II Y ( 2 T . x ) - 2 Y ( T , x) II may b e small while Ilp(T, x) 11 is not small. Still, by tabulating Y ( 2 T , x ) - 2 Y ( T , x), candidates f o r 8 ( 6 , x) can be found and tested by comparing the values of Y ( T , x ) / T for several of them, since these should all approximate f ( x ) . In practice, this method is no worse
81
COMPUTATION BY EXTRAPOLATION
than the direct calculation in (i), and in periodic cases, it reliably gives r ( x ) after calculation over one period.
Jr(s’x)(~, 7
Thus from either (i) or (ii), we use the approximation 1 B(8,x) proceed to integrate (8) using this approximation.
X ) ~ Tfor
and
2.2 THE EXTRAPOLATION METHOD , Then, in particular, We pick T to be a 8-translation number of p ( ~x).
for x = 5 + O ( h ) . The existence of such a value of T follows from viewing ( 1 3 ) as the statement that T is an approximation to a 8-translation number (compare (12) with O(8) = hp). Such a T value must be found, perhaps using one of the methods in Section 2.1 or additional knowledge about a specific problem being studied. Once a T value is found which satisfies (13). we define E’ = h / T as in Section 1 , and then we calculate x ( h , ~ ’ / 2 )and x ( h , E ’ ) from (8) by a pth-order numerical method. It follows from formulas (9) and ( 1 1 ) that
We refer here to the Remark in Section 1 to emphasize that the approach represented by the extrapolation method is important both computationally and theoretically. Note that if T differs by the quantity A from the approximation B ( 8 , x) of the 8-translation number, then from (14) we see that the corresponding difference in the associated values of
F.C. HOPPENSTEADT and W.L. MIRANKER
82
x(h,
E)
is O( h 2 / T 3 ) A .
The dependence of this estimate on h is illustrated in Table 2-1.
2.3 COMPUTATIONAL EXPERIMENTS: A LINEAR SYSTEM The linear initial value problem
_ du - ( ( I / E ) A + B)u, dt
~ ( 0=) 4
is taken into the problem
dv dt by the change of variables u =
=
e-Al/eBeAl/ry,
v(0) =
d‘/‘v. We take veR4 and 5
5
= (1, 1, 1, 1)
T
Running values of Y(T, 6) are computed using a quadrature increment of AT = .01. The tolerances for each of the methods (i) and (ii) are denoted by Si and ail, respectively, and the corresponding values of T at which the associated computations halt are denoted T i and T l i , respectively. The calculations are carried out for two different matrices:
where w is a parameter specified below. B is taken to be
can be determined in closed form, which we denote by B(1, w) and B(2, w ) , The average respectively. These are compared with B’s obtained by methods (i) and (ii); e.g., B(1, w) is compared with B i ( l , w) and B i j ( l , w) in Table 2-1. The averaging method gives e B h[ as an approximation to x ( h , E ) . Table 2-1 compares these approximations for the three ways of determining B. The extrapolation formula (14) gives an approximation to x ( h , E ) . Results of two utilizations of this formula are presented in Table 2-1 respectively, where E ’ ~= h / T i and as well. These are denoted by E x ( E ’ ; )and E ’ = ~ ~h / T j i . Note that no value of E is prescribed for these computations. This demonstrates the effectiveness of singular perturbation methods in supplying approximations which are , The latter in turn depends uniformly valid for all E smaller than some prescribed value E ~ say. only on the accuracy desired of the leading term of the expansion as an approximation to the full solution (for the particular differential equation at hand).
COMPUTATION BY EXTRAPOLATION
a. Approximations to
83
%
Approximating vector
Approximation eiht
1.192
1.614
1.614
1.192
eztht
1.178
1.604
1.587
1.222
eittht
1.198
1.603
1.603
1.198
Ex( .0006)
1.199
1.607
1.638
1.168
Ex(.002)
1.193
1.617
1.617
1.192
c. Approximations to x ( h ,
E),
h = .05 Approximating vector
Approximation eBh<
1.099
1.277
1.277
1.099
eijtht
1.091
1.276
1.269
1.110
eisjht
1.100
1.276
1.276
1.100
Ex( ,0003)
1.102
1.273
1.287
1.088
Ex( .oo1)
1.102
1.213
1.287
1.088
Table 2-1 Case 1. A = A (1, %)
a4
F.C.
a. Approximations to jj
HOPPENSTEADT and W . L . MIRANKER
r, = 404 28
6i = .01
80, YJ -9.167
.ooo Bji(2,
Bi(2, 14)
9.167 '28.167 -290.556 -60.4R4 28.167
6,, =
-52.069 289.444
9.104 -2R.02R -289.P46 28.092
0.000
0.000
-2.167
2'9.833
,002
0.000
0.000 -29.833
2.167
*.Of3
b. Approximations to Approximation
9.026
,028 .039
-60.117
52.809 -288.484
-2.154 -29.711
9.167
-60.443
52.888 -289.437
29.7b8
0.000
0.000
-2.166
29.388
-1.97G
0.000
0.000
-29.833
'2.167
€1, h = .1
3ht
-55.4
-26.5
-1.31
-5.98
egiht
-55.2
-26.6
-1.28
-5.97
e&&
-55.4
-26.5
-1.31
-5.98
Ex(.002)
1.09
-13.9
-.701
Ex (-001
1.546
-13.6
-.761
-.873 -1.01
Approximating vector
Phhl
-22.4
-10.9
1.12
-1.55
ehhl
-22.3
-10.9
1.12
1.54
3ithE
-22.4
-10.9
1.12
-1.55
Ex(.oo1)
-14.2
-9.82
1.12
-1.54
Ex(.ooo~)
-14.3
-9.77
Approximation
3
9.165 -28.166 -290.599 28.166
Approximating vector
Approximation
r,, = 7 5 1
.965
-.814
Approximating vector
Jh(
-9.46
-4.46
1.38
-.0363
Exx(.00025)
-8.46
-4.35
1.34
.0557
Table 2-1 Case 2. A = A ( 2 , %)
COMPUTATION BY EXTRAPOLATION
85
2.4 DISCUSSION Existing stiff differential equation solving routines, such as Gear’s can degrade markedly when applied to problems having highly oscillatory solutions since computations must continually be made using very small increments. On the other hand, methods like those presented here can require extensive a prrori preparation of the system to be solved. Numerical implementation of the averaging procedure requires the determination of T I (Section 2.1) which is used in direct approximation of 7 or determination of T I , (Section 2.l.ii) as an approximate translation number for the integral V ( T , x). The ratio h / E is a measure of the stiffness, while 1/(Eh) measures the work involved in direct computation of solutions. On the other hand, T , / A T ( A T is the increment used in the averaging quadrature) measures the work needed to calculate 7, and T , , / A T is a measure of the work in calculating the approximate translation number. While the method based on approximation of 7 in Section 2.li is reliable, it is costly even for a periodic function f. The work in the computation of translation numbers varies from minimal (e.g.. for a periodic function) to an amount which offers no improvement when resonances occur in the integral Y. Finally, note that formula (14) shows the error arising in the extrapolation procedure decreases as a power of h; e.g., replacing h by h/2 implies the error changes from h 2 / E to h 2 / ( 4 ~ ) . Consequences of this fact are illustrated in Table 2 where Computations are carried out for the two or three values of h. Using 11. 1I to denote the norm in R 4 , then from Case 2 of that table we find: -
IIe.l8(
- ~ x ( . 0 0 1 )11 = 59.5,
-
11e.05B( - E x ( . 0 0 0 5 ) jl = 8.1,
IIe.025B(-Ex(.00025) 11 = 1.1. REFERENCES Hoppensteadt, F.C. and Miranker, W.L., An extrapolation method for stiff differential equations, RC 7697, IBM Research Center, Yorktown Heights, NY (1979). Hoppensteadt, F.C., Properties of solutions of ordinary differential equations with small parameters, Comm. Pure Appl. Math., XXlV (1971) 807-840. Miranker, W.L., Numerical methods of boundary layer type for stiff systems of differential equations, Computing, 11, (1973) 221-234. Hindmarsh, A.C., Gear’s ordinary differential equation solver, UCID-30001 (Rev. 3). Lawrence Livermore Lab., Livermore, CA 945500 (December 1974). Persek, S.C. and Hoppensteadt, F.C., Iterated averaging methods for systems of ordinary differential equations with a small parameter, Comm. Pure Appl. Math., XXXI (1978) 133-156.
This Page Intentionally Left Blank
A N A L Y T I C A L AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S . A x e l s s o n , L . S . F r a n k , A . vdn d e r S l u i s ( e r l s . ) @ N o r t h - H o l l a n d P u b l i s h i n g C o m p a n y , 1981
PROPER APPROXIMATION OF A NORMED SPACE AND SINGULAR PERTURBATIONS Denise Huet U.E.R. de Mathematiques U n i v e r s i t e de Nancy I C.O. 140, 54037 Nancy C@dex Equipe de recherche associee au C.N.R.S.
no 839
T h i s paper d e a l s w i t h l i n e a r e l l i p t i c s i n g u l a r l y p e r t u r b e d d(E,nE, , S) o f problems. I b u i l d a p r o p e r a p p r o x i m a t i o n a Banach space E , i n t h e sense o f F. Stummel. I show t h a t t h e usual known convergence theorems and e r r o r e s t i m a t e s , i n L2 o r Lp Sobolev spaces, a r e s p e c i a l cases o f general p r o p e r convergence theorems.
1. INTRODUCTION Throughout t h i s paper, E denotes a s t r i c t l y p o s i t i v e parameter and a l l l i m i t s a r e taken as E + 0 I f E i s a H i l b e r t space [ r e s p . a normed space1 we denote by , t h e s c a l a r p r o d u c t [ r e s p . by II . l l E , t h e norm1 i n E I f E and F
.
a r e normed spaces,
of
E
into
.
F
(if
1< p < t
u E L (R) such t h a t P
i s the multi-index
q
alq\
Dqu =
91 ax292
axl subspace If
p = 2
denotes t h e space o f a l l continuous l i n e a r o p e r a t o r s
R be a smooth open subset o f lRn , l e t r be a non n e g a t i -
Let
ve i n t e g e r and l e t functions norm
.
X(E,F)
...
qn axn
set
We denote by Dqu E
q2,
) ; ;(a)
ey(Q)o f t e s t
, we
(ql,
.
H ~ ( Q =)
L (n) P
..., qn),
we s e t
141 = q1 +...
denotes t h e c l o s u r e , i n
Wr(R)
P
f u n c t i o n s w i t h compact s u p p o r t i n
wl(n) .
L e t m and m ' be two i n t e g e r s w i t h B , the d i f f e r e n t i a l operators : 1 Ipl,Iql 6 m
m > m'
>0
R
. We
+
9,
, of
et the
( c f . [81). denote by
C
and
( - l ) I p l DP(cpq Dq)
II (-1) Ipl,Iql 6 m' where t h e c o e f f i c i e n t s
Wr(R) t h e Sobolev space o f t h e P I q l 4 r , equipped w i t h t h e
for
Dp( bpq Dq)
c and b a r e smooth enough i n P9 Pq t h e s e s q u i l i n e a r forms a s s o c i a t e d w i t h C and B :
R
, and by c and b
D. HUET
88
The f o r m c [ r e s p . We s e t B, = CC t B
b l i s well defined i f and b, = E C t b
.
u,v E Hm(Q)
[resp.
.
u,v E H " ' ( n ) I
I now g i v e t h r e e s i m p l e examples o f e l l i p t i c s i n g u l a r l y p e r t u r b e d p r o b l e m s . F o r t h e sake o f s i m p l i c i t y , I r e s t r i c t m y s e l f t o t h e D i r i c h l e t b o u n d a r y c o n d i t i o n s . Example 1.1. L e t C and B be t h e o p e r a t o r s d e f i n e d by (1.1). We suppose t h a t B, and B a r e c o e r c i v e i n t h e f o l l o w i n g sense : t h e r e e x i s t p o s i t i v e c o n s t a n t s a, B , y such t h a t I b E ( u , u ) I 3 a, I l u II
(1.3)
H'"(Q)
f o r any
I b ( u , u ) I 3 Y I I u II 2 Hrn'(Q) L e t h be g i v e n i n t h e d u a l space problems :
H-"'(Q)
uE E i m ( Q )
,
BEuE = h
(1.5)
u E l?'(Q)
,
Bu = h
.
that
u,
I t i s known ( s e e [23 and [61)
h E L2(Q) 1
IIU,
-
u II
Hm' (Q)
=
0
(E
0
u E H"'
.
of
l?(Q)
in
Hm'(R)
1
(n)
@(Q)
u E
.
We c o n s i d e r t h e D i r i c h l e t
.
(1.4)
Moreover, i f , f o r i n s t a n c e ,
f o r any
+BIlUll2 Hm' ( Q )
+
,
u
m' P 0
, we
,E
~ "
u,
+
0
in
Hm(n)
have t h e e r r o r e s t i m a t e
&) *
Example 1.2. k!e c o n s i d e r t h e same p r o b l e m as i n example 1.1, b u t we suppose t h a t t h e o p e r a t o r s C and B a r e G i r d i n g e l l i p t i c i . e . t h e r e e x i s t c o n s t a n t s B1 , B2 , a l > 0 , a2 > 0 such t h a t
2
Re b(u,u) t B2 I I u II L2(n)
5 a2 IIU II
Hrn'(Q)
f o r any
0
u E H~
1
(n)
.
F. Stummel i n [lo1 o b t a i n e d t h e same r e s u l t s as i n example 1.1 ( s e e a l s o [51)
.
.
We suppose t h a t t h e o p e r a t o r C ( n o t n e c e s s a r i l y o f Example 1.3. L e t 1 c p c m t h e f o r m ( 1 . 1 ) ) i s e l l i p t i c i n t h e u s u a l sense, o f o r d e r 2m , and t h a t i t s p r i n c i p a l symbol c ( x , c ) s a t i s f i e s t h e Agmon c o n d i t i o n
.
89
A NORMED SPACE AND S I N G U L A R PERTURBATIONS
5 # 0 and any x
f o r any r e a l v e c t o r
u,
and l e t
E
(see [ l l and [ l l l ) . L e t
fl
h E L
be t h e s o l u t i o n o f t h e O i r i c h l e t problem
P
(n)
(1.8) I t i s known ( [ 3 1 , [41) t h a t
U,
+
h
in
L (0) and P
u,
E
--t
0
in
WZm(K?) P
.
The r e s u l t s o b t a i n e d i n t h e above t h r e e examples a r e i d e n t i c a l . B u t t h e p r o o f s , g i v e n i n t h e r e f e r e n c e s , a r e q u i t e d i f f e r e n t . The purpose o f t h e f o l l o w i n g s e c t i o n s i s a c o n s t r u c t i o n o f a s u i t a b l e a p p r o x i m a t i o n o f a Banach space and t h e statement o f p r o p e r convergence theorems which c o n t a i n t h e above r e s u l t s as s p e c i a l cases. 2. PROPER APPROXIMATION OF A NORMEO SPACE Let
nE,
be a f a m i l y o f normed spaces. We
t h e C a r t e s i a n p r o d u c t o f t h e spaces
nEE ( i . e . uE E E E f o r a l l
of
(EE), >
be a normed space and l e t
E
denote by
D e f i n i t i o n 2.1. L e t
and
(u,)
-
l e n t i f and o n l y i f l l u E
(u,)
an element
> 0).
E
(v,)
vE II
and by
E,
--t
0
EE
.
nEE
be i n
.
They a r e s a i d t o be e q u i v a -
D e f i n i t i o n 2.2. L e t R : u !-+ R(u) be a l i n e a r map o f E i n t o t h e s e t o f t h e We say t h a t e q u i v a l e n c e c l a s s e s , such t h a t R(u) # 0 f o r a l l u E E &(E, nEE , R) i s a p r o p e r a p p r o x i m a t i o n o f E i f and o n l y i f we have
.
l l U & II
--t
IIU IIE
EE
for all
u E E
and f o r a l l
(REu)
We denote by D e f i n i t i o n 2.3.
Let
elements
,
(E,
+
u E E
E ; R)
(2.1)
{
Remark 2.1.
E R(u)
(u,)
.
an element o f t h e e q u i v a l e n c e . c l a s s
nE, , R)
d(E, uE E
E ,
E
.
R(u)
be a p r o p e r a p p r o x i m a t i o n o f
> 0 , t h e p r o p e r convergence
u,
.
E
For
L, u
i s d e f i n e d by 4
Ut
i.e.
uE
u (EE u,
+
+
E ; R)
E R(u)
u (E,
+
lluE
i f and o n l y i f
E
E ; R)
l l u E IIE + I I u l l E
implies
be a normed space. F o r
I I - [ I E = 1 1 . I I E , and f o r
REu II
6
E
> 0
, we
+
0
EE
.
F o r more d e t a i l s on t h e above d e f i n i t i o n s see [91. Example 2.1. L e t
-
.
set
E,
-
u IIE
=
E ,
u E E
&
T ( u ) = {(u,)
(2.2) Then
&(E,
nEc , T )
E nEE ; IIu,
-
u II
= EE
i s a proper approximation o f
llu,
+
0
1 .
E called the t r i v i a l
-
90
approximation, and usual sense.
uc
D. HUET
u
(EE
-t
, T)
E
i f and o n l y i f
uE
u
-+
in
i n the
E
Example 2.2. (see [ l o ] and 151). L e t D and E be two H i l b e r t spaces such t h a t D cC We suppose t h a t D i s dense i n E and t h a t t h e i d e n t i t y mapping o f D into i s continuous. We s e t E c = D , f o r E > 0 , and
i
= E ( u , v ) ~t ( u , v ) ~
(u,v)E
(2.3)
for
E D
U,V
&
.
equipped w i t h t h e s c a l a r p r o d u c t (2.3) i s a H i l b e r t space. We now EE d e f i n e l i n e a r o p e r a t o r s RE E %(E, EE) by
Then
(v,U)E = ( v
(2.4 R
Let
u E E
be d e f i n e d , f o r
u E E
for all
I
,
.
v E D
and a l l
by
(2.5 We have
d(E, l l E E , R)
Theorem 2.1.
1)
2) I f cp E D
, one
3)
uE
4u
(EE
E llEE
, R)
E
+
Let
R&U
u E E
-t
and l e t lluE
-
u
, lemme
.
(u,)
@PI E&
uE
+
01
u
in
E
in
D
-,
. and
uE -, 0
cl/'
in
D.
1.1, t h a t
E ,
in
-
; IIu&
i f and o n l y i f
P r o o f 1 ) I t was shown i n [51 (2.7)
E
has
R(cp) = {(u,)
(2.6)
i s a proper approximation o f
R & U -,
o
.
u E E
f o r any
E R(u) ; (2.3) and ( 2 . 5 ) g i v e :
REu llE
+
0
and
c1j2
JluE
-
REu llD
+
.
0
Therefore l l u e 11'
E&
=
1 1 ~ ~( u' C ~- REu + REu) 1l0 2 + llu,
2) I t f o l l o w s from (2.3) and (2.7) which proves ( 2 . 6 ) .
t h a t , f o r any
(2.8)
Ilu
-
wIlE
E
s6
nEc ;
cp
IIE 2
E D , IIcp
-
+
IIu llE 2
REcpllE
+
0
.
I t f o l l o w s from Stummel's r e s u l t s (see
R(u) = {(u,)
R&u + R,u
&
3) i s an obvious consequence o f ( 2 . 3 ) and ( 2 . 7 ) Remark 2.2. uE:t
-
V 6 > 0
and I l u E
-
,
[91, ( 4 . 1 . 6 ) )
3 cp E D
REcpII d 6 E&
c0 > 0
and
for
t h a t , f o r any
E
6 co1
such t h a t
.
91
A NORMED SPACE AND S I N G U L A R PERTURBATIONS
Exam l e 2 . 3 . I now want t o e x t e n d t o Banach spaces t h e c o n s t r u c t i o n o f example h and E be two Banach spaces such t h a t D c E . We suppose t h a t D i s dense i n E and t h a t t h e i d e n t i t y mapping o f D i n t o E i s c o n t i n u o u s . The f a m i l y o f Banach spaces
EE = D
(2.9)
i s d e f i n e d by
(EE)
IIu l l E
where
. We
a > 0
DE
denote by
llE
1111
t h e space
(2.10)
, TIEc , s )
&(DE
Then,
seep
we can t a k e Lemma 2.1.
vIlE
+
,
01
&
.
D
E
U,
i s a p r o p e r a p p r o x i m a t i o n o f t h e normed space f o r any
E
,
s)
(EE
cp
'pE+
.
E
= cp
-
; lIwE
~ ( c p )= { ( W E ) €FEE
by
s(@)
H
+
DE
0
z
DE
.
i f and o n l y i f
cacpE
D
in
0
-+
.
1 1 . IIE
D equipped w i t h t h e norm
I d e f i n e t h e map cp
L o o k i n g a t p a r t 2) o f theorem 2.1,
in
IIu I I D
= ca 6
and
and
.+ 'p
p ',
T h i s lemma f o l l o w s f r o m (2.9) and (2.10). u E E
We now d e f i n e f o r
i
(2.11)
S ( u ) = {(u,)
E
Ilu - v l l E
6
,3
v 6 > 0
;
, and lluE
'p
E D
and
- scvllE 4 6
co > 0
for
such t h a t
E Q to)
E
Then, i t i s known, s i n c e
E
a p p r o x i m a t i o n of 4.1.9). Remark 2.3.
nEE
i s dense i n
which s a t i s f i e s
D and
If
0
E
, that
E
S('p)
nEc , S )
&(E,
f o r any
= s(q)
a r e H i l b e r t spaces, and
'p
i s a proper
.
E D
21 , t h e
a =
. ,
(See [91
R
mappings
and
S d e f i n e d by ( 2 . 5 ) and (2.11) r e s p e c t i v e l y s a t i s f y R(u) = S ( u ) f o r any u € E Therefore, i n t h i s case, d ( E , n E E , R) and d ( E , nEE , S ) a r e i d e n t i c a l . d ( E , nEE , S )
Theorem 2.2. L e t
(2.9), in
D
Proof
exists any
E
(2.10),
(2.11). Then
and
-t
uE
u
in
1) Suppose t h a t cp E 0
4 c
0
and
. Hence
HUE
- u
E
.
uE*
111.1
IIu II
E
=
a'
uE4u
u
IIE 6 lluc
- u IIE
+
0
(EE
(EE
+
E
, S)
26
.
-
-,E , S) . l e t - QIIQ
SC'pllE
t
IISE"
&
t
E
i f and o n l y i f
6 > 0
-
.
d e f i n e d by a E u E -, 0
By ( 2 . 1 1 ) , t h e r e 6
and l l u E - sE'pII
such t h a t IIu
c0 > 0
6
Therefore
be t h e p r o p e r a p p r o x i m a t i o n o f
.
6
5
for
+
0
i n 0.
EE
+ II'p - u l l E
"HE &
IIsEcp - v l l E
for
E Q
c0
E
.
Now, i t f o l l o w s f r o m remark 2.1 t h a t
l l u c IID t l l u c l l E
-t
IIu l l E
.
Hence
cauE -, 0
in
u
and
D
.
FE
2) L e t
u E E , (u,)
E nEE . We suppose t h a t
uE
-t
in
E
cauC
D. HUET
92
We want t o show t h a t (u,) 6 IIcp UII 6 4 . Therefore
lIuc if
-
'PllE
6 c
a
Example 2.4. L e t We denote by G,, family
a
(u,)
-
lI(P11D + lluc
E S(u)
u l l E + IIu
IIu l l H
.
f o r any --f
E
IIu llH
> 0
.
u E G
f o r any
E G we d e f i n e E nHE ; 1P'l-;
q(Q) = {(cp,)
QllH
+
.
01
&
, nHE , q)
d(GH
q(cp)
(cpp,) E
i s a proper approximation o f
Q(u)
(2.14)
HE = E
E
.
cp E G
nHc
-
11
q,cp
Q
6
;
v
,3
6 > 0
f o r any
cp
E Q E
&
, Q)
and
E G
> 0
such t h a t
0
.
1
i s a proper approximation o f
E~
H
.
Example 2.3 i s a p a r t i c u l a r case o f example 2.4 w i t h
Remark 2.4.
,
since, f o r
u E H , Q ( u ) by
= {(u&) E
llut
d ( H , r[H&
GH
:
We now d e f i n e , f o r any
Then
.
&
(2.13)
for
'PIIE d 6
.
{
Then
-
such t h a t
H be a norrned space and l e t G be a dense subspace o f H t h e space G equipped w i t h t h e norm o f H We c o n s i d e r a
HE = G
(2.12)
cp
E
cp E D
exists
o f normed spaces such t h a t :
(HE)
F o r any
110 +
llUE
. There
6 > 0
6
i s small enough. Hence
c
. Let
E S(u)
-
Indeed IIu l l E = c a l l u l l D + I I u I I E
+
IIu l l E f o r any
H = E
u E D
E
.
,G
= D
,
The purpose o f t h e n e x t s e c t i o n i s t h e statement o f t h e main theorems o f t h i s paper. 3. PROPER APPROXIMATION THEOREMS
Theorem 3.1.
Let
F
, F, , E
proper a p p r o x i m a t i o n o f example 2.4. i)
A
ii)
AE
(3.1)
F AE E %(HE
Let
i s surjective, and
A
Ac
.
, be normed spaces and l e t d(F,TTFE , P) G , H , HE , d ( H , r [ H E , Q) be t h e same
> 0
Let
, FE) , A
E .%(H, F )
i s bijective,
t
and suppose t h a t
> 0 ;
s a t i s f y the consistency condition :
AEq
L+
Acp
(FE
+
F
, P)
f o r any
cp E
G
,
be a as i n
A NORMED SPACE AND SINGULAR PERTURBATIONS
At
i i i ) the operators
K > 0
a constant
A
Then,
uc-
4 l l A t u IIF
Ht
1) L e t
u E G
+ l l A u IIF
ACuE-
.
, Q)
l l A E u II
there exists
.
f o r any
u E G
Au ( F t
F
and a n y
E
.
> 0
E
i s b i j e c t i v e , and
u ( H E -, H
___ Proof
satisfy the inverse s t a b i l i t y condition i.e.
such t h a t
K IIu II
(3.2)
93
.
From ( 3 . 1 ) ,
,
implies
P)
.
Au ( F t -, F , P )
Q
, I I u llH
From ( 2 . 1 2 )
FE Therefore, t a k i n g t h e l i m i t
Atu
-,
+
.
IIu IIH
Thus
E
o f (3.2) y i e l d s
K I I u IIH < IlAu l l F f o r any
.
u E G
Since
u E H
holds f o r any
.
6 > 0
. We
IIIAIII = IIAIIx(H,F))
.
Au ( F E -, F , P )
There e i x s t s
A E %(H,
and
H
, this
We w a n t t o show t h a t
-
IIcp
such t h a t
E G
cp
u llH 4
have
< llU& -
l l u c - qtcpllH
F)
inequality
i s injective.
A
ACut+
2) We suppose t h a t Let
i s dense i n
G
and
&
Il(p
It f o l l o w s f r o m ( 2 . 1 3 ) t h a t
-
Q I I H + Ilcp t
-
qE(pllH
311IAIIl
.
.
q,cpII
.
0
+
ut E Q ( u ) (where
6K
Ht From ( 3 . 2 ) and ( 3 . 1 )
&
-
llue
Thus,
qtcpllH 6 6
C o r o l l a r y 3.1.
E
Let
if
, be Banach spaces and l e t d ( F , n F t , P ) b e D , E , Ec , & ( E , nEE , S ) be t h e same as EX(EE, F t ) , A E X(E, F ) and suppose t h a t
F , FE ,
a proper approximation o f i n example 2.3. L e t i) ii)
At
f o r any
and cp
A
E D
a copstant
A
t
.
> 0
Let
i s bijective,
t >
0
.
s a t i s f y the consistency condition i.e.
AEcp
r-)
(FE
Acp
-,
F
.
iii)The o p e r a t o r s Then
At
F
i s s u r j e c t i v e , At
A
.
uc E Q ( u )
i s s m a l l enough and
c
satisfy the inverse s t a b i l i t y condition i.e. there exists
At
K > 0
, P)
K IIu I I E S IIA,u
such t h a t
i s b i j e c t i v e and
Acue
E
r-t
Au ( F E
+
I1 F
F&
f o r any
, P)
implies
u E uE-
D
.
u(EE
+
E
, S)
.
T h i s c o r o l l a r y f o l l o w s f r o m t h e o r e m 3 . 1 and r e m a r k 2.4. Theorem 3.2. L e t
F , FE
proper approximation o f example 2.3.
Let
operators o f
EE
of
t
,
E
> 0
b e Banach spaces and l e t
F
.
Let
D , E , EE ,
A EIf(E, F) into
and dense i n
E
Fc
.
and l e t
, whose
A&
domain
&(E,
d.(F , n F t , P ) be a be t h e same as i n
nEE , S )
b e a f a m i l y o f l i n e a r unbounded G
i s a subspace o f
D ,
independant
94
D. HUET
We suppose t h a t i)
A
ii)
A&
A
and
(3.3)
i
A&
there exists any
, P)
F
+
f o r any
u E G
K > 0
> 0
.
A C u E G Au ( F c
+
and f o r
t
&
,
F
Thus,
A& E
,
%(HE
FE)
K I I u I1
A E
and
+ 1)
6 (K
A
i s i n j e c t i v e since
We now suppose t h a t
. Let
(u,) E S(u) 6 > 0
if
E
llF
.
IIA,u
.
Since
vIIE
cp
-
Let
IIF
for &
uc+ u ( E c
-.
E
,
S)
.
u E G ; ( 3 . 4 ) and (3.5) y i e l d
. We
.
IlAu llF
(FE
+
F
.
H
i s dense i n
G
.
, P)
I t f o l l o w s from theorem 3 . 1 t h a t
s t i l l have t o prove t h a t
. Then,
E Gc0
uc E Q ( u )
6 IIu
llA,u
I t f o l l o w s from ( 2 . 9 ) and ( 3 . 3 ) t h a t
AEuC 4 Au
, Q)
u E o u (Hc .+ H
> 0
&
K I l u IIH < ( K t 1 )
-
.
HE
We now t a k e t h e l i m i t o f ( 3 . 6 ) .
IIu
g(H,F )
<
implies
P)
We i n t r o d u c e t h e f o l l o w i n g normed spaces
Thus,
E
E G
cp
K IIu llE
such t h a t
Proof.
(3.6)
,
FE
onto
G
s a t i s f y the inverse s t a b i l i t y condition i . e .
i s b i j e c t i v e and
A
(Ft
Acp
c-1
i i i ) the operators
(3'4)
est bijective o f
s a t i s f y the consistency condition
A&Q
Then
A&
i s surjective,
,
"IIH
there exists
cp
E G such t h a t
and
-
SE~IIE
6
i s s m a l l enough. T h e r e f o r e
S) i . e . ( E E -,E q(cp) c s(cp) . L e t
u& 4 u
b y (2.10) and ( 2 . 1 3 ) ,
llUE (u,)
E S(u)
.
lluC
&
-
qEcpllH 4< 6
,
&
Theorem 3.3. Assume t h e h y p o t h e s i s o f theorem 3.2. Furthermore, we suppose t h a t t h e r e e x i s t s a f a m i l y of o p e r a t o r s PE E X ( F , FE) and c o n s t a n t s K1 , K2 such that i) ii) iii)
(P,v) IIP,v
E P(v)
II
F&
I l A i 1 PEh
If u E E
f o r any
4 K1 l l v llF
-
v E F
.
f o r any
v E F
.
1 A- h IIE 6 K2 ca I I A - l h 1ID f o r a l l
we denote by
&
uc
t h e unique s o l u t i o n o f
h E F
such t h a t
A-'h
E D
.
A NORMED SPACE AND SINGULAR PERTURBATIONS
EG
u
(3.7)
&
Acut
M1
Then, t h e r e e x i s t c o n s t a n t s
P,
=
.
Au
M2
and
95
such t h a t
I
(3.9)
IIU&
-
u IIE =
(3.10)
IIU&
-
u I I 6~ M~ ca I I U I I ~ f o r any
Proof.(2.9),
(3.4),
o(1)
u E D
( 3 . 7 ) and assumption i i ) y i e l d
K l l u E I I E s K IIuE IIE d K K1 IllAlll I I u I I E &
and ( 3 . 8 ) f o l l o w s . From assumption i ) and ( 3 . 7 ) ,
ACuEL* Au
(Fc
-+
F
, P)
and
( 3 . 9 ) f o l l o w s from theorems 3.2 and 2.2.
We now suppose t h a t
u E D
.
(3.10) f o l l o w s f r o m assumption i i i ) , where
h = Au
.
C o r o l l a r y 3.2. The h y p o t h e s i s a r e t h e same as i n theorem 3.3 except i i i ) which i s rep1 aced by IIP&AV - A ~ C II P
iii')
F&
I I ~ I , I ~ f o r any cp E D
c K~
.
Furthermore we suppose t h a t G = D . Then t h e c o n c l u s i o n o f theorem 3.3 i s s t i l l valid.
Proof.
h E F
Let
K IIA;' =
u, = A - l h E D
and suppose t h a t PEh
-
A - l h II
IIPEAu, - AEu,II
F&
E&
<
llPch
-
.
By (3.4) and i i i ' ) one has
AEA-'h II
F&
4 K2 ca I I A - l h IID
which i s t h e assumption i i i ) o f theorem 3.3. Remark 3.1. Theorem 3.3 i s an e x t e n s i o n o f theorem 1.2 o f [51, o b t a i n e d i n t h e D and E a r e H i l b e r t spaces a = 7 ( c f . example 2.21, F = E ,
case where FE = E
&
,P
= R
PE = R E
( d e f i n e d by ( 2 . 5 ) )
( d e f i n e d by ( 2 . 4 ) ) .
Remark 3.2. The e s t i m a t e s , o b t a i n e d i n theorem 3.3 can be improved by i n t e r p o l a t i o n as i t was done i n [21 and [ 5 1 , [61
.
4. APPLICATIONS Example 4.1. Let
c
D
Let
[resp.
bl
,
E
, EE ,
d(E,
nEE , R)
be t h e same as i n example 2.2.
be a c o n t i n u o u s s e s q u i l i n e a r f o r m on
D
suppose t h a t c and b a r e G 8 r d i n g - e l l i p t i c on D and E t h e r e e x i s t s a compact, s t r i c t l y p o s i t i v e s e s q u i l i n e a r f o r m kE on (4.1)
El
and p o s i t i v e c o n s t a n t s a
IIu IID2 4 Re c(u,u)
a
,p
t kD(u,u)
such t h a t f o r any
u E D
[resp.
El
.
We
respectively i.e. kD on D [ r e s p .
96
D. HUET
2 [ r e s p . ( 4 . 2 ) B I I u l l E 6 Re b(u,u) t kE(u,u) We s e t
bc =
EC
.
t b
Let
,
AE E g ( E E
(4.3)
bc(v,u)
= ( v , A,u)
[resp.
b(v,u) = (v, A u ) ~
Ec)
A E %(E,
[resp.
for
u,v E EE
for
u,v E E l
EE
[lo]
I t was p r o v e d i n
(see a l s o [ 5 ] )
that
, F&
, P
F = E
c o r o l l a r y 3.1, w i t h
= EE
A&
E)l
be d e f i n e d b y
. A
and
.
R
=
.
u E El
f o r any
s a t i s f y t h e assumptions o f
T h e r e f o r e , theorem 1.1 o f [ 5 1 i s a
s p e c i a l c a s e o f c o r o l l a r y 3.1.
0
.
1
D = $(n) , E = Hm (a) L e t c and b be d e f i n e d b y ( 1 . 2 ) and s a t i s f y ( 1 . 6 ) . Thus c and b s a t i s f y ( 4 . 1 ) and ( 4 . 2 ) . Furthermore, (1.4) [resp. ( 1 . 5 ) l i s e q u i v a l e n t t o
I now go back t o example 1.2. L e t
(4.4)
u&
[resp. (4.5)
u E E
where
g
A&U& =
EE
RE9
-
AU = 91
i s a s u i t a b l e element o f
applying corollary 3.1 y i e l d s 2.1 t h a t
uc+u
in
$‘(a)
E =
uE
associated w i t h
E
u (Ec
+
, R)
E
uE
and
0
+
h
.
Since
Rcg
4
,
g
and i t f o l l o w s f r o m t h e o r e m in
D = @(a)
.
.
Example 4.2. L e t V and W be t w o H i l b e r t spaces s u c h t h a t V c W We suppose t h a t V i s dense i n W and t h a t t h e i d e n t i t y m a p p i n g o f V i n t o W i s continuous. L e t c [resp. b l be a continuous s e s q u i l i n e a r form on V
.
Wl
[resp. a, B , y
Let
. We
bc = cc t b
suppose t h a t t h e r e e x i s t p o s i t i v e c o n s t a n t s
such t h a t
(4.6)
2 2 I b E ( u , u ) l s, ac IIu l l v t 13 11u l l w
(4.7)
2 I b ( u , u ) I >, Y I I u l l w
d
We d e n o t e b y v
a(u, v ) u E D(&),dLu
d
where
= cdb t
N
t h e unbounded o p e r a t o r i n
I
. For
any
i s a constant. Therefore, there e x i s t s
&(F
G = D ( J ~ )= ( D =
,nF,
, P)
=
v)
IIU
I I ~ 6) II
= {U
E V ;
,
K > D such t h a t
u llW
dEcp +
c ( E = w)
d(W,nW,
D(&)
domain i s
.
Furthermore, i t i s obvious t h a t Let
, whose
, one has ( s e e e.g. [61 , p. 135)
u E D(&)
~ ( c l ” I I 1 1~” t
(4.9)
W
on V e q u i p p e d w i t h t h e norm II. l l w 1 . F o r a n y b ( v , & u ) = a(v,u) f o r any v E V Let
i s continuous i s d e f i n e d by
-t
.
.
u E W
f o r any
u E V
f o r any
a =
in
cp
,
f o r any
W
F = W
,
u E D(A)
f o r any
cp
.
E D(4)
.
,
, T ) ( t h e t r i v i a l a p p r o x i m a t i o n o f W ) . Then, we c a n
a p p l y t h e o r e m 3.2 and t h e o r e m 3.3, t o
AE = d c
,A
= I
and we o b t a i n t h e m a i n
97
A NORMED SPACE AND S I N G U L A R PERTURBATIONS
theorem o f [ 2 1 Now l e t
.
0
1
V = @(Q) , W = Hm ( Q )
.
Let
c
and
b
be d e f i n e d by ( 1 . 2 ) .
Suppose t h a t ( 1 . 3 ) h o l d s . Thus, problems ( 1 . 4 ) and ( 1 . 5 ) a r e e q u i v a l e n t t o (4.10)
u&
J1,
E&
UE
=
and t h e r e s u l t s o b t a i n e d i n example 1.1 f o l l o w from theorem 2 . 1 and remark 3.2.
.
L e t 1 < p < + m L e t C be a l i n e a r d i f f e r e n t i a l o p e r a t o r . Assumpa r e t h e same as i n example 1.3. L e t ( 6 . ) , j = 1 ,..., m be a J normal system o f l i n e a r d i f f e r e n t i a l o p e r a t o r s w i t h c o e f f i c i e n t s d e f i n e d on t h e Example 4.3. t i o n s on C
W2m(Q; ( B . ) ) t h e subspace o f WZm(Q) o f a l l f u n c t i o n s u P J P B.u = 0 , j = 1, 2, ..., m , on t h e boundary aR o f C . ( I n example J a j-1 1.3 , E. = , where v denote t h e normal t o a a ) ( s e e [ l ] ) . I t was proved J F i n [l] t h a t , f o r any E small enough, f o r any u E W2m(Q; ( 6 . ) ) P J
boundary. We denote by satisfying
I t i s obvious t h a t , f o r any
D = WZm(Q; ( B . ) ) , P J
Let
= L
P
P
(Q) ,
7
a = 1
, -
t Ccp
+
cp
-, cp
in
,nFc
P ) = Jb(L
=
.
L
.
-,E
Lp(R)
(a) and l e t P be t h e t r i v i a l a p p r o x i m a t i o n o f ,F
(a) , Lp(R) , T) P Then, we can a p p l y c o r o l l a r i e s 3.1 and 3.2, t o AE = - E C + I , A = I Therefore u c h 1.3, t h e problem ( 1 . 8 ) i s e q u i v a l e n t t o Acu& = h &(F
,
E
cp E Wzm(0; ( B j ) )
.
L
(a).
P I n example
(a) , S ) and t h e r e s u l t s f o l l o w f r o m theorem 2.2. P Remark 4.1. The r e s u l t s o b t a i n e d i n s e c t i o n s 2 t h r o u g h 4 were announced i n [71 (Ec
= L
L - b e h a v i o r ( p # 2 ) o f t h e s o l u t i o n uE P boundary v a l u e problem a s s o c i a t e d w i t h t h e e q u a t i o n c C u E + B ut = h Remark 4.2. The s t u d y o f t h e
C
and
B
are e l l i p t i c l i n e a r operators o f order
2m
and
2m'
with
.
o f some
, where # D ,
m'
i s , a t my knowledge, an open problem. I n a f u t u r e paper I hope t o a p p l y p r o p e r convergence theorems t o t h a t problem. BIBLIOGRAPHY
[11 Agmon, S., On t h e e i g e n f u n c t i o n s and on t h e e i g e n v a l u e s o f g e n e r a l e l l i p t i c boundary v a l u e problems, Comm. Pure. App. Math 15 (1962) 119-147.
[21 Greenlee, W.M., Rate o f convergence i n s i n g u l a r p e r t u r b a t i o n s , Ann. I n s t . F o u r i e r 18 (1969) 135-192. 131 Huet, D., Sur quelques problemes de p e r t u r b a t i o n s s i n g u l i P r e s dans l e s espaces Lp , Rev. Faculdade Cienc. L i s b o a 11 (1965) 137-164. [ 4 1 Huet, D.,
Remarque s u r un theoreme d'Agmon, B o l l . U.M.I.
2 1 (1966) 219-227.
[51 Huet, D., P e r t u r b a t i o n s s i n g u l i P r e s de problemes e l l i p t i q u e s , L e c t u r e notes i n Math. S p r i n g e r Verlag, 594 (1977) 288-300.
D. HUET
98
[61 Huet, O . , Decomposition s p e c t r a l e e t o p e r a t e u r s , Presses U n i v e r s i t a i r e s de France, P a r i s , 1977. [ 7 ] Huet, D., C. R. Acad. Sc. P a r i s 289 (1979) 69-70
and
595-596.
[8] L i o n s , J . L . , and Magenes, E., Problemi a i l i m i t i non omogenei, Scuola Norm. Sup. d i P i s a , 15 (1961) 39-101.
[91 Stummel, F., D i s k r e t e Konvergenz l i n e a r e r Operatoren, Math. Ann. 190 (1970) 45-92. [ l o ] Stummel, F., S i n g u l a r p e r t u r b a t i o n s o f e l l i p t i c s e s q u i l i n e a r forms, L e c t u r e Notes i n Math. S p r i n g e r Verlag, 280 (1972) 155-180.
1113 V i s i k , M.I. and L y u s t e r n i k , L.A.,
E l l i p t i c problems w i t h a parameter and p a r a b o l i c problems o f general t y p e , Uspehi Mat. Nauk 1 9 (1964) 53-155 and Russian Math. Surveys 19 (1964) 53-159.
A N A L Y T I C A L AND NlJMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS IN A N A L I ' S I S 5'. A x e l s s o n , L . S . P r a n k , A . v a n d e r S l u i s O N o r t h - H o l l a n d P u b l i s h i n g C o m p a n y , 1981
leas.)
AN ANALYSIS OF SOME FINITE ELEMENT METHODS FOR ADVECTION-DIFFUSION PROBLEMS Claes Johnson and Uno Navert Department of Computer Science Chalmers University of Technology 412 96 Goteborg, Sweden
We analyze two ways of improving the performance of the usual Galerkin finite element method when applied to a stationary advection-diffusion problem with small diffusion having a non-smooth solution. The first method consists in postprocessing the usual Galerkin solution by smoothing which increases the accuracy in regions where the exact solution is smooth. The second method is the result of applying the usual Galerkin method to a modified advection-diffusion problem obtained by adding extra diffusion in the streamline direction and compensating by modifying the right hand side. We.prove error estimates and give the results of some numerical experiments. INTRODUCTION The usual Galerkin finite element method when applied to an advection-diffusion problem may give an oscillating approximate solution which is not close to the exact solution in e.g. the L2-norm in case the exact solution is non-smooth and h > E/a where h is the mesh size, a is the velocity of the advection and E the diffusion coefficient. In this note we shall analyze two different ways of improving the performance of the Galerkin finite element method for such problems. Roughly speaking, the first method consists in post-processing the (oscillating) Galerkin finite element solution by a smoothing process, thus obtaining a non-oscillating approximate solution and reducing the L2-error. The success of the smoothing procedure relies on an error estimate in a Sobolev norm with negative index (cf.16)). The second idea, first proposed by Hughes and Brooks 1 5 1 , is to apply the Galerkin finite element method to a modified advection-diffusion problem obtained essentially by adding a suitably chosen diffusion term to the original problem. This "artificial viscosity" involves diffusion only in the direction of the streamlines and no "crosswind" diffusion. The resulting finite element schemes have an "upwind" character (cf.[2]) and produce non-oscillating solution but smear sharp fronts relatively little in the cross-wind direction. This is in contrast to the performance of more classical upwind or artificial viscosity 99
100
C.
JOHNSON and U. NAVERT
schemes obtained by adding diffusion in all directions which in general smear too much. Another important feature of the streamline diffusion method is that it is possible to compensate for the added diffusion by modifying the right hand side of the equation thus obtaining a higher-order accurate method. Such a modification cannot be done in practice if diffusion is added in all directions and classical upwind or artificial viscosity methods are only first order accurate. To briefly indicate some main features of the two methods to be considered below let us consider an advection problem with vanishing (or very small) viscosity. In such a problem information is propaga-
ted downstream in the direction of the characteristics (streamlines). If we apply the usual Galerkin method to such a problem we obtain a discrete model where information may be propagated also in other directions with relatively little damping. This is manifested by a discrete solution which oscillates in case the exact solution is nonsmooth and which is very sensitive to the outflow boundary conditions specified. However, by adding diffusion in the streamline direction one obtains a discrete model where the propagation is damped in directions transverse and opposite to the streamlines. In this way, we thus obtain a discrete analogue which better models the essential features of the continuous problem. In this note we shall consider a stationary constant-coefficient model problem with non-smooth solution. For simplicity we consider in the analysis only the case of vanishing viscosity. However, the problem considered is a representative model for linear advectiondominated flows with the main difficulties still present. For extensions of the results presented here to time-dependant and variable coefficient problems with non-vanishing viscosity, see [ 8 1 . An outline of the note is as follows: In Section 1 we introduce the model problem and in Section 2 we consider the usual Galerkin method together with smoothing. We prove in the case of continuous piecewise linear trial functions on a general triangulation that the global L 2 -error is at least of order O(h 'I4) and that with smoothing this can be improved to almost (3(h5'4) in regions where the exact solution is smooth. In Section 3 we analyze the streamline diffusion method and prove in the same case as in Section 2 a global L -estimate of order (3(h'I4) and a local L2-estimate of order (3(h3$) to-
ADVECTION-DIFFUSION
101
PROBLEMS
gether with a local L2-estimate of the derivative of the error in the streamline direction of order O(h). The results of Sections 2 and 3 can be extended to the case of higher order polynomial approximation with the corresponding increase of powers of h in the local estimates. Finally, in Section 4 we present some numerical results which indicate rates of convergence which are the same as or slightly better than the theoretical minimum rates just mentioned. Below C will denote a positive constant, possibly different at different occurences, which does not depend on the mesh parameter h. Further, for s a positive integer and R a bounded region we let H S ( R ) denote the usual Sobolev space with norm closure of C;(n) in the norm 1 1 - l l s , R .
11
s,n
and HE ( R )
is the
1. A MODEL PROBLEM As a model problem we shall consider the following stationary advec-
tion-diffusion problem with constant coefficients:
(l.lb)
-E-
auE + u n = gnx an X au' an
(1.lc) where
E-=
o
on on
Find
uE such that
r-,
r+u r0
is a small positive parameter, R is a bounded domain in the au Further, the boundary r is (x,y)-plane with boundary r and ux = -ax' divided into an inflow part r - l an outflow part r', and a characteE
ristic part Y o defined by (cf. Fig. 1.1)
~r
:nx(x,y) < 01 , E r :nx(x,y) > 0 1 , E r :nx(x,y) = 0 ) , where n = (nxln ) is the outward unit normal to r . We note that the Y boundary condition ( .lb) corresponds to a given total inflow and that the outflow condition ( 1 . 1 ~ )is of Neumann type. Below we shall a l s o briefly comment on the following Dirichlet boundary conditions as alternatives to the natural boundary conditions (l.lblc): (1.2) (1.3)
uE= g uE= o
on on
r-, r+ u ro.
C.
102
JOHNSON and U. NAVERT
ro Fig. 1.1 The r e d u c e d problem o b t a i n e d by s e t t i n g s = 0 i n ( 1 . 1 ) r e a d s a s f o l - lows : (1.4)
lx u
+ u = f
in
R,
u = g
on
r .
-
W e n o t e t h a t i n t h e r e d u c e d problem t h e boundary c o n d i t i o n s a r e g i v e n
o n l y on t h e i n f l o w p a r t
r - . The c h a r a c t e r i s t i c s of t h i s problem a r e
o r i e n t e d i n t h e d i r e c t i o n of t h e p o s i t i v e x - a x i s . I n g e n e r a l ( c f . [ 4 1 ) t h e s o l u t i o n of
( 1 . l a ) t o g e t h e r w i t h a combina-
t i o n of t h e boundary c o n d i t i o n s g i v e n by ( 1 . 1 b , c ) , w i l l have a boundary l a y e r a t t h e boundary I'+ U Y o .
l a y e r i s i n general of order O ( E ) a t
r+
( 1 . 2 ) o r (1.3)
The w i d t h of t h e
f ) along and C ( E
ro. Further,
i f t h e i n f l o w d a t a g h a s a jump d i s c o n t i n u i t y t h e n t h e s o l u t i o n w i l l
O(Ef ) i n t h e i n t e r i o r o f R ( i n t e r n a l l a y e r ) p a r a l l e l t o t h e s t r e a m l i n e s . I n t h i s case t h e s o l u t i o n u of t h e rehave a l a y e r of w i d t h
duced problem ( 1 . 4 ) w i l l have a jump d i s c o n t i n u i t y i n fi a c r o s s a characteristic. L e t u s now r e t u r n t o t h e model problem
( 1 . 1 ) . The f i n i t e e l e m e n t
methods f o r t h i s problem t o be i n t r o d u c e d below w i l l be b a s e d on t h e 1 f o l l o w i n g v a r i a t i o n a l f o r m u l a t i o n o f ( 1 . 1 ) : Find us E H (R) s u c h t h a t (1.5)
Ea(uE,v) + ( u i , v ) + (us,v)
-
< u E I v > -= ( f , v ) - < g , v > v v E H1(Q),
where a ( v , w ) = 1 Qv-vw d x d y , R ( v , w ) = I vw dxdy, R -= 1 vw n X d s .
r-
W e n o t i c e t h a t t h e boundary c o n d i t i o n s a r e imposed weakly i n ( 1 . 5 ) .
103
ADVECTION-DIFFUSION PROBLEMS
Remark. If the inflow data g and r- are smooth then it is possible auE in to compensate up to a term of order O ( C2 ) for the extra term can (l.lb) as compared to (1.2) by modifying the right hand side of (1.lb). In this case we may thus, neglecting terms of order O ( E2 ) , consider also the Dirichlet inflow condition (1.2) to be included in the formulation (1.5). However, if we replace the Neumann outflow condition ( 1 . 1 ~ )by the Dirichlet outflow condition (1.3) we will have to impose this condition strongly in the corresponding variational formulation.,
2. THE USUAL GALERKIN METHOD WITH SMOOTHING For simplicity, let us suppose that R is convex with polygonal boundary. Let Th = { K } be a family of triangulations of Q. indexed by the positive parameter h E (0,l) representing the maximum of the diameters of the triangles of Th. We shall assume that the usual minimum angle condition ( [ 3 1 ) is satisfied, i.e., the angles of the triangles K E Th are bounded below by a positive constant independent of h. Let us introduce the finite element space
vh
=
tv E H'(R): v is linear on K,
K E T ~ I ,
and let us then formulate the following discrete analogue of (1.5): Find u i E Vh such that (2.1)
(u;,v)-Ea (u;1,v)+ (u;lrx,v)+
Taking here v = u; (2.2)
lIv~;ll
c
r,
(f,v)--
VVEVh.
we easily obtain the following stability estimate: +
1IU;lI
+
Iuil 5 C(IIfl/ + 141
with C an absolute constant. Here
r'
=
(1.11
r
- ) r
is the L2(R) norm and for
1vlr, = ( /v2 Inxlds)+.
r' For simplicity we write 1v/ instead of Ivlr. Below, we assume that f E L2(R) and that 141 - < From the estimate (2.2) we obtain
-.
r uniqueness and thus also existence of a solution of (2.1).
We want to analyze the method (2.1) in the case E < h. In particular E may be very small. For simplicity we shall in the rest of this section consider only this case and we may then effectively put E
= 0 in (2.1). Thus
, we
shall analyze the following finite element
104
C. JOHNSON and U. NiVERT
method for the reduced problem ( 1 .4): (2.3)
Find
u h E Vh such that
= u(f,v)-~Uh,x'~~+~uh'v~-~ hlv~~
V V E Vh'
If the solution u of the reduced problem ( 1 .4) is smooth we easily find (cf. (2.4) below) that lIu-uhll 5 Ch. Here we want to analyze the case when u is discontinuous having a jump discontinuity across a characteristic corresponding to an internal layer in the full problem (1.1).
It is easy to guarantee immediate success of the finite element method (2.3) also in this case simply by choosing the finite element mesh to follow the streamlines where u is discontinuous. In practice, however, one does not in general have this freedom of adapting the mesh and thus we do not here assume any special properties of the triangulation Th except the minimum angle condition. Let us now prove some error estimates for the method (2.3) for the reduced problem (1.4) in the case the solution u of (1.4) is discontinuous. We have with e = u-uh: Theorem 2.1. There is a constant C independent of u such that (2.4)
IIelI
+
5 C(IIu-v/I + Iu-vI
lei
+
vv E Vh.
l~ux-v~l)
If u is peicewise smooth with a discontinuity across a characteristic
then
inf (11u-ql + Iu-vI + IIux-vx/I) 5 Ch 1 /4 . vEh' Proof. Taking v = uh-w with w E Vh in (2.3) and in the corresponding (2.5)
equation for the exact solution u we easily obtain (2.4) using the stability estimate (2.2) with E = 0. To indicate a proof of (2.5) suppose that u has a jump discontinuity across the x-axis, or more precisely, let us assume that
I
u(x,y) =
1
for
y > 0,
0
for
y < 0.
Let us now choose v E Vh to interpolate the function (XIY) E 1R
U(X,Y) = e(y),
2
,
at the nodes of Th, where 8 E C ' (R)is a transition function to be defined below satisfying 8 (y) E [ 0,1] for y E P I e(y) =
I
1
for
y > d,
0
for
y < 0,
and d > 0 is the width of the transition region D = I (x,y)E n:O
5 y 5 dl.
105
A D V E C T I O N - D I F F U S I O N PROBLEMS
S i n c e v a g r e e s w i t h u o u t s i d e t h e r e g i o n D h = i (x,y) E 0:-h
5
y < d+hl
o f w i d t h ( d + 2 h ) and v t a k e s v a l u e s b e t w e e n 0 and 1 , i t i s c l e a r t h a t (2.6)
< C(d+2h), llu-q12 + Iu-vl2 -
t h a t ux : 0
.
N e x t , t o e s t i m a t e I [ u -v 1 1 w e r e c a l l x x and t h a t vx = 0 o u t s i d e D h . By a T a y l o r ' s e x p a n s i o n
where C depends o n l y on
I?
u s i n g t h e f a c t t h a t fix = 0 it e a s i l y f o l l o w s t h a t , a s s u m i n g 0 " t o be piecewise continuous, P
where C d e p e n d s o n l y on t h e minimum a n g l e o f K . t h u s h a v e w i t h Ti = { K E T h : K
+ Ch2 I: J [ max Kt Ti K (x,yE K I f w e now c h o o s e d = h' e(y) = y
2
1
D
Recalling (2.6) we
# 81,
l e " ( y ) 11 2dx d y .
and
(3d-2y)h
we easily find, since
n
E K,
-3/2
18" ( y )
for 0
I 5
6h-l,
5
y
5
d,
that for h
5
1,
which proves t h e d e s i r e d r e s u l t . Remark 2 . 1 .
Theorem 2.1 c a n e a s i l y b e g e n e r a l i z e d t o t h e f u l l p r o b l e m
( 1 . 1 ) w i t h d i s c r e t e a n a l o g u e ( 2 . 1 ) . I n t h i s c a s e t h e Neumann o u t f l o w 1
boundary l a y e r c o n t r i b u t e s t o t h e e r r o r w i t h a t e r m of o r d e r O ( E ' ) and a n i n t e r n a l l a y e r w i t h a t e r m of o r d e r ( 3 ( ~ ' / ~ ) .
Remark
2.2.
I f w e r e p l a c e t h e Neumann o u t f l o w c o n d i t i o n
t h e Dirichlet outflow condition
( 1 . 1 ~ )by
( 1 . 3 ) and i n t h e c o r r e s p o n d i n g
d i s c r e t e analogue use a f i n i t e element space of functions vanishing
r + w e c a n n o t o u t o f a n a b s t r a c t e s t i m a t e o f t h e form ( 2 . 4 ) o b t a i n a p o s i t i v e rate of convergence. This i s because i n a D i r i c h l e t o u t -
on
f l o w l a y e r a t r + d e r i v a t i v e s o f t h e e x a c t s o l u t i o n o f order k b e h a v e -k l i k e O ( E ) ( c f . [ 41 1 . However, away f r o m t h e o u t f l o w boundary the exact
s o l u t i o n i s a f f e c t e d by d i f f e r e n t o u t f l o w b o u n d a r y c o n d i t i o n s o n l y up t o a t e r m o f o r d e r W E ) . M o r e o v e r , s i n c e t h e w i d t h o f t h e o u t f l o w boundary l a y e r a t
i s o f o r d e r O ( E ) and h >
E,
such a l a y e r cannot
b e r e s o l v e d by t h e c h o s e n d i s c r e t i z a t i o n and t h u s w e c a n n o t r e a l l y
106
C. JOHNSON and U. NAVERT
hope t o model d i f f e r e n t o u t f l o w c o n d i t i o n s o n
rt
in the discrete
model. Hence, o n e may as w e l l c h o o s e t h e o u t f l o w boundary c o n d i t i o n t h a t i s b e s t from c o m p u t a t i o n a l p o i n t o f v i e w , i . e . , t h e Neumann au = 0 on rt. A l a y e r a t Y o h a s t h e w i d t h O ( Ef ) outflow condition an and t h u s s u c h a l a y e r may b e r e s o l v e d t o some e x t e n t i f h < E'. The problem c o r r e s p o n d i n g t o a D i r i c h l e t c o n d i t i o n on Y o i s s i m i l a r t o
a problem w i t h a n i n t e r n a l l a y e r and c a n b e t r e a t e d a c c o r d i n g l y . From Theorem 2 . 1
it follows t h a t t h e g l o b a l L2(R)-error f o r t h e
G a l e r k i n method ( 2 . 3 ) i s a t l e a s t of o r d e r O(h1'4)
(the error i n the
( 0 ) - p r o j e c t i o n o n t o Vh i s of t h e o r d e r (!)(hi)). One c o u l d hope t o 2 g e t a h i g h e r r a t e of c o n v e r g e n c e away from t h e jumps i n u . However,
L
numerical experiments
( c f . S e c t i o n 4 ) show t h a t t h e a p p r o x i m a t e
s o l u t i o n o b t a i n e d by t h e u s u a l f i n i t e e l e m e n t method ( 2 . 3 ) i s o s c i l l a t i n g g l o b a l l y and t h a t t h e r a t e of c o n v e r g e n c e i n t h e L2-norm o v e r r e g i o n s where u i s smooth i s n o t b e t t e r t h a n t h e r a t e o f c o n v e r g e n c e i n t h e g l o b a l L2-norm. L e t u s now p r o v e an e r r o r estimate i n a S o b o l e v norm w i t h n e g a t i v e
index u s i n g a d u a l i t y argument. For s
L
0 w e i n t r o d u c e t h e norms
where 0' i s a bounded r e g i o n . B e l o w w e s h a l l w r i t e 0 ' c c 0 t o mean 0 and t h a t t h e d i s t a n c e from t h e c l o s u r e o f R ' t o t h e com-
t h a t R' c
plement o f n i s p o s i t i v e . Theorem 2 . 2 .
If
0' c c 0 t h e n t h e r e i s a c o n s t a n t C i n d e p e n d e n t o f
e such t h a t
2
P r o o f . Given X E H o ( n ' ) l e t cp b e t h e s o l u t i o n o f t h e p r o b l e m
I t i s e a s y t o see t h a t t h e r e i s a c o n s t a n t C i n d e p e n d e n t of X s u c h
that
F u r t h e r , by (2.7), cph
=
o
on
r+,
( 2 . 3 ) and (1.4 ) w e have f o r any cp
E Vh w i t h
ADVECTION-DIFFUSION PROBLEMS
By c h o o s i n g find that
I
vh E
(erx)
107
Vh t o be t h e i n t e r p o l a n t of cp and u s i n g
1 5 chllell I I X I I ~ , ~ ~
V
(2.8) we
2
X E HO(Q'),
which p r o v e s t h e d e s i r e d e s t i m a t e . By a smoothing p r o c e s s it i s p o s s i b l e t o o b t a i n an L 2 ( R ' ) - e s t i m a t e s i m i l a r t o ( 2 . 6 ) . To d e s c r i b e t h i s i n somewhat more d e t a i l , l e t K ( x , y ) be a smooth f u n c t i o n w i t h compact s u p p o r t s u c h t h a t
(cf.[6])
I K ( x , y ) d x dy = 1 , n,m
/ K ( x , y ) x n y m dx dy = 0 ,
2
0 , n+m
< r,
where n,m and r a r e i n t e g e r s , and d e f i n e
K6
=
1 7 K(5,:)
>
6
8
0.
I n t r o d u c i n g t h e smoothing o p e r a t o r S 6 S 6 v ( X , y ) = K,*v(X,y) =
d e f i n e d by
i K6 (x-x,y-y)v(x,y)dx dy,
we have i f R' c c n, f o r 6 s u f f i c i e n t l y s m a l l , (2.9)
II'-S6V11L2(R')
<
C6rllVllr,R
-
Now, l e t u s p o s t p r o c e s s t h e f i n i t e e l e m e n t s o l u t i o n uh by a p p l y i n g and l e t u s compare S 6 u h and u i n a r e g i o n R ' c c R where u i s S6
smooth. W e have u-S6uh = u-S 6u + S 6 e . By ( 2 . 9 ) i s f o l l o w s t h a t f o r 6 s u f f i c i e n t l y s m a l l , (2.10)
11~-s6u/IL2(Q') < C ~ ~ I I U ~ I r, , ~ ~ ~
where R' c c
a'' c
c Cl and
then Is6e(xry) which shows t h a t
I 5
u i s smooth i n R". F u r t h e r , i f ( x , y ) €
Q',
I 1 ~ l l _ ~ , n ~ ~ / I ~ g r~ ~ - ~ ~ ~ ~ ~ ~ / l ~ , n ~ ~
108
C.
JOHNSON
and U. NAVERT
Combining (2.10)and (2.11) we get
Equilibrating terms we obtain 1
so that finally
This proves that if r is large then we get after smoothing almost the same exponent for the L2-error as for the negative norm error. The drawback with r large is that 6 increases with r which means that the averaging procedure will involve many meshpoints (cf. also [ I ] ) .
3. THE STREAMLINE DIFFUSION SCHEME We shall now consider a modification of the usual Galerkin finite element method obtained by adding extra diffusion in the streamline direction. Let us then first consider the case
E
=
0, i.e., the
reduced problem
and let us consider the following modification of this problem: (3.2)
1-6uxy + ux + u = f in 6 u _ + u = g on
R,
-
l',
where 6 > 0 (below we shall take 6 = h) , obtained by adding the diffusion term -6u and perturbing the boundary conditions. We notice xx that the extra diffusion -6uxx acts only in the streamline direction. If we apply the usual Galerkin method to the modified problem (3.2) we obtain an improved discrete model as compared to the usual Galerkin method (2.3) where propagation of effects in directions opposite to and transverse to the flow direction is damped. In order to be able to easily compensate for the added diffusion by perturbing the right hand side we shall below start from the following modification of (3.2):
PROBLEMS
ADVECTION-DIFFUSION
(3.3a)
-6 ( u x + u ) x
+
U
+ u = f : - 6 f x + f
X
-6(ux + u ) + u =
(3.3b)
4-
+ g
-6f
109
in on
0,
-
r ,
We n o t i c e t h e p r e s e n c e o f t h e d i s s i p a t i v e t e r m -6uxx i n ( 3 . 3 a ) , t h a t t h e r i g h t hand s i d e o f s i d e of blem
( 3 . 3 ) i s a m o d i f i c a t i o n o f t h e r i g h t hand
( 3 . 2 ) and f i n a l l y t h a t t h e s o l u t i o n u o f t h e o r i g i n a l p r o -
( 3 . 1 ) i s i n f a c t a l s o t h e s o l u t i o n of
(3.3). Multiplying (3.3a)
by v E H ' ( Q ) and i n t e g r a t i n g t h e terms
( ~ ( U ~ + U ) ~ ,and V )
b y p a r t s , w e see t h a t t h e s o l u t i o n u o f
(3.3) has
following v a r i a t i o n a l characterization:
such t h a t
1
A(u,v) = L(v)
(3.4)
Find u
(6fx,v)
(formally) t h e
V v E H (Q),
where A ( u , v ) = 6 ( u + U , V ) + ( u X r V ) + (u,V) x x L(v) = (f,v+6vx)
-
-
We c a n now f i n a l l y m o t i v a t e d by of
( 3 . 1 ) t o be s t u d i e d below:
.
(3.4) formulate t h e d i s c r e t e analogue
Find uhE
A(uhrv) = L(v)
(3.5)
-,
Vh s u c h t h a t
v v E Vh,
w i t h Vh a s a b o v e . L e t u s s t a t e t h e method c o r r e s p o n d i n g t o ( 3 . 5 ) i n t h e
Remark 3 . 1 .
case
E
(3.6)
> 0: Find u: A
E
E V h--s u c h t h a t
E
(uh,v) = L(v)
v v E Vh/
where A€(W,V)
with
= E(VW,VV)
-
+ 6(wX+w,vx) + (wX,v) + (Wrv)
-
<w,v>-r
6 = max(0,h-E).
The problem ( 3 . 6 ) i s t o b e c o n s i d e r e d t o b e a d i s c r e t e a n a l o g u e o f t h e p r o b l e m ( 1 . 5 ) . L e t u s n o t e t h a t t h e s o l u t i o n uE o f ( 1 . 5 ) s a t i s f ies (3.7)
A€ (uE,v) = L ( V )
+
ET(AU€
vv E H
,vX)
1
I n p a r t i c u l a r w e n o t i c e t h a t t h e r i g h t hand s i d e o f from t h a t o f smooth.
( 3 . 6 ) w i t h a term o f o r d e r
For a n a n a l y s i s o f
( Q ) .
(3.7) d i f f e r s
(3(~8) i n r e g i o n s where u i s
( 3 . 6 ) see [ 81.
110
C.
Remark 3 . 2 . (l.la),
JOHNSON and U. NAVERT
The c o r r e s p o n d i n g d i s c r e t e a n a l o g u e o f t h e problem
( l . l b ) together with t h e D i r i c h l e t outflow condition
(1.3) reads:
Find
WE E
Wh
such t h a t
A E (wL,v) = L ( v )
(3.9)
v v E Wh,
where
w h = I V E vh:
v
= o
L e t u s now a n a l y z e t h e method
on
r+ u ro1.
(3.5) c o n s i d e r e d as a d i s c r e t e
( 3 . 1 ) and l e t u s t h e n from now on c h o o s e 6 = h .
analogue of
We have
which p r o v e s e x i s t e n c e o f a u n i q u e f u n c t i o n uh E Vh s a t i s f y i n g ( 3 . 5 ) i f 6 = h < 1 . By a r g u i n g a s i n t h e p r o o f o f Theorem 2 . 1 w e o b t a i n t h e f o l l o w i n g g l o b a l e s t i m a t e o f t h e e r r o r e = u-uh, where u s a t i s f i e s ( 3 . 1 ) : There i s a p o s i t i v e c o n s t a n t C independent of u such
Theorem 3.1.
< 4
that for h
I n p a r t i c u l a r , i f u i s p i e c e w i s e smooth w i t h a jump a c r o s s a c h a r a c t e r i s t i c , then t h e r e i s a constant C such t h a t
L e t u s now p r o v e a n estimate o f t h e e r r o r away from a jump o f u
across a c h a r a c t e r i s t i c i n R and away from t h e o u t f l o w boundary r + u Y o . T o b e s p e c i f i c l e t u s assume t h a t u h a s a jump d i s c o n t i n u i t y across t h e x - a x i s
n’
regions
= {(x,y) E
L e t us define
nl
c
n2
c
n
,
and t h a t u i s smooth i n t h e non-empty
>
n:y < 0)
e x c e p t p o s s i b l y close t o
(see F i g . 3 . 1 ) t h e subdomains n l and n 2 ,
with boundaries
r+u r 0 .
ADVECTION-DIFFUSION
r.
=
rI
= { (x,y) E
where 0 < a 2
r ia r e
PROBLEMS
I (x,ai) E n i u I (x,bi) E G I u
r
- :ai
< al < b l < b2
< y < bil
111
-
ri u
+
Ti'
i = 1,2,
and t h e d i s t a n c e s between
r +l I r 2+
and
positive.
Y
Fig. 3.1 L e t u s assume t h a t t h e e x a c t s o l u t i o n u i s smooth i n 0 2 . F u r t h e r ,
l e t u s assume t h a t t h e t r i a n g u l a t i o n Th i s q u a s i - u n i f o r m so t h a t t h e f o l l o w i n g i n v e r s e e s t i m a t e h o l d s f o r K E Th and v E Vh: IIvIIH1 ( K ) '
(3.9)
5
Ch
-1
IIv/IL2 ( K )
'
W e c a n now s t a t e and p r o v e t h e main r e s u l t o f t h i s s e c t i o n . Here
11 -11
d e n o t e s t h e L2 (ni)-norm.
Theorem 3 . 2 .
Under t h e a b o v e a s s u m p t i o n s w e have f o r h
Ile.Jn
1
< 4,
5 Ch.
P r o o f . By ( 3 . 4 ) and ( 3 . 5 ) w e have t h e f o l l o w i n g e r r o r e q u a t i o n (3.10) L e t now JI
I
ah:H
A(e,v) = 0 JI
vv E
be a smooth " c u t - o f f
vh' f u n c t i o n " d e f i n e d on D such t h a t
1 in R $= 0 i n R \ R 2 and J1, 5 0 i n R . F u r t h e r , l e t 1' 1 (il) + Vh b e t h e u s u a l i n t e r p o l a t i o n o p e r a t o r and d e f i n e
112
I - = u -
- uh so t h a t e = I- + 8 h ( c f . [31 ) w e h a v e
e
and
nhu
= II u
s t a n d a r d estimates
I1$nl1+
(3.11b)
2 1$17I-zCh
/uI
r
II+e -
l.IS,O
2 < Ch 2 Ch Iu12,1)2 -
hi1 ( $ n ) A I 5
(3.11a)
where
JOHNSON and U. NAVERT
C.
< Ch 2,r- + hi1
nh(+e)lI
Iqe12,K
(3.12b)
/$e
By
,
, 2
2
'
( $ e l x - ~ ~ ( ~ 1 e 5) ~Chl I(Cl$e12,K)', K
d e n o t e s t h e seminorm o f d e r i v a t i v e s o f o r d e r s i n t h e
e i s l i n e a r on e a c h
S o b o l e v s p a c e Hs(o). S i n c e
so t h a t ( c f .
2
and 8 E Vh.
D])
5
K w e have
Cilelll,K
using a l s o t h e inverse estimate
-nh($e)
lr-
5
(3.9),
.
Ch/81r-
I t i s e a s y t o check t h a t
2 h Ile.J*
E
= A(e,$e)
-
1-h
+
2 lel
2
2
IIelI*
+
h(ex,$,e)
+
1 -h ~ ( e , $ .e ) ,
where
Thus s i n c e $x
5
0 and h
and t h e f a c t t h a t e = E
5
A(e,$n)
< 1 w e have u s i n g also (3.10) w i t h v
I-
+ 0
+
A(e,$e-nh($e))
+
hile.Jln IIeII 2
R2
= ~ ~ ( $ 8 )
.
R e c a l l i n g (3.11) and ( 3 . 1 2 ) w e t h e r e f o r e conclude t h a t E
5 Ch 2 lleAIJI
+
c4le I I I I d n2
R e p l a c i n g h e r e 8 by e-n fact that
+ Ch311e
R2
+
11
+ Ch211elI$ + Ch 2 l e ( $ + N l e A l n
n2
2
cNelln I I e l I 2
R2
+
Ch e l
-
r2
+
1IelI
h311ell
R2
R2
and u s i n g a g a i n (3.1 ) t o g e t h e r w i t h t h e
.
A D V E C T I O N - D I F F U S I O N PROBLEMS
113
L
+ C h / e l2r 2
=
T h u s , s i n c e J,
+
Ch 3 l e l r 2
1 i n R,
Finally, since
a1ex/
w e have
41eJ
I t e r a t i n g t h i s i n e q u a l i t y s i x t i m e s on a s e q u e n c e o f d o m a i n s r e l a t e d
as a l and
R2
a n d u s i n g t h e s t a b i l i t y e s t i m a t e g i v e n by
(3.8) t o
f i n a l l y bound norms o v e r R w e o b t a i n t h e d e s i r e d r e s u l t . Remark.
Theorem 3.2 c a n e a s i l y b e g e n e r a l i z e d t o t h e case 0
<
C
< h
w i t h e i t h e r Neumann o r D i r i c h l e t o u t f l o w b o u n d a r y c o n d i t i o n s . Remark. I n t h e case o f a smooth s o l u t i o n w e o b t a i n g l o b a l e s t i m a t e s of t h e same o r d e r a s i n Theorem 3 . 2 . T h u s , w i t h t h e e x t r a d i f f u s i o n
term w e loose o n l y 2
r a t e (3(h
).
&
i n t h e L2-norm a s compared t o t h e o p t i m a l
F o r t h e u s u a l G a l e r k i n f i n i t e e l e m e n t method o n e g e t s
w i t h d i r e c t e s t i m a t e s O ( h ) i n t h e L2-norm,
i . e . a loss o f o n e power 2 o f h . However, i n s p e c i a l cases o n e g e t s e x p e r i m e n t a l l y C)(h ) . 8 Remark. By a d u a l i t y a r g u m e n t o n e c a n p r o v e t h e f o l l o w i n g n e g a t i v e
norm e s t i m a t e f o r ( 3 . 5 ) : l l e l l - l , R t < Ch2 i f u i s smooth i n fi'c c .R
114
4.
C . JOHNSON and
u. N ~ V E R T
NUMERICAL RESULTS
We have made some numerical experiments with the different methods discussed above on the model problem: B - V ~+ u = u o
in f i f u = g o n r ,
1
where B = (2,1), q = uo on T - f 0 = (0,l)% (0,l) is the unit square and ( x - 2 ~ ) ~ for y < 0.51 + 0 . 5 ~ ~ uo = ( x - 2 ~ +) ~1 for y > 0 . 5 1 + 0 . 5 ~ . The exact solution is u = u o . We used piecewise linear ( T 1 ) and for the streamline upwind scheme a l s o piecewise quadratic (T2) finite element spaces on a triangulation of the form
'4 h
-
I
-
R'
+X
Fig. 4.1
Notice that the grid is not oriented to follow the characteristics. We obtained the following results. Here n ' is the subregion indicated in Fig. 4.1.
115
&roi
!
! : : : l!
I Degree of smoothing h = 1/8
h = 1/16
No smoothing
0.163
0.121
No m t h i n g
0.172- 10-1
0.120-10-1
1 smoothing s t e p
*O. 127- 10-1
0.521-10-2
0.438.10-2
2 smoothing s t e p s
0.136-10-1
"0.455- 1Ow2
0.204-1 0-2
3 m t h i n g steps
I
1
h = 1/32
I
0.949.10-1 o.lll~lo-l
I
0.600*10-2 *0.175.10-2
4 m t h i n g steps
0.753- 1 0-2
rate of h0*35
h1.38
0.214-
I
< l m t :Ype
Error L2(R)
'
h = 1/8
h = 1/16
h = 1/32
0.167
0.128
0 ..971- 10-1
Estin-ated rate of convergence
ho. 4 h2. 2
T1 BH1 (0)
' 1
0.186-101
0.137.101
0.139*101
.o
BH ( n ' )
0.175
hl
L2(0)
0.119
ho. 4 h2.6
T2 0.738 H1 ( G I )
0.241-10-1
I
0.569~10-~
h2. 1
C. JOHNSON and U. NZVERT
116
Comments. Each smoothing step consists of multiplication with the mass-matrix. To estimate the rate of convergence after smoothing we used the numbers marked with a star. This is intended to account for the fact that according to the smoothing procedure in Section 2 , the support of the smoothing kernel should cover more meshpoints 1 as h gets smaller. Further, f3H ( w ) indicates the norm
REFERENCES J.H. Bramble and A.H. Schatz, High order local accuracy by averaging in the finite element method, Math. Comp. 31 ( 1 9 7 7 ) 94
-
111.
,
I. Christie and A.R. Mitchell, Upwinding of high order Galerkin methods in Conduction-Convection problems, Int. J. for Num. Meth. in Eng. 1 2 ( 1 9 7 8 1 , 1 3 8 9 - 1 3 9 6 . P.G. Ciarlet, The finite element method for elliptic problems, (North Holland, Amsterdam 1 9 7 8 ) . W. Eckhaus, Asymptotic analysis of singular perturbation problems, (North Holland, Amsterdam 1 9 7 9 ) .
T.J. Hughes and A. Brooks, A multi-dimensional upwind scheme with no crosswind diffusion, Technical report, California Institute of Technology, 1 9 7 9 . M.S. Mock and P.D. Lax, The computation of discontinuous solutions of linear hyperbolic equations, CPAM vol. 31 ( 1 9 7 8 ) , 423
-
430.
J.A. Nitsche and A.H. Schatz, Interior estimates for RitzGalerkin methods, Math. Comp. 2 8 ( 1 9 7 4 ) , 9 3 7 - 9 5 8 . U. NZvert, Finite element methods for advection-diffusion problems, Technical report, Chalmers University of Technology, to appear.
Note : For negative norm estimates for hyperbolic problems and smoothing, see also M. Luskin, A finite element method for first order hyperbolic systems, Technical report, Mathematics Department, The University of Michigan, 1 9 8 0 .
A N A L Y T I C A L AND NUMERICAL APPROACHES TO A S I M P T O T I C PROBLEMS I N A N A L Y S I S S . A x e l s s o n , L . S . F r a n k , A . van d r i S l u l s ( e d s . ) @ North-Holland Puhlishinq C o m p a n y , 1981
SHORT TIME ASYMPTOTIC BEHAVIOR FOR PARABOLIC EQUATIONS
Yakar Kannai Department of T h e o r e t i c a l Mathematics The Weizmann I n s t i t u t e of S c i e n c e Rehovot ISRAEL
We s t u d y t h e a s y m p t o t i c b e h a v i o r , o f f t h e d i a g o n a l , as t + O , of t h e fundamental s o l u t i o n of a second o r d e r p a r a b o l i c e q u a t i o n , and o b t a i n by a n a l y t i c methods b o t h l o c a l and g l o b a l r e s u l t s . We t r e a t b o t h t h e fundamental s o l u t i o n of t h e Cauchy problem and t h a t of some mixed i n i t i a l boundary v a l u e problems. We a l s o i n v e s t i g a t e t h e o s c i l l a t o r y b e h a v i o r , o f f t h e d i a g o n a l , as X + m , of t h e s p e c t r a l f u n c t i o n of a h i g h e r o r d e r e l l i p t i c o p e r a t o r . Some open problems a r e a l s o e x h i b i t e d .
1.
INTRODUCTION
We w i l l p r e s e n t i n t h i s l e c t u r e ( i n a somewhat l e i s u r e l y manner) some r e s u l t s on t h e a s y m t o t i c b e h a v i o r , a s t + O , away from t h e d i a g o n a l , of t h e fundamental s o l u t i o n s of p a r a b o l i c e q u a t i o n s , and t h e r e l a t e d b e h a v i o r of t h e s p e c t r a l f u n c t i o n s ( a s h + m ) f o r e l l i p t i c o p e r a t o r s . We w i l l a l s o w x h i b i t s e v e r a l open problems.
,
f o r d e f i n i t e n e s s , b e a s e l f - a d j o i n t p o s i t i v e e l l i p t i c o p e r a t o r of o r d e r c o e f f i c i e n t s , d e f i n e d on a compact n-dimensional manifold M w i t h d e n s i t y dx , Then t h e compact o p e r a t o r .-At ( t > 0) o u t boundary, having a C" i s a n i n t e g r a l o p e r a t o r w i t h a C" k e r n e l . T h i s k e r n e l w i l l b e denoted by u(x,y,t). Regarded as a f u n c t i o n of x and t , u i s a s o l u t i o n of t h e parab o l i c equation
Let
A
rn w i t h
Cm
satisfying the i n i t i a l condition (1.2)
l i m u(x,y,t) = 6(x;y) t+O+
,
where 6 ( x ; y ) i s t h e D i r a c k e r n e l . Let dEX d e n o t e t h e s p e c t r a l measure a s s o c i a t e d w i t h t h e s p e c t r a l r e s o l u t i o n of A , Then Eh i s a l s o a n i n t e g r a l k e r n e l , denoted by e ( x , y ; h ) , and u ( x , y , t ) i s t h e Laplace o p e r a t o r w i t h a C" t r a n s f o r m of e(x,y;X) ; (1.3)
u(x,y,t) =
o/
" -th e
de(x,y;X)
,
t > O
.
For t h e second o r d e r c a s e m = 2 , w e can d e s c r i b e p r e c i s e l y t h e a s y m p t o t i c beFor m > 2 , we were a b l e , up t o now, t o o b t a i n h a v i o r , as t + O + , of u ( x , y , t ) a s X + m ( t h i s beo n l y less p r e c i s e r e s u l t s c o n c e r n i n g t h e b e h a v i o r of e(x,y;X) h a v i o r i s n o t w e l l known i f x # y ) . A s i s w e l l known, t h e b e h a v i o r of u ( x , y , t ) f o r l a r g e A are c l o s e l y r e l a t e d t o f o r small t and t h e b e h a v i o r o f e(x,y;A) each o t h e r . Work i s now i n p r o g r e s s about t h e b e h a v i o r of u ( x , y , t ) , i f m > 2 , by a s t u d e n t , K. T i n t a r e v , b u t we have no c o n c r e t e r e s u l t s as y e t .
.
117
118
Y. KANNAI
The method of proof involves transformation formulas 191, which relate solutions of various differential equations (of different types) to each other. The (disiA1lms (here Allm detribution) kernel w(x,y,s) of the unitary operator z notes the positive m-th root of A ) is given as the Fourier transform of e (x,y ;Am) : ishl/m de(x,y;X) = /eisXde(x,y;Am) (1.4) w(x,y,s) = le
.
On the other hand, w satisfies (as a function of pseudo-differential equation
with the initial condition (1.6)
w(x,y,O)
=
6(x;y)
x
and
s ) the hyperbolic
.
For general m one uses delicate properties of Fourier transforms [13,15] to If m = 2 , derive properties of e(x,y;X) from (1.4) and known properties of w then a simple inversion of the Fourier transform (1.4) and substitution in the Laplace transform (1.3) yield the result
.
Using the explicit Hadamard construction of fundamental solutions for second order hyperbolic equations [ 4 , 8 ] and some other special properties of second order equations, one can obtain easily from (1.7) representation formulas for u The formal asymptotic formulas for u (m=2) were derived by J . K . Cohen and R.11. Lewis [31 without rigorous proofs; V.P. Maslov discusses the matter briefly in his book [171. Complete proofs were given by S . A . Molchanov in [la]; those proofs, however, are highly probabilistic. The method of [l], while completely analytic, involves "harder" analysis than our relatively simple and "soft" argument. This lecture is organized as follows. The local theory (x near y) for second order operators is expounded in Section 2, where we present also the transmutation technique. I n Section 3 we discuss global results for second order operators. Here we discuss also the behavior in the presence of a boundary and treat some aspects not covered in [141. I n Section 4 we characterize (following essentially 1151) Fourier transforms of locally analytic functions, and introduce
.
I n Section 5 the theory of Section the "pseudo-differential operator" e-x2'D' 4 is applied to the study of the asymptotic behavior (as X + m ) of e(x,y;X) for A number of open problems are listed in Section 6 . x#y
.
2.
SECOND ORDER EQUATIONS - LOCAL THEORY
Our point of departure is the transmutation formula (1.7). That formula was arrived at heuristically by inverting the Fourier transform (1.4) and substituting - in (1.3): 2 2 u(x,y,t) = o/m emtXde(x,y;X) = o/m e-tX de(x,y;X ) = Q
o/m
(2.1)
e-thL(2n)-1[
-m
Ime-isXw(x,y,s)ds]dX
(2n)-l-mlm e-tX2 [-,Im e-isXw(x,y,s)ds)dX (2n)-l
/m[-m/m -m
= =
e-tX2-isXdh]w(x,y,s)ds = 1
J4nt -,I
m
-s
2 /4t
w(x,ys)ds
.
119
PARABOLIC EQUATIONS
Instead of justifying the various steps in (2.1), establishing (1.7) for positive self-adjoint operators A only, we will demonstrate the validity of (1.7) directly in a way which will establish it (slightly modified) for a much wider class of operators A
.
In fact, let A be a realization of the formal differential operator -a(x,a/ax) given (locally) by
We assume that M is a smooth connected n-dimensional manifold which is either (i) compact, or (ii) a manifold with a smooth boundary, or (iii) a complete open (thus unbounded) manifold. I n case (i), A is the unique realization of -a(x,a/ax) in L2(II) , with Dom(A) = H2(M) I n the other cases, we postulate the existence of a formally self-adjoint differential operator ao(x,a/ax) having the same principal part as a , and a strictly negative self-adjoint realization -A0 , such that Dom(A) = Dom(A0) and A-AO is bounded relative to -A0 Here A and A 0 operate (unboundedly) on the Hilbert space L2(M) of functions which are square integrable with respect to the Riemannian density induced on M by the principal part of a(x,a/ax) . I n case (ii) we assume further that the domain of . , Dom(Ao) , is given by regular elliptic boundary conditions for ao(x,a/ax) A (for example, Dom(A0) could be given by Dirichlet or Neumann boundary conditions). I n all cases we obtain from standard hyperbolic theory existence of solutions for the "wave" equation
.
.
(2.3)
= -Aw as2 with distribution Cauchy data (at least if the data belong to Dom(Ai) for some r E R ) . Moreover, the solutions satisfy energy inequalities with exponential dependence on Is1 , Hence it can be shown that w(x,y,s) , a solution of ( 2 . 3 ) with w(x,y,O) = 6(x,y) (the value of (a/as)w(x,y,s) is not really relevant, as long as it is admissible), can be restricted to a line x=const ,y=const , s o as to yield a distribution acting continuously on functions v(s) , whose growth
is estimable by
O(e
-ES2
) (for some E > O ) .
Thus
But
for
t>O
.
Hence
-A-
Jq,t -=
so
rm e-'
2 /4tw(x,y,s)ds
=
a at
J4nt -=
rm e-'
2 /4tw(x,y,s)ds
'that the right hand side of (1.7) solves the heat equation (1.1).
we get that lim - rm e- S'/4tw(x,y,s)ds (2.4) t+O+ LTz -=
,
Noting that
7 ,
=
-aJ
m
G(s)w(x,y,s)ds
=w(x,y,Q) = 6(x,y)
By uniqueness we get (1.7). Note that w(x,y,s)
could be the distribution kernel of
COS(A~/~S) as well as
120
Y . KANNAI
1/2s t h a t of eiA I f w e t a k e t h e former c h o i c e we o b t a i n f o r w t h e Cauchy d a t a w(x,y,O) = 6 ( x , y ) , ( a / a s ) w ( x , y , O ) = 0 , and w e can r e p l a c e t h e i n t e g r a l
.
-co
f
m
i n ( 1 . 7 ) by
20/m (of t h e same i n t e g r a n d ) i f
xfy
.
Denoting by
R(x,y,s)
t h e ( u s u a l ) fundamental s o l u t i o n f o r t h e wave e q u a t i o n , we o b t a i n t h e r e l a t i o n aR 2 2 I n t h e s i m p l e s t one d i m e n s i o n a l c a s e , M = R ' , a = a / a x w(x,y,s) = s ( x , y , s ) 6 (x-y+s) + s (x-y-s) and w ( x , y , s ) = Hence 2 2 1 j m e-s2/4t [6 (x-y+s) + 6 (x-y-s)_l d s = __ 1 .-(x-y) /4t (2.5) u ( x , y , t ) = __ 2
.
J47it
.
J47it
-m
a s we know o n l y too w e l l . (But J. Hadamand was a p p a r e n t l y unaware of t r a n s mutation formulas such a s ( 1 . 7 ) , a s i n [81 h e l a b o r e d t o o b t a i n t h e formula ( 2 . 5 ) from h y p e r b o l i c e q u a t i o n s , o b t a i n i n g i t from a complicated l i m i t of t h e a% = a2u fundamental s o l u t i o n f o r t h e t e l e g r a p h i s t ' s e q u a t i o n E at2 a t ax2 as E - * o * ) We w i l l u s e t h e c l a s s i c a l 3 . Hadamard c o n s t r u c t i o n 181 f o r R ( x , y , s ) , i n t h e form given i n Courant-Hilbert [ 4 ] . R e c a l l t h e r e p r e s e n t a t i o n
+ au
~
where d ( x , y ) i s t h e d i s t a n c e between x and y i n t h e Riemannian m e t r i c on M , gk(xrY) a r e Cm f u n c t i o n s determined by means of t r a n s p o r t e q u a t i o n s , , 6 ( j ) den o t e s t h e j - t h d e r i v a t i v e of t h e D i r a c 8 - f u n c t i o n f o r j 2 0 w h i l e f i ( - J ) ( r ) = T(j-l)-'rj-'H(r) for j 2 1 The s e r i e s (2.6) r e p r e s e n t R ( a t l e a s t l o c a l l y f o r x n o t t o o f a r from y i n i n t M and s 2 0 n o t t o o l a r g e ) i n t h e s e n s e that the difference
.
(The can b e made a r b i t r a r y smooth i n s and x , i f N i s chosen l a r g e enough. f a c t t h a t (2.6) i s v a l i d o n l y f o r s 2 0 w a s overlooked by t h e a u t h o r i n [ 1 4 ] and was p o i n t e d o u t t o him by K. T i n t a r e v ; t h e a u t h o r i s g r a t e f u l t o K. T i n t a r e v f o r t h i s remark. Note t h a t R ( x , y , s ) can be o b t a i n e d € o r a l l s by m u l t i p l y i n g t h e r i g h t band s i d e of (2.6) by s i g n ( s ) ; t h a t t h e p r o d u c t makes s e n s e can b e s e e n by c o n s i d e r i n g t h e wave f r o n t s e t s of t h e f a c t o r s . ) I f N i s chosen so t h a t E , then
%
.
Indeed, t h e f a c t t h a t t h e Cauchy d a t a a t s = O of where r+ d e n o t e s max(r,O) R is s u p p o r t e d a t x = y i m p l i e s t h a t R ( x , y , s ) = 0 f o r I s 1 < d ( x , y ) A l l the ( f r a c t i o n a l p o s i t i v e o r n e g a t i v e ) d e r i v a t i v e s of 8 v a n i s h i d e n t i c a l l y f o r f u n c t i o n (of t h e v a r i a b l e s ) n e g a t i v e v a l u e s of t h e argument. Hence t h e CP+' RN(x,y.s) v a n i s h e s f o r 0 S s < d ( x , y ) and ( 2 . 8 ) f o l l o w s . The a s y m p t o t i c s of t"e-d2(xyY)'4t f o r some u ( x , y , t ) which we would l i k e t o d e r i v e a r e of t h e form real a Simple c o n s i d e r a t i o n s show t h a t i n i n t e g r a l s of t h e form
.
.
i t i s only t h e b e h a v i o r n e a r s = d ( x , y ) which c o n t r i b u t e s s i g n i f i c a n t l y t o t h e a s y m p t o t i c b e h a v i o r as t + O , t h e i n t e g r a t i o n over v a l u e s of s g r e a t e r t h a n , s a y , d ( x , y ) + l , a r e n e g l i g i b l e i n comparison t o e -[d2(x,y)+E1/4t f o r a certain positive
E
.
Rewriting ( 1 . 7 )
as
PARABOLIC EQUATIONS
121
L
utilizing (2.8) and the above remark, we see that the second integral is estimable
.
by O(t p+1/2e-d2(xyy)/4t) as t+O+ Concerning the first integral on the right hand side of (2.9), observe that the sum contains either Dirac measures, their derivatives (of fractional positive orders) concentrated at s = d(x,y) , or positive powers of [s-d(x,y)]+ , multiplied by rational functions of s (regular at s = d(x,y) ) . The highest derivative of a Dirac function appearing in the sum . Hence the biggest term (as t 0, ) will be of the order is 6(n-1)’2
-.1
-f
t(l-n)/2e-d2(x,~)/4t
=
t-(n/2)e-d2(x,~)/4t
JO.rrt evaluated, either
.
The other
can be
explicit13 or using the classical Laplace integral method, noting that the function s /4 attains a strict maximum at the end point s = d(x,y) of the interval and l/t plays the role of the large parameters. Thus we can find u,(x,y) , j =O,l, , such that for every p ,
...
(For further details, see [14].) the asymptotic formula
+o+
Since p
t
can be taken arbitrarily, we obtain
2 (4nt)-n/2e-d (x9y)/4t ‘ j = O
.
u j. (x,y) tj t The formula (2.10) was discovered in [4], without a discussion of the error term. Note that the functions uj(x,y) are determined by means of transport equations; the latter are different from those appearing in the computation of the gk in (2.6). Note also that we have derived (2.10) under the assumption that x is not too far from y (If x = y then the exponential factor disappears; the result on the diagonal is well-known. The formula (2.9) is not valid, strictly speaking, if x = y , but it is easily seen that (2.10) is valid uniformly as y + x . )
(2.10)
u(x,y,t)
.
3.
SECOND ORDER EQUATIONS - GLOBAL THEORY
Here we shall describe asymptotic formulas for u(x,y,t) valid without the nearness assumption made in Section 2. We begin by recalling certain well-known facts from the differential geometry of a complete Riemannian manifold M without boundary. It is well-known that any two points of M can be joined by a minimizing geodesic. Also, every geodesic is infinitely extendable. Hence for every y E M , the exponential map expy :Ty(M) M is a well-defined mapping of the tangent Let y E M and let T be a unit vector in Ty(M). space of M at y onto M Let g(t) be the geodesic (parametrized by arc length) issued from y in the direction of T For small values of t , t = d(y,g(t)) Let r be the smallest positive number such that r # d(y,g(r)) (if no such r exists, then we set r = m ) . The point g ( r ) is said to be the cut point OF y in the direction of T (or along the geodesic g(t) ) . The set of all cut points of y is called the cut locus of y and is denoted by C(y) Then d2(x,y) E Cm(M\C( ) ) , and ex mapps the open subset R C Ty(M) given by ( 0 = IT E T (M) : lrTll < r(T/IITir , where g(r) is the cut point in the direction oy T/IITII ) diffeomorphically onto M\C(y) In particular, the Jacobian of exp does not vanish on R A critical value of expy is called a conjugate poine of t A point z is a cut point of y along a geodesic g(t) issuing from y if either z is the first conjugate point of y in the
.
-+
.
.
.
.
.
.
122
Y . KANNAI
geodesic o r i f t h e r e e x i s t s a t l e a s t one g e o d e s i c h d i s t i n c t from g j o i n i n g (Note t h a t y and z , such t h a t d ( z , y ) i s e q u a l t o t h e a r c l e n g t h of h t h e s e p o s s i b i l i t i e s a r e n o t mutually e x c l u s i v e . )
.
The l o c a l c o n s i d e r a t i o n s of S e c t i o n fundamental s o l u t i o n f o r t h e Cauchy We w i l l f i x now i f x E M\C(y) There e x i s t open s u b s e t s U1,U2,V C following hold:
.
2 and t h e Hadamard c o n s t r u c t i o n of a problem f o r t h e wave-equation are v a l i d o n l y y and assume t h a t - x i s a c u t p o i n t of y M w i t h x E U 1 , y E Ug , s u c h t h a t t h e
(3.1) There e x i s t s a p o s i t i v e E such t h a t f o r a l l we have 2 2 2 d (x,z) + d (z,y) 5 d (x,y)/2 + E
x E U1
,y E
U2
,
z f V
.
,
.
(3.2) No z E V is a c u t p o i n t of any x E U1 (y E U2) a l o n g t h e minimizing g e o d e s i c j o i n i n g z w i t h x ( z w i t h y ) , and t h e a s y m p t o t i c formula (2.10) h o l d s ( o r (x,y) E (vxU2) ) . uniformly i n ( x , y ) E UlxV To i l l u s t r a t e t h e i d e a s , We r e f e r t o [ 1 4 ] f o r t h e c o n s t r u c t i o n of U1,U2,V-. c o n s i d e r t h e c a s e M = Sn = { x E Rn+': IlxlI=1} , x = (-1,O 0) , 0) Then w e can s e t y = (1,0,
...,
.
=
Ix E
u1
,...,
s"
:X
I <
01
, u2
= {x E
s"
:X
I >
01
,v
=
{xE
s"
:-1/2 <
X I
< 1/21
(Thus U1 and U2 a r e t h e upper and lower hemi-sphere, r e s p e c t i v e l y , whereas i s a band around t h e e q u a t o r . )
*
V
A f a i r l y general asymptotic representation r e s u l t i s the following
x
THEOREM 3.1: L e t , 7 , M I U 1 , U2 and V be a s above. Then t h e r e e x i s t Cm , such t h a t f u n c t i o n s v. ( x , z ) ( v . ( z , y ) ) d e f i n e d i n U1xV(VxU2) , j = 0 , 1 , .
I
u(x,y, t)
as
t+O+
,
-
J r~:~(2~t)-mvle-[d2(x,z)+d2(Z,y) 1/2t
uniformly i n
( x , y ) E U1xU2
.
.
The proof s t a r t s by o b s e r v i n g t h a t t h e semi-group p r o p e r t y of i n particular that (3.4)
..
u ( x , y , t ) = Mfu(x,z,t/2)u(z,y,t/2)dz
.
u(x,y, t ) = v/u(x,z, t / 2 ) u ( z , y , t/2)dz
+
u(x,y,t)
implies
Hence (3.5)
M , V I ~ ( ~t /, 2~) u, ( z , y , t / 2 ) d z
.
The f i r s t i n t e g r a l i n t h e r i g h t hand s i d e of (3.5) can be e v a l u a t e d a s y m p t o t i c a l l y ( a s t + 0 + ) by a p p l y i n g (2.10) v i a (3.2) t o u ( x , z , t / 2 ) and t o u ( z , y , t / 2 ) and then i n t e g r a t i n g over V , o b t a i n i n g i n t h i s f a s h i o n t h e sum i n t h e l e f t hand s i d e of ( 3 . 3 ) w i t h a n e r r o r l i k e t h e one i n t h e r i g h t hand s i d e of ( 3 . 3 ) . W e a r e t h u s l e f t w i t h t h e problem of e s t i m a t i n g t h e second i n t e g r a l i n t h e r i g h t hand s i d e of ( 3 . 5 ) . If
M
i s compact, t h e n t h e t r a n s m u t a t i o n formula (1.7) i m p l i e s t h a t
uniformly i n p,q E M t h e d e s i r e d bound. I f
,
.
f o r every E > O Combining (3.6) w i t h ( 3 . 1 ) , w e o b t a i n M i s n o t compact t h e n ( 3 . 6 j i s s t i l l v a l i d f o r p , q i n
123
PARABOLIC EQUATIONS
a compact subset of M - say, if p,q E A , where A = I z E M :d(x,z) Ia,d(z,y) I a] and a is a sufficiently large real number. On the other hand, the hyperbolic theory leading to (3.6) yields also the global L2 estimate
valid for every E > O if q ranges in a compact subset of M , and by considering A* instead of A one obtains
We now decompose the second integral in the right hand side of (3.5) into an integral over (MLV) fl A , and an integral over (M\V)\A For the integral over (MLV) n A we utilize (3.61, and for integration over (M\V)\A we use (3.7) and (3.8) along with the Cauchy-Schwarz inequality, thus absorbing those terms in the right hand side of (3.3).
.
If we have more specific information on the relationship between 2 and 7 , we can obtain more explicit information from (3.3). As an illustration, consider the case of a cut point which is not a conjugate point. (This happens for example on the circle S1 and on the torus.) In this case 7 is a regular value of exp-
.
Hence the set
(expZ)-l(y)
(finitely many) N
C
Tx(M)
unit vectors
T1
is discrete, and in particular there are
,...,TN E
T-(M)
with
7
1 6 i 1 N , (that N > 1 follows from the fact that the exponential map
exp-
V
C
ti
M
is a cut point of
is locally invertible near containing ?
exist neighborhoods U 1 , U 2
d(Z,y)Ti
-t
x,y E U1xU2
T(M)
.
, 1 1 i s N , such that Si(x,y) E Tx(M)
It follows that each pair
desic curves g(t,i,x,y) ( g(t,i,x,y) geodesics
=
x,y E U1xU2
expxtSi(x,y)/ lSi(x,y)
I),
7 ,
ii ) and
Thus there
smooth maps and
expxSi(x,y) = y
can be joined by
N
for
geo-
1 6 i C N , the functions
being smooth functions of their arguments g(t,i,x,?)
.
=
7 , respectively, and an open set
and
such that (3.1) and (3.2) are satisfied, and N
: U1xU2
exp.-(d(x,y)Ti)
t , x and y
, and the N
are all the minimizing geodesics joining X
7
with
.
Without loss of generality we may assume that for all x,y E U1xU2 , the geodesic minimizing the distance between x Set di(x,y) di(Z,y)
=
=
d(Z,y)
unit circle and
, 16 i E N
ISi(x,y)I and
d(x,y)
?=n/2
=
and y
.
is one of
We find that
g(t,i,x,y)
,
1IiIN
di(x,y) E Cm(UlxU2)
.
,
.
min di(x,y) in U1xU2 (Thus, if M is the 16iSN take dl(x,y) =y-x , d2(x,y) =a-(y-x) .)
, 7 =3n/2 , we
We want to express u(x,y,t) asymptotically (in U1XU2)by means of the functions di(x,y) For this purpose we assume (by decreasing U1 and U2 if necessary) that V contains not only the midpoints of minimizing geodesics joining x and y for x,y E U1xU2 , but all points of the form g (di(x,y)/Z,i,x,y), 1 5 i I N , i.e., all midpoints zi(x,y) of each of the N geodesics joining x and y constructed above. Under these circumstances, the only local minima of the exponent d2(x,z) +d2(z,y) in z E V are zi(x,y) , 1 6 i C N , and the value of All those minima are nonthe exponent at the i-th minimum is dZ(x,y)/2 degenerate. Applying Laplace asymptotic integral to the integral in (3.3) and noting that a multiplicative factor of the order ctnl2 is introduced, we see that there exist N families of functions v~,~(x,Y), v ~ 6 ,C (U1xU2) ~ , 1 s i 1 N , k=0,1, , such that
.
.
...
124
Y . KANNAI
as
t +O+
~-n/2
-a;(x,>
=
O(t
,
uniformly in U1xU2
i=l
Note that the functions di(x,y)
. are local solutions of the ciconal equation
Thus, the functions di(x,y) "branch" at x = z , y = j , and the formula (3.9) exhibits a "Stokes" phenomenon of the leading term passing from one di(x,y) to another one. The integral in (3.3) can be analyzed i n other cases as well, see [14]. The ndimensional integration over V can always be replaced by an (n-1)-dimensional integration (compare also [61); in many cases lower dimensional integrations suffice (thus, if 7 is a cut point which is not a conjugate point, the zero-dimensional integration (3.9) suffices), but there are cases (like Sn ) where we need all n-1 dimensions (in Sn the integral over V reduces asymptotically to an integral over the equator).
We pass now to the case (ii) of a Riemannian manifold with a non-empty boundary M If Z , y E int M and all minimizing geodesics joining x near X to y near 7 stay away from the boundary M , then the former analysis applies. New phenomena occur if either (i) X or 7 are near (or at) the boundary, or if (ii) X and 7 cannot be joined by a minimizing geodesics. Here we will discuss in detail only case (i) (for case (ii) see [14]).
.
The simplest case (analogous somewhat to the case of a cut point which is not a conjugate point) occurs if % E aM , 7 E iiit M and M is locally geodesically convex in the sense that there exist neighborhoods U1 ,U2 of f , 7 respectively, such that every two points x,y E U1,U2 can be joined by a (smooth) minimizing geodesic contained (except for possibly one end point) in int M Assume also that 7 is not a cut point of Z , and that the minimizing geodesic joining % and 7 makes a non-zero angle with aM at X Let u(x,y,t) be a fundamental solution of (l.l), where A is the zero Dirichlet data realization. Then u(x,y,t) = 0 for x E a M and all t > O , I n this case we set (as our new "branch" of a solution for the ciconal equation (3.10))
.
.
.
inf [d(x,z) +d(z,y)] E aM Then e(x,y) is a well defined smooth function in U1xU2 , e(x,y) = d(x,y) for x E aM n U1 , and e(x,y) , regarded as a function of x , is a solution of (3.10) (differing from d(x,y) i n U1 int M ) . (The function e(x,y) yields the "reflected" distance between x and y .) The characteristic curves of the nonlinear equation (3.10) with respect to the solution e(x,y) are the geodesics joining z to x , where z is the point in aM n U1 , where the infimum in (3.11) is attained. Hence transport equations can also be set with respect to e(x,y) It follows that in U1xU2 the fundamental solution for the mixed problem for the wave equation (with zero Dirichlet data) is given by (3.11)
e(x,y)
=
z
.
(3.12) where
R(x,~,s) = R~(x,Y,S) Rl(x,y,s)
-
R2(X,Y,S)
is given asymptotically by (2.6)
rewritten as
PARABOLIC EQUATIONS
125
where the functions gZ,k(x,y) are determined by means of the transport equations (corresponding to e(x,y) ) and the initial conditions g2,k(X,Y) = gl,k(X,Y) Applying the transmutation formula (1.7) to (a/as)R(x,y,s) for x E aM n U and using (3.13f and (3.14), we obtain (using a suitable finite propagation of supports argument) the asymptotic formula
.
Similar results can be obtained for the Neumann realization or the Robin realization (see [14]). If we let 7 tend to aM (in particular, if we want 7 to be near 5 ) then diffraction phenomena become important. Using a useful observation due to Seeley [191, we can say something in this case too. I n fact, the formula (3.15) still holds if (3.16) d(x,y) < C[dist(y,aM)] 112 for a certain constant C > 0 , as no reflected ray from x to y would hit the boundary once more (between x and y ) if (3.16) holds. .Hence the eiconal equation and the transport equations (relative to e(x,y) ) make sense in this case too. The recent advances in the theory of propagation of singularities (close to the boundary) for solutions of hyperbolic mixed initial boundary value problems (due to K . G . Anderson, J . Chazarin, G.I. Eskin, R. Melrose, M.E. Taylor, and others) have not led, s o far, to explicit formulas for w(x,y,s) (even in the second order case) - only micro local formulas are known. We can therefore argue only implicitly and obtain a !general formula" similar to the one obtained in Theorem 4.6 in [14]. Thus let R(x,y,s) -be a parametrix for a mixed initial boundary value problem in the sense that R(x,y,s) contains all the singularities of the fundamental solution w(x,y,s) where w(x,y,O) = 6(x,y) , (a/as)w(x,y,O) = 0 and w(x,y,s) satisfies (as a function of x , s) the boundary conditions. We assume also that the speed of light for the mixed problem is equal to one. Then $(x,y,s) = w(x,y,s)-R(x,y,s) is a C" function of all its arguments, and $(x,y,s) = -R(x,y,s) if I s 1
.
.
follows in the same way that the formula (4.28) is derived in [14]. 4.
FOURIER TRANSFORMS ANALYTIC NEAR THE ORIGIN
.
Then its Fourier transform w(t) Let w(x) be a temperate distribution on R1 is a well-defined tempered-distribution, and smoothness properties of w are reflected by the growth of w - at least as far as global smoothness is concerned. Things become more complicated when local smoothness is at issue: a distribution w may be very nice, say, at a neighborhood of the origin, but slightly singular
126
Y . KANNAI
elsewhere -and the growth properties of w will be determined by the singular behavior. For example, the function w(x) = l/(l-x) is C" (in fact analytic) in (-l,l), but its Fourier transform, when regarded as a distribution (using the principal value for the singularity) is G(t) = e-itH(t) - thus 6(t) does not even decrease as It I + a While C" behavior can be localized (by the use of cutoff functions), the situation is especially serious if we want to investigate real analyticity, as analytic functions are "rigid". Appropriate sequences of cut-off functions have been used [ 1 2 ] , but we prefer to l o o k at w and w themselves. It becomes intuitively clear, if one considers examples such as the one presented above, that the rate of oscillation of w(t) plays a crucial role in determining the analyticity of w near the origin. Recall that w(x) can be continued analytically to a strip IIm z I < a so that w(z) is bounded in that strip, if,
.
.
The oscillatory character of k(t) and only if, Iw(t)l = O(e-altl) as It1 + m (if w is analytic near the ofigin) can be expressed in terms of exponential decrease of moving averages of w , action of w on certain functions of minimal exponential type, or exponential decay of analytic continuation of (parts o f ) ;(t) One can also construct "germ analytic-hypoelliptic" "pseudo-differential'' operators.
.
We will now give a precise statement of some typical results. Recall that w is real analytic near the origin if and only if the two points (O,l),(O,-l) ( E T*(R1) \ O ) are not contained in WFA(u) , the analytic wave front set of u Hence it suffices to determine whether or not (0,l) E WFA(u) , say.
.
THEOREM 4.1:
The following statements are equivalent for w E S'(R1) :
(091) @ WFA(U) ; (ii) For every probability measure
(i)
(4.1)
u([t,m))
(4.2)
/e-ixtdu(t) = O ( / X ~ - ~ )as
exists an (4.3)
E
>O
O(e-ct)
=
t+m
supported on
,
[O,m)
such that
for a certain constant c > 0
,
1x1 + m
,
and
for a certain constant a > 0 , there
such that
Iw(t)dp(")(t)
volution power of
as
p
=
p ;
O(e-En)
as
, where
n+m
p(n)
denotes the n-th con-
wE
(iii) The functionals , defined by ;,(I$) = o/" e-Etw(t)cp(t)dt (with AG(t) modified if necessary near the origin and the integral meaning action of w on emEt(p(t) ) converge a s E + O + , for all functions cp for which there exists a sector larg z I i a such that cp is of minimal exponential type in I arg 21 < a a>o ;
,
(iv) There exist functions Gl(z) , ;2(2) such that ;1(z) ( G z ( 2 ) ) is holomorphic in the sector O < arg z < n / 2 (O>arg z>-n/2) and continuous in the halfclosed sector O6arg z < n / 2 (Oaarg z > - n / 2 ) . There exist non-negative integers n . m such that
O < a < n / 2 there exists a
(4.5)
Gl(reie) =
sector 0
f
8ia
0(2
(0 a
6>O
-6r sin 8
such that (G2(re
ie
= o(e
6r sin 0
) ) as
r+a
,
in the
e 3 -a).
Each of the conditions (ii)-(iv) is a way of expressing precisely the idea that w(t) is either exponentially decreasing, or that w(t) oscillates at least as raplidly as a trigonometric function of t Note also that, in the reverse direction, (ii) and (iv) may be weakened (thus, for example, if (4.3) holds for one u satisfying (4.1) and ( 4 . 2 ) , then (4.3) holds for all such p and (0,l) @ WFA(W) ) . (As these are proceedings of a conference in asymptotics, I
.
127
PARABOLIC EQUATIONS
venture to mention that this observation can be regarded as a Tauberian theorem.) The complete proof of Theorem 4 . 1 may be found in [ 1 5 ] (some of the arguments go back to [ 2 ] and [131). Here we shall only sketch the main highlights. The main ideas of the proof of the equivalence of (i),(ii),(iii) and (iv) (and the formulation of (iv)) simplify if we make the additional assumption that w E Ll(R1) If (i) holds, then there exists a positive number d such that the restriction of w(x) to (-2d,2d) is equal to the boundary values (in distribution sense) of a function holomorphic in {z : IzI < 2d.Im z < 01 , and let u s denote this function too by w(z) Let D1 denote the line segment {-d(1-t+it),O s t s l } directed from -d to -id , and let D2 denote the line segment {d(l-t-it) ,O i t s 11 , directed from -id to +d By Cauchy's integral theorem
.
.
.
(4.6)
Set
= ( ~ I T ) - ~ / ~ [ - ~ I - ~ + I+dIm]e-ixtw(x)dx 2 D1
k;(t)
$,(z)
=
j2(z)
=
(~IT)-'"[-~I-~ + /le-ixzw(x)dx D1
IT)-^/^[,2I+d/m]e-ixzw(~)dx
,
.
.
Then ( 4 . 4 ) holds with n = m = O The estimate ( 4 . 5 ) follows for $1 , say, once we observe that for z = reie , 0 6 E < IT/^ , and x E (-m,d) U D1 , we have Re(-ixz) s -rd min(sin 8,cos 0 ) The estimate ( 4 . 5 ) then follows by setting 6 = d min(1,cos a/sin a ) Let (iv) hold, with n = m = O , and let Q ( Z ) be holomorphic in a certain sector zI ia such tha,t for every X > 0 there exists a
.
.
< m w:th lcp(z) I 6 for z in the sector. Then the integral defining w,(cp) (in (iii)) converges classically. Using Cauchy's integral theorem along with the estimates on ~ ( D ( z I) and the estimates ( 4 . 5 ) , weAsee that w,(Q) is equal to the sum of two integrals, one that of the function v~(z)e-~~~(z) integrated on the ray arg z = a / 2 , and the other that of vZ(z)e-EZq(z) , integrated over arg z = - a / 2 On those lines the integrals converge by Lebesgue's dominated convergence theorem as E + O + , and (iii) follows. Note that
K(A)
.
the appearance of the operator
tn(
in ( 4 . 4 ) is due to the fact that if
l-$)m
w k? L1(R1) then the well-known structure theory for temperate distributions implies the existence of a function v E Ll(R1) and two non-negative integers n , m such that
and clearly
(0,l) E WFA(w)
if and only if
(0,l) E WFA(v)
.
The condition (iii) implies, by the principle of uniform boundedness, that the limit functional lim wE may be defined as a continuous functional on the space of holomorphic functions cp in a certain sector larg zI 6 a satisfying there the estimate cp(z) = O(eXIZI) in that sector. I n particular the limit operates on functions of the form z -t eiXz if 1x1 is sufficiently small. In this fashion the'inverse Fourier transorm of the restriction of w to the positive axis is extended to a holomorphic function in an open neighborhood of the origin. Setting g(z) (4.7)
-m
=
/e-iztdp(t)
Im w(x)gn(x)dx
and using Parseval's formula, we may rewrite ( 4 . 3 ) as
= O(e-En)
as
n+m
.
If (i) is true, then ( 4 . 1 ) implies that g(z) is holomorphic in the half-plane Im z < c , and ( 4 . 2 ) implies that ( 4 . 7 ) is equivalent to the exponential decrease
.
-6/6 w(x)gn(x)dx for any 6 > O But g ( 0 ) = 1 while Ig(z) I < 1 for z # O , Im z S O Deforming the integration path from the real interval [-6,6] to the circular arc z = 6eie , IT 6 8 s ZIT , we obtain ( 4 . 7 ) . (Strictly speaking we did it
of
.
128
Y . KANNAI
only under additional integrability assumptions on w , but it is a simple matter to remove those assumptions.) If (ii) holds, then (under additional assumptions of local integrability of w ) not only does ( 4 . 7 ) hold, but also (4.8)
w(x)gn(x)dx
=
.
O(e-€")
as
n+m
for all 6 > O Multiplying ( 4 . 8 ) by einz , summing and interchanging the integration and the summation, we see that the function
is holomorphic in a full neighborhood of the origin. Noting the similarity between ; ( z ) and a Hilbert transform of w , we can deduce (i) from the analyticity of (In [ 1 3 ] the authors multiply ( 4 . 8 ) by g-"(z) to obtain the holomorphic function
w.
A similar reasoning enables us to prove the following result.
THEOREM 4 . 2 :
The operator P : E'(R')
+
D'(R')
defined by
maps functions which are real analytic near the origin into functions holomorphic in a horizontal strip IIm 21 i 6 for some 6 > 0 , and if Pu is real analytic near the origin then u is also real analytic near the origin. Clearly (Pu)(x) is real analytic except perhaps at x = 0 (getting "more" analytic as 1x1 gets larger). Inserting the definition G(5) = (21~)l/~u(e-~~E) in (.4 . 9 ), we find that 2 2 (PU)(X) = u+ (x+ix ) - u-(x-ix ) where u+(x) = eixS;(C)dS , u-(x) = -m /O e-ix5u(S)dS , and the theorem follows upon noting that the maps invertible near the origin. Recall [lo] that the operator (1+x2(5()-*
.
for all n
l+x2151
z
+
z+iz
L
and
z
+
z-iz
L
are
is also 3 P Soelliptic, l is the hence pointwise limit of such so
is The "symbol" e-
symbols. But we were unable to find any class of pseudo-differential operators in which 5.
e - x 2 1 C 1 is invertible. Note that for x # O
this symbol is in S
-m
.
OFF DIAGONAL BEHAVIOR OF SPECTRAL FUNCTIONS OF ELLIPTIC OPERATORS
In the present section we assume, in addition to the assumptions made in Sectionl, that the elliptic operator A is defined on a real analytic manifold and has analytic coefficients. Then the kernel w(x,y,s) is not only smooth for x # y and s small, but is actually real analytic there. This fact follows from the analyticity of the initial data (1.6) and the theory of propagation of analytic singularities and wave front sets for solutions of the pseudo-differential equation (1.5), a theory which should be analogous to that for C" singularities [ 5 ] . Alternatively, observe that w(x,y,s) is a solution of the differential eauation 1 a mw (5.1) AW = (--) i as
.
129
PARABOLIC EQUATIONS
Denote by
WFf(u)
the analytic wave front set of
Then the microlocal character of
Allm
, WF!(u)
u E D'(M)
implies that
WF!(A1/m~)
.
E T*(M)\O
c WF:(u)
Applying this successively to (a/as)jw(*,y,O) (yfixed), we see that if x # y By Holmgren then (a/as)Jw(x,y,O) is real analytic, as a function of x uniqueness theorem and Cauchy-Kowalewsky theorem, w(x,y,s) is analytic in (x,s) for x # y and Is( sufficiently small. We may now apply the theory of propagation of analytic singularities (in MxRl ) for solutions of the differential equation (5.1) with analytic coefficients and simple characteristics [12], to conclude that w(x,y,s) is analytic in x , s if x f y and Is1 is s o small, that no null bicharacteristic of A1lm-Ds issued from (y,O,S,.r) ( ET*(MxR1) ) passes over (x,s) (This argument is due to L. H6rmander; note that we need the theory of propagation of singularities only in order to know how far we can go in the s direction before losing analyticity, as the mere analyticity near s = O follows from Cauchy-Kowalewsky and Holmgren.) It follows from (1.4) that the equivalent conditions (ii),(iii) and (iv) of Theorem 4.1 hold for de(x,y;Xm) Picking e.g. (ii), we get:
.
.
.
THEOREM 5.1: Let be a probability measure supported on the conditions (4.1) and (4.2). Then there exists an E > O
(5.2)
/e(x,y;Xm)dU(")(X)
=
/e(x,y;X)du(n)(X1/m)
=
O(e-En)
[O,") and satisfying such that
as
n+-
.
noted i n Section 4, formulas such as (5.2) (or the other statements (iii) and (iv) of Theorem 4.1) imply that, as a function of X , e(x,y;h) is either very small for large X , or else that e(x,y;A) is highly oscillatory. For example, if M = R ~and A = (l/i)(d/dx)m , then
As
Note that we have used here only one direction of Theorem 4.1 (that (i) implies (ii)), but it follows from the other direction ((ii) implies (i)) that we cannot get from the analyticity of w(x,y,s) more information than that contained in (5.2). Note also that in the second order case, discussed in Section 2, we have used the fact that w(x,y,s) = 0 for Is1 < d(x,y) (actually we used the vanishing of the even part of w ) . Thus the Fourier transform (1.4) has a gap, or de(x,y,h2) possesses a spectral gap. It i s well-known [ Z O ] that a s ectral gap implies that (the even extension to X < O of ) the measure de(x,y,h ) is highly oscillating. Here we use the weaker "analytic gap".
5
Weaker oscillation results (rapid decrease instead of exponential decrease) can be obtained if the coefficients of A are only C" (see [151).
6. OPEN PROBLEMS I n this section we suggest certain possible directions for further research along
the lines outlined in the preceding sections.
6.1. Small parameters. Consider the function u(x,y,t;~) fundamental solution of the differential equation
defined as the
The asymptotic behavior of u(x,y,t;~) , when both t and E tend to 0, t , € > O , has been studied via probabilistic methods [16]; to the best of my knowledge the problem has not been approached in a purely analytic fashion. The transmutation method requires knowledge of the fundamental solution of the hyperbolic equations
130
Y. KANNAI
Singularly perterbed hyperbolic equations, such as (6.2), were not investigated ( s o far) in detail in the literature. 6.2. Degenerate equations. What happens if we remove the assumption that A elliptic and require instead only that Zy,j=l aij(x)SiSj
(6.3)
>, 0
for all
x E M
,5
is
,
E Rn
(i.e. we replace a positive definite form by a positive semi-definite one)? Using probabilistic methods, it was shown [7] that the operator still exists in a certain sense, and in several cases information can be obtained on the kernels u(x,y,t) . An approach like the one taken in Section 2 of the present paper necessitates the study of the degenerate hyperbolic equation (2.3), where we assume only that the principal part of a(x,a/ax) satisfies (6.3). 6.3. Local analyticity o f Fourier transforms in several variables. No fully satisfactory extension of Theorems 4.1 and 4.2 to n dimensions have been found till now. If we take for p in (4.3) a Gaussian measure (this case is not covered precisely by Theorem 4.1(ii), but the difference is not crucial, see Remark 3.3 in [15]), then (4.7) reads - m r m e - i x ~ - xw(x)dx 2~ = O(e -€') as 5 - t - , (6.4) and the existence of E > 0 such that (6.4) holds is equivalent to (0,l)fWFA(w). For n-dimensions, it was essentially proved in [2] that ( 0 , t ) f WFA(W) if and only if ther.e exists a positive E such that
.
.
in the set O S p L X Here < , > denotes the inner product in Rn One of the difficulties in extending part (ii) of Theorem 4.1 to n-dimensions lies in the fact that behavior of the Hilbert transform differs makedly, depending on whether or not n = l Thus, let w(z) be holomorphic in Im z > O and let w(z) have sufficiently nice boundary values on the real line. Then the origin is a removable singularity for the function w(z) if and only if the origin is a removable singularity for the function w+(z) defined by
.
w+(z)
=
w(x)dx !7 , Im z > 0
.
But in two variables, there exists a function w(zl,z2) i = 1 , 2 , such that the function
holomorphic in
might be extended analytically from Im zl,Im 22 > 0 to being a removable singularity for w where ul (z l ) u2(z2)
.
Im z i >0,
C2 without the origin
For example, let w(zl,z2)
=
u1(z1)u2(z2),
is holomorphic in Im z l > 0 and has a singularity at
zl=O
, and
is an entire function such that u2 (XI
r---x- z
dx : 0 for
r=l .
Im z > O
.
Note that the symbol of the pseudo-differential operator mapping w equal to if n 2 2
H(Ei)
to w+
If n = 1 then this symbol is smooth in T*(R1) ' 0
then the symbol is discontinuous in the set
U~=l{S : 5,
= 01
.
is
, but
Note that
131
PARABOLIC EQUATIONS
the symbol is (micro) elliptic in the conic set get analytic hypoellipticity in this set.
‘
{ S : S i > O , lCiiln}
Similarly, it is not known whether the operator P (Pu)(x)
=
Rn/e
i<x,S>-<x,x> 5
,
, and we can
defined by
‘j(S)dt
mapps only functions which are holomorphic near the origin into such functions. ix 5 - x ? ~ j does have this property.) e j j (It is clear that the symbol
n;=l
’
REFERENCES Babich, V.M. and Rapaport, J u . O . , Short time asymptotic behavior of the fundamental solution of the Cauchy problem for a second order parabolic equation. (Russian) Problems in Math. Phys. 7 (Izdat. Leningrad Univ., Leningrad, 1974) 21-31. Bros, J. and Iagolnitzer, D., Local analytic structure of distributions I Generalized Fourier transformations and essential supports, reported by D. Iagolnitzer in Hyperfunctions and Theoretical Physics, Lecture Notes in Math. 449 (1975) 121-132. Cohen, J.K. and Lewis, R.M., A ray method fot the asymptotic solution of the diffusion equation, J. Inst. Math. App. 3 (1967) 266-290. Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. 11 (Interscience, New York, 1962). Duistermaat, J.J. and HGrmander,L., Fourier integral operators 11, Acta Math. 128 (1972), 183-269. Duistermaat, J.J. and Guillemin, V.W., The spectra of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975) 39-79. Friedman, A., Stochastic Differential Equations and Applications, Vols. I and I1 (Academic Press, New York, 1975). Hadamard, J . , Lectures on Cauchy’s problem in linear partial differential equations (Dover, New York, 1952). Hersh, R., The’method of transmutation, Partial Differential Equations and Related Topics, Lecture Notes i n Math. 446 (1975) 264-282. HGrmander, L., Pseudo-differential operators and hypoelliptic equations, Amer. Math. SOC. Symp. Pure Math. 10 (1966) Singular integral operators, 138-183. HGrmander, L., Fourier integral operators I, Acta Math. 127 (1971) 79-183. II8rmander, L., Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971) 671-704. Iagolnitzer, D. and Stapp, H.P., Macroscopic causality and physical region analyticity in S-matrix theory, Comm. Math. Phys. 14 (1969) 15-55.
1 Kannai, Y., Off diagonal short time asymptotics for fundamental solutions of diffusion equations, Comm. P.D.E.
2 (1977) 781-830.
132
Y . KANNAI
[15] Kannai, Y., Local analyticity of Fourier transforms, J. D'Analyse Math. 34 (1978) 162-193. [16] Kifer, Ju.I., On the asymptotic behavior of transition densities of processes with small diffusions, Theor. Probability Appl. 2 (1976) 513-522. [17] Maslov, V.P., Theorie des Pertubations et Methodes Asymptotiques (DunordGauthier-Villars, Paris, 1972). [18] Molchanov, S.A., Diffusion processes and Riemannian geometry, Russian Math. Surveys 30 (1975) No.1, 1-63. [19] Seeley, R., A sharp asymptotic remainder estimate f o r the eigenvalues of the Laplacian in a domain of R3, Advances in Math. 29 (1978) 244-269. [20] Shapiro, H.S., Functions with a spectral gap, Bull. h e r . Math. SOC. 79 (1973) 355-360.
A N A L Y T I C A L AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS Ih' A N A L S S I S S . A x r l s s o n , L . S . F r a n k , A . v a n d c r Sluis ( e o s . ) d North-Hul l a n d P u b l i s h i n g C o m p a n y , 1981
DIFFERENCE APPROXIMATION FOR A SINGULAR PERTURBATION PROBLEM WITH TURNING POINTS Bruce Kellogg
R.
*
I J n i v e r s i t y of Maryland C o l l e g e P a r k , Maryland
The numerical s o l u t i o n of t h e sing111.ar p e r t u r b a t i o n problem pu' + qu = f , -1 < x < 1 , w i t h u ( + l ) g i v e n , i s cons i d e r e d . We suppose t h a t q ( x ) > 0 , and t h a t p ( x ) v a n i s h e s a t a f i n i t e number of p o i n t s , t h e s t a g n a n t p o i n t s . C e r t a i n o t h e r h y p o t h e s e s a r e made. An e r r o r e s t i m a t e i s g i v e n f o r t h e Southwell-Allen e x p o n e n t i a l d i f f e r e n c e scheme € o r t h i s problem. The e r r o r e s t i m a t e i s uniform i n E
-EU"
+
.
INTRODUCTION
We a r e i n t e r e s t e d i n d e v e l o p i n g n u m e r i c a l schemes f o r s o l v i n g s i n g u l a r p e r t u r b a t i o n problems which do n o t r e q u i r e mesh r e f i n e m e n t s n e a r t h e boundary l a y e r . Such methods have been d e v i s e d f o r one d i m e n s i o n a l s i n g u l a r p e r t u r b a t i o n problems witho u t t u r n i n g p o i n t s [1,2]. I n t h i s paper we a p p l y one such scheme, t h e SouthwellA l l e n d i f f e r e n c e scheme, t o a problem w i t h t u r n i n g p o i n t s . We s t u d y t h e d i s c r e t i z a t i o n e r r o r a s s o c i a t e d w i t h t h e a p p r o x i m a t i o n , Under c e r t a i n assumptions we o b t a i n a n e r r o r e s t i m a t e t h a t i s u n i f o r m l y v a l i d f o r a l l E €(0,1]. Our p r o o f s , a s i n [l], u s e ,comparison f u n c t i o n s and t h e maximum p r i n c i p l e . Very l i k e l y our r e s u l t s a r e n o t b e s t p o s s i b l e , e i t h e r a s r e g a r d s t h e r a t e of convergence i n t h e e r r o r e s t i m a t e , o r t h e h y p o t h e s e s t h a t we make.
We c o n s i d e r t h e boundary v a l u e problem -Eu"
+
pu'
+
qu
given
u(+l)
(1.1)
,
f
=
-1 < x < 1 ,
.
We suppose t h a t p ( x ) , q ( x ) , f ( x ) a r e smooth f u n c t i o n s i n -1 < x < 1 , and t h a t W e a l s o suppose t h a t p(+l)-# 0-, and t h a t E > 0 and q ( x ) > 0 , -1 5 x 5 1 p(x) v a n i s h e s o n l y a t a f i n i t e number of p o i n t s i n (-1,l) We c a l l t h e z e r o s of p ( x ) s t a g n a n t p o i n t s of t h e problem. I f p(x*) = 0 , we suppose t h a t We s a y t h a t x* i s a n a t t r a c t i v e ( r e p u l s i v e ) s t a g n a n t p o i n t i f p'(x*) # 0 p'(x*) < 0 (p'(x*) > 0 )
.
.
.
.
I t i s n o t h a r d t o s e e t h a t t h e r e i s a unique s o l u t i o n of t h e problem (1.1). Physic a l l y , u ( x ) may r e p r e s e n t t h e c o n c e n t r a t i o n of a s u b s t a n c e which i s d i f f u s i n g and c o n v e c t i n g i n a f l u i d . With t h i s i n t e r p r e t a t i o n , E i s t h e d i f f u s i o n c o e f f i c i e n t , p ( x ) i s t h e f l u i d v e l o c i t y , f ( x ) i s a n e x t e r n a l s o u r c e of t h e s u b s t a n c e , and qu r e p r e s e n t s a l o s s t e r m . Our i n t e r e s t i n t h e problem i s when process. may b e shown t h a t a s s o l u t i o n u of t h e reduced problem
It
*
E E
i s s m a l l and c o n v e c t i o n dominates t h e flow 0 , t h e s o l u t i o n u converges t o t h e
-+
Work s u p p o r t e d i n p a r t by t h e N a t i o n a l I n s t i t u t e s of H e a l t h . I33
134
R.B.
pu'
+
$(-l) ;(1)
=
KELLOGG
qu = f
- l < x < l
= u(-l)
if
p(-1)
u(1)
if
p(1) < 0
> 0
.
The problem ( 1 . 2 ) c o n s i s t s of a f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n , and 0 , 1, o r 2 boundary c o n d i t i o n s , depending on t h e s i g n of p ( + l ) N e v e r t h e l e s s , i t may b e shown t h a t ( 1 . 2 ) h a s a unique c o n t i n u o u s s o l u t i o n .
.
PROPERTIES OF THE SOLUTION For our e r r o r a n a l y s i s we r e q u i r e e s t i m a t e s f o r t h e d e r i v a t i v e s of t h e s o l u t i o n of (1.1) t h a t a r e uniform i n c One may show t h a t a t each p o i n t x € (-1,l) such t h a t p ( x ) # 0 , t h e d e r i v a t i v e s of u s a t i s f y
.
I u ( ~ ) ( x ) I5 c ( x , k )
(2.1)
u
.
where c ( x , k ) depends on p. q , and f , b u t does n o t depend on E The p o s s i b l e p o i n t s of d i f f i c u l t y a r e t h e end p o i n t s , x = +1 , and t h e s t a g n a n t p o i n t s . I f p ( 1 ) < 0 , t h e r e i s a boundary c o n d i t i o n a t x = 1 f o r t h e reduced problem, and i t may be shown t h a t ( 2 . 1 ) h o l d s a t x = 1 . . I n t h i s c a s e , t h e r e i s no boundary I f p ( 1 ) > 0 , t h e arguments i n [l] g i v e a d e q u a t e i n f o r m a t i o n layer a t x = 1 t o bound t h e d e r i v a t i v e s . A similar t r e a t m e n t may b e given a t x = 0 I t remains t o c o n s i d e r t h e s t a g n a n t p o i n t s . I f x* i s an a t t r a c t i v e s t a g n a n t p o i n t , i t may a g a i n b e shown t h a t ( 2 . 1 ) h o l d s a t x = x* I f x* i s a r e p u l s i v e s t a g We s h a l l assume t h a t n a n t p o i n t , we s e t a = - p ' ( x * ) / q ( x * ) > 0 , and B = l/a
.
.
.
.
a > 1 f o r each r e p u l s i v e s t a g n a n t p o i n t
(2.2)
.
We then have Lemma 1. Let x* be a r e p u l s i v e s t a g n a n t p o i n t . c o n t a i n i n g X* and c o n s t a n t s Ck > 0 , k = 1 , 2 ,
There i s an open i n t e r v a l I t h a t , i f u solves(1.2),
..., such
We s k e t c h t h e i d e a s i n t h e proof of t h i s lemma. Suppose, f o r s i m p l i c i t y , t h a t t h e s t a g n a n t p o i n t i s a t x = 0 . The dominant b e h a v i o r of t h e s o l u t i o n n e a r x = 0 i s given by t h e s o l u t i o n of t h e homogeneous e q u a t i o n -Ez"
- p'(0)xz'
+
q(0)z = 0
o r , a f t e r d i v i d i n g through by
q(o), and r e d e f i n i n g
(2.3)
-62''
- axz'
+
E
,
z = 0 ,
.
Suppose z ( x ) solves ( 2 . 3 ) i n ( - l , l ) , w i t h z ( t 1 ) g i v e n independent of E Set t = x ( a / ~ ) l / ,~ z ( x ) = Z ( t ) Then Z ( t ) = e x p ( - 1 / 4 t 2 ) z ( t ) , where z i s a 1/2 We a l s o have z ( + ( a / ~ ) l / ~= ) p a r a b o l i c c y l i n d e r f u n c t i o n of o r d e r R i(tl) exp ( h a / € ) R e p r e s e n t i n g z ( t ) i n terms of t h e s o b a s i c p a r a b o l i c c y l i n d e r f u n c t i o n s , a s given i n [ 3 , 19.31 and u s i n g t h e p r o p e r t i e s d e s c r i b e d t h e r e , we a r e led to the desired inequality.
.
.
+
.
THE DIFFERENCE APPROXIMATION
Let N be a p o s i t i v e i n t e g e r , and l e t t h e r e be g i v e n a uniformly spaced c o l l e c t i o n of mesh p o i n t s on [-1,1], w i t h mesh s p a c i n g h = N - l The mesh p o i n t s a r e We u s e t h e d i f f e r e n c e o p e r a t o r s D ' u i = ( u i + l - u i - l ) / Z h , D 2 U i i h , -N 5 i 5 N u i - l ) / h 2 , and we d e f i n e y ( t ) = t c o t h t . We s h a l l c o n s i d e r (ui+l - 2ui t h e Southwell-Allen d i f f e r e n c e o p e r a t o r Lh , d e f i n e d by,
+
.
.
135
SINGULAR PERTURBATION PROBLEM
where p i = p ( i h ) , q i = q ( i h ) , e t c . The o p e r a t o r Lh i s of p o s i t i v e t y p e , i n t h e f o l l o w i n g s e n s e : i f U and V a r e 2 mesh f u n c t i o n s such t h a t U+N 5 V ~ N, and LhUi t h e n U i LVi f o r -N < i < N sequence, t h e mesh f u n c t i o n U i
5 LhVi ,
.
-
N ,
-N < i
(The proof of t h i s i s given i n [l].) d e f i n e d by
A s a con-
(3.1) h a s a unique s o l u t i o n . We a r e i n t e r e s t e d i n e s t i m a t i n g t h e e r r o r e i = ui - Ui , where u i s t h e s o l u t i o n t o (1.1) and U i s t h e s o l u t i o n t o (3.1). For t h i s , w e s h a l l e s t i m a t e t h e t r u n c a t i o n e r r o r ~i = Lh(u-U)i = Lhui - f i , and we s h a l l i n t r o d u c e comparison f u n c t i o n s t o c o n v e r t e s t i m a t e s f o r ~i i n t o e s t i m a t e s f o r ei * To e s t i m a t e ~i we u s e t h e f o l l o w i n g i n e q u a l i t i e s , which s h a r p e n t h e correspondi n g i n e q u a l i t y i n [l, Lemma 3.31:
x. X.
X.
1- 1
) I u " ( t ) Idt
1-1
1-1
xi+l ( X i + p ) lu"(t) d t ;
+ ch-llPilj xi
xi+l
We s h a l l u s e Lemma 1 t o e s t i m a t e ~i n e a r a r e p u l s i v e s t a g n a n t p o i n t . s t a t e an e a s i l y proved i n e q u a l i t y . Lemma 2 .
Let
u > 0 , and l e t
.
g ( x ) = (x+l);'
I
Ig(x+h) - g(x-h) Ig(x)-ll
Then f o r some
,
< ch(x+l)-'-'
5 c(x+xbg(x) ,
c > 0
We f i r s t
,
o.h5x/2, x > 0
.
To o b t a i n s u i t a b l e bounds on t h e t r u n c a t i o n e r r o r , we a l s o must assume:
(3.4)
each r e p u l s i v e s t a g n a n t p o i n t i s a l s o a mesh p o i n t .
Using t h i s assumption we have Lemma 3 . L e t x* b e a r e p u l s i v e s t a g n a n t p o i n t . There i s a n i n t e r v a l I cont a i n i n g x* and a con t a n t c > 0 independent of h and E s u c h t h a t , i f x i E I , and i f d = , and z i = \xi-x*l ,
(3.5)
(3.6)
bil I c
h6+h2-BdB + c z (h+6)2
'
{h6'+h1+' (h+6)2
+
(h+6)2-B
1
,
zi = 0 , h
.
R.B. KELLOGG
136
~
Proof.
zi
If
2 2h , we
u s e Lemma 1 and Lemma 2 t o e s t i m a t e xi+h zi+h
5
Iu"'(t)Idt
E
~5
(t+A)8-3dt
x.-h
z .-h
< c5 2 [(zi-h+6)B-2 - (zi+h+5)@-']
-
=
2 .
c@[(1+
5
2.
1-hp-2 -
($+1+h)B-2 5 1
6
< c6B -
. 1?6 .
2.
(++ 1)@-3
< ch6 2 (z~+LF)'-~
-
.
This bounds t h e f i r s t term on t h e r i g h t s i d e of ( 3 . 2 ) . ed s i m i l a r l y , g i v i n g ( 3 . 5 ) . Suppose now t h a t
xi =
X*
+
h
.
The second term i s t r e a t -
Then we have
xi+l l u " ' ( t ) l d t 5 c5
E
( t + ~ S ) ~ - ' d t< c(h6+h2-B6B)(h+6)B-2
,
X. 1- 1
0
'i-1
.
+
which proves ( 3 . 6 ) , i n t h e c a s e xi = x* h I f x = x* - h , a s i m i l a r argument a p p l i e s . I f xi = x* , pi = 0 , and t h e remainfng term i n ( 3 . 2 ) i s e s t i mated a s above, f i n i s h i n g t h e proof of t h e lemma.
A COMPARISON FUNCTION By a comparison f u n c t i o n we mean a mesh f u n c t i o n V such t h a t L Vi > 0 , -N < These f u n c t i o n s a r e u s e d , t o g e t h e r w i t h tke maximum p r i n i < N , and VtN > 0 It is evident t h a t the function c i p l e , t o c o n v e r t bounds on T t o bounds on e E d e f i n e d by E . 3 1 i s a comparison f u n c t i o n . I n [l] w e c o n s t r u c t a comparison f u n c t i o n t o handle t h e boundary l a y e r . I n t h i s s e c t i o n w e g i v e a comparison function t o handle t h e e r r o r a t a repulsive stagnant point.
.
.
Suppose, f o r s i m p l i c i t y , t h a t x* = 0 i s a r e p u l s i v e s t a g n a n t p o i n t , a n d l e t a > 1 be t h e number a s s o c i a t e d w i t h t h i s p o i n t . L e t a > a , B = l/a < B , and let P > 0 Set -
.
S ( x ) = (1x1
+
.
& @
We s h a l l c o n s t r u c t o u r comparison f u n c t i o n u s i n g t h e mesh f u n c t i o n
We s h a l l assume t h a t (4.1)
a < a*
=
1.035
.
We c o n j e c t u r e t h a t t h i s assumption i s n o t e s s e n t i a l .
Si = S(xi)
,
SINGULAR PERTURBATION PROBLEM Lemma 4.
For
xi
0
i n a neighborhood of
(4.2)
xi # 0 ,
and
’-’
-
I I
5
LhSi
,
c1 (a xi h+c) ( X ~ + E ’ ] )
0 , /Lhsi/
For x i o u t s i d e any neighborhood of a r e independent of E and h
.
137
.
5 c2
. The c o n s t a n t s
c 1 , c2
.
P r o o f . We s h a l l prove ( 4 . 2 ) f o r x > 0 Suppose, w i t h o u t l o s s of g e n e r a l i t y , t h a t q ( 0 ) = 1 . S i n c e S ( 4 ) ( x ) < 0 , we have from [1,(3.7)],
.
D2S(xi) < S ” ( x i ) Since
y(t)
,
c3(l+ltl)
with
c3
.654 , t h i s g i v e s
=
To h a n d l e t h e f i r s t d e r i v a t i v e term, w e s t a r t w i t h t h e i n e q u a l i t y
(l+z)B - (1-2)B
< -
202
where
+
c 22,
0 <
4
-
-
-
z = ~/(x.+E’)
We r e c a l l t h a t t o 0 , -piB/x
Since
, we
-piD’Si
(4.4)
c4 1) = 0
-pi
< 1
,
> 0
.
arrive a t the inequality
x piBSi
5 -1
-
B*
We s e t
= .965
1-1 = 2/(2-8)
!j c4pih
:(
-
x + E ’ ) ~ ~. ~
.
cZ(1) = -.6137, w e s e e t h a t t h e r e i s a -
suffices.
,
’0
,
B”5Bil
This proves ( 4 . 2 ) .
and u s e t h e i n e q u a l i t y (x+e’)B-2 2 c ( x e-z+€z)-i
Then we may f i n d c o n s t a n t s (4.5)
=
,
f o r x i > 0 , xi n e a r 0 For x i s u f f i c i e n t l y c l o s e Using t h i s , and combining ( 4 . 3 ) and ( 4 . 4 ) , we o b t a i n
B(l-B)c3 - C 4 G )
In f a c t , l y proved
< 1
.
c 4 = c4(B) = 28 - 28 Setting
2
c5 > 0
Lh(S+cgE)i
B* < 1 s u c h t h a t
.
The rest of t h e lemma i s r e a d i -
.
such t h a t
2
I I
c 6 ( a xi h+E )
2-B+€2
+ c 7 , . i # 0 .
X.
This i n e q u a l i t y g i v e s t h e d e s i r e d comparison f u n c t i o n t o h a n d l e t h e r e p u l s i v e s t a g nant point. ERROR ESTIMATE
We now s k e t c h t h e d e r i v a t i o n of uniform e r r o r e s t i m a t e s f o r o u r d i f f e r e n c e approximation. For t h i s , we c o n s t r u c t a comparison f u n c t i o n V , and we show t h a t t h e r e I ~ i l5 Ch’LhVi From t h i s and t h e a r e p o s i t i v e c o n s t a n t s 0 and c such t h a t l e i 1 5 cheVi , which i s f a c t t h a t Lh i s of p o s i t i v e t y p e , i t t h e n f o l l o w s t h a t t h e d e s i r e d e r r o r e s t i m a t e . The comparison f u n c t i o n i s chosen t o be a p o s i t i v e l i n e a r combination of t h e f u n c t i o n s S , one f o r each a t t r a c t i v e s t a g n a n t p o i n t , t h e f u n c t i o n E , and ( i f n e c e s s a r y ) t h e comparison f u n c t i o n s used i n [l] t o hand l e t h e boundary l a y e r . I n t h i s p a p e r , w e d i s c u s s o n l y t h e e r r o r a r i s i n g from a r e p u l s i v e s t a g n a n t p o i n t , a s t h e e r r o r a r i s i n g from t h e bounda’ry l a y e r i s known t o be O(h)
.
.
138
R.B. KELLOGG
To o b t a i n o u r e r r o r e s t i m a t e s , w e s h a l l r e q u i r e an i n e q u a l i t y r e l a t i n g geometric programming and l i n e a r programming. Lemma 5. L e t Bi 5 0 and e i j 2 0 , b e g i v e n , where Suppose t h e s e t of l i n e a r i n e q u a l i t i e s
2
n
, 15
j
5m
.
lziLrn,
y a e . < e i ,
(5.1)
15 i
j = l j i~ -
m
has a solution.
For p o s i t i v e q u a n t i t i e s
Hence, s e t t i n g
,
wi
w.= J
t o prove ( 5 . 3 ) , i t s u f f i c e s t o prove n Bi n w.1 i= 1
Since
kn
and numbers
n
n
< c
cii
satisfying (5.2),
eij
w i=l i
-
w i E (O,l),
.
1
(5.3)
Proof.
Then t h e r e i s a c o n s t a n t c > 0 such t h a t f o r a l l n 8. m n 8.. n Wil 5 c n WilJ i=l y = l i=l
’
m
ci.
.
n w J j
j=1
i s a monotone f u n c t i o n , i t s u f f i c e s t o p r o v e , f o r some n m eiLn wi 5 a.knW. + c = a j e i j e n wi , -i=l j=1 J J i,j
1
1
1
or C(ei-
i
Since
kn wi
5 0 , t h i s inequality
l a j e i j ) k n wi j
5c
.
i s i m p l i e d by ( 5 . 1 ) , proving t h e lemma.
We now s t a t e Lemma 6 . L e t x* be an a t t r a c t i v e s t a g n a n t p o i n t . Then t h e r e a r e c o n s t a n t s 8 > 0 and c > 0 , independent of h and E , such t h a t f o r a l l x i i n a neighborhood of x*, x i # x*, 5 Ch%h(SfE)i The c o n s t a n t 6 may be t a k e n t o be any number < ( 1 / 2 ) 6
.
ITI~
.
The proof r e q u i r e s a d e t a i l e d examination of t h e bounds f o r -ii Lemma 3 . I n t h i s , Lemma 5 i s a u s e f u l t o o l . The r e s t r i c t i o n 0
,
a s g i v e n by
(5.4) 2 t o s o l v e t h e v a r i o u s systems of l i n e a r i n e q u a l i t i e s t h a t a r i s e . A t the stagnant point, Our a n a l y s i s i s complete, e x c e p t f o r t h e p o i n t x i = x* LhS < 0 , One c a n , however, show t h a t , a t t h e s t a g n a n t p o i n t x i = x* ,
.
The proof of t h e l a t t e r i n e q u a l i t y may be accomplished, f o r example, w i t h t h e h e l p of Lemma 5.
139
SINGULAR PERTURBATION PROBLEM
We now i n t r o d u c e t h e d i s c r e t e Green’s f u n c t i o n i n a neighborhood of t h e a t t r a c t i v e s t a g n a n t p o i n t x* . Suppose t h e s t a g n a n t p o i n t i s a t x* = 0 . We d e f i n e a mesh f u n c t i o n G by 1, i = O (LhG)i = O , - n < i < n , i s 0 , and
G+,
-
=
0
.
Here
x +n -
a r e f i x e d mesh p o i n t s n e a r
0
.
I t may he shown t h a t
.
where c 1 and c2 are i n d e endent of E T h i s i n e q u a l i t y comes from making t h e change of v a r i a b l e s t = X / E l 2 , and s t u d y i n g t h e d i f f e r e n c e e q u a t i o n on t h e new mesh.
P
We u s e , i n a neighborhood of t h e s t a g n a n t p o i n t , t h e comparison f u n c t i o n
From t h e p r e c e d i n g , we s e e t h a t f o r c Hence, f o r i h , I T I i 5 Eh’LhVi
+o
.
l a r g e enough, h u t independent of
,
c
and
Thus, we have s k e t c h e d t h e proof of t h e f o l l o w i n g theorem Theorem. Suppose each r e p u l s i v e s t a g n a n t p o i n t s a t i s f i e s (2.2), ( 3 . 3 ) and ( 4 . 1 ) . L e t 0 < 1 / 2 min B , where t h e minimum i s t a k e n o v e r a l l r e p u l s i v e s t a g n a n t p o i n t s . Then /e. where
c > 0
I
5
ch
0
,
xi
i s independent of
# a repulsive stagnant point, h
and
E
.
REFERENCES [l]
Kellogg, R.B. and Tsan, A . , A n a l y s i s of some d i f f e r e n c e a p p r o x i m a t i o n s f o r a s i n g u l a r p e r t u r b a t i o n problem w i t h o u t t u r n i n g p o i n t s , Math. Comp. 32 (1978) 1025-1039.
[2]
Berger, A . E . , Solomon, J . M . , Ciment, M . , , L e v e n t h a l , S.H. and Weinberg, B . C . , G e n e r a l i z e d O C I schemes f o r boundary l a y e r problems, t o a p p e a r i n Math. Comp.
[ 3 ] Abramowitz, M. and Stegun, I . A . ,
Handbook of Mathematical F u n c t i o n s (U.S. Government P r i n t i n g O f f i c e , Washington, D.C. 1 9 6 4 ) .
This Page Intentionally Left Blank
A N A L Y T I C A L AND NLI.YZRICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S . A x e l s s o n , L . S . F r a n k , A . van der S l u i s (eds.) @ N o r t h - H o l l a n d P u b l i s h l n y Company, 1981
STABILITY AND CONSISTENCY ANALYSIS OF DIFFERENCE METHODS FOR SINGULAR PERTURBATION PROBLEMS Jens Lorenz Fakultat fur Mathematik Universitat Konstanz Konstanz Bundesrepublik Deutschland The well known principle "stabilty and consistency imply convergence" is often used to study the behaviour of discretisation processes. In this paper we consider singular perturbation problems and ask for convergence uniformly in the perturbation parameter e . The principle "stability uniformly in E and consistency uniformly in E imply convergence uniformly in E " will be used to study difference methods for a mildly nonlinear two-point boundary value problem of singular perturbation type * INTRODUCTION Consider a "continuous" problem depending on a parameter e E (1)
(Oreo]
T u = 0, u < U c l R R &
where u is an unknown function out of some function space U . As a specific example we shall take the boundary value problem (2a) T u = - E U " + a(x)u' + b(x,u) = 0, X E [0,1], e 2 (2b) u E U = { V E C [0,11: v(0) = v(1) = 0). Assume that (1) has a unique solution called u e E U , and let (3)
Telhu = 0, u E U h
=
lRRh
be a discretisation of (1) with a unique solution called u E r h E U h . Q h c Qdenotes a finite mesh with the discretisation parameter h. We use the notation of finite difference methods, but the general ideas also apply to finite element equations with a slight change of terminology. If E is kept fixed, then the discretisation ( 3 ) is called (asymptotically) stable with respect to norms I1 11, I1 11' if there exists ho(c) > 0 and CO(&) > 0, such that the following stability inequality holds (4)
IIU-~IISC~(E) IITc,hU-TE,hVIIt Vu,vEUh,
(The norms I1 11,
11
VhE (OfhO(&)l.
formally depend on h, but we suppress this 141
142
J . LOREN2
notationally. ) Let uE l h E lRRh = Uh denote the restriction of the continuous solution u to the grid ah; thus TErhuClhis the vector of & truncation errors, and the discretisation ( 3 ) is called consistent of order p (for E fixed) if (5) IITCrhu&IhII' s C 1 (c)hp. Evidently, we can conclude from ( 4 ) and ( 5 )
I I U ~ ~ ~ - U ~ , IITt,h~cIhII' ~ ~ I ~ C ~ ~ ( C~o ( ) & ) C 1 ( ~ ) hfor P
O<
hsho(&),
i.e. stability and consistency imply convergence for each E fixed. It turns out that these well established asymptotic concepts of stability, consistency, and convergence are rather useless in the study of discretization methods for singular perturbation problems if the concepts are applied only for E fixed. For example, if the ordinary difference method, which replaces u " and u' by central differences, is applied to problem ( 2 ) , there results a discretisation which is asymptotically stable, consistent, and convergent for each fixed E , but the method produces useless results for c << h. The asymptotic concepts of stability, consistency, and convergence for fixed E only get significant for h << E . The main reason for the failure of many standard discretisations applied to singular perturbation problems is a lack of stability of the discrete equat ons ( 3 ) for E << h; i.e. a stability inequality IIu-vII
S T A B I L I T Y A N D CONSISTENCY ANALYSIS
143
approximation of u
by the discrete solutions u is not needed E ,h inside the layers. In this case the truncation error may be great
inside the layers, and it is only important that the truncation error does not spread from the layers to other parts of the region. Results of this type are given in [I] for some linear systems and are discussed in [ I 1 3 for the upwind scheme and for Samarskii's scheme [18lapplied to a single linear equation. But it should be noted that often the reduced problem to a singular perturbation problem ( 1 ) is known, and in this case one may just solve the reduced problem numerically to obtain a discrete approximation of u outside the & layers for small E . Methods based on this idea are discussed in [8,121. Here we shall assume that b) u is of interest outside and inside the iayers.In this case the & truncation error should be small everywhere. There are two possibilities to obtain a small truncation error inside the layers, namely to choose a fine mesh there or - what is less trivial - to choose a difference formula reflecting the behaviour of u inside the layers. E
We shall discuss both possibilities for a problem of the form (2) where the essential assumption la(x) I > a > O will be made. A method usingauniform grid which reflects the boundary layer behaviour of uE to first order has been discussed by Il'in [lo] and later in [11,151. For earlier applications to flow-problems see [31. We shall establish uniform consistency for Il'in's method in the mildly nonlinear case if the truncation error is measured in the discrete 1 -norm. This implies 1 uniform convergence, and thus generalizes convergence results given in [10,11,15]. The discrete values u
can easily be interpolated to c ih yield a continuous function Ipo (u ) satisfying & ,h I I I ~ o ( u ~ ,-~ ~ )& l L l C h . Here C is independent of
E
and h [131.
In principle it is possible to design difference methods on an equidistant grid which converge, uniformly in c , of an order greater than one. To show this, we establish quadratic convergence, uniformly in E , for a method based on a Mehrstellen-formula, but we need restrictive assumptions for the continuous problem. The main difficulty to prove uniform convergence is to estimate the consistency error uniformly. Here knowledge of the boundary layer behavior of uE is needed. But it should be noticed that this knowledge is used only to prove theorems about the numerical method, and is not needed to perform the method itself.
144
J . LOREN2
tdamerical examples show that difference methods converging uniformly in E are especially well suited for moderately stiff problems, i.e. for c S h [13]. If E < c h, all reasonable methods on an equidistant grid essentially only solve the reduced equation approximately. Finally we give an error estimate for difference methods on nonequidistant grids. Here the boundary layer behaviour of u & is not used in the proofs, merely estimates for the derivatives of uE will be needed. 1. THE CONTINUOUS PROBLEM In this section we present results about the continuous problem needed in the analysis below. Complete proofs of all theorems are given in [13]. It is always assumed that E > O , aEC[O,l], bEC([O,llxlR) and
b(x,u) rb(x,v) for u t v , x E [0,1], u,vElR. 2 Tc is given as in ( 2 a ) for u E C [ 0 , 1 ] and Ru
= (u(O),
~(1)).
Further smoothness assumptions and sign conditions on a(x) will come into the discussion later. Theorem 1 2 The operator u + (T&u,Ru) from C [0,1] to C[0,l]xJR2is bijective and inverse monotone, i.e. 2 T u l T v, R u < R v * u I v V u , v E C [0,1]. &
&
Especially, the problem ( 2 ) has a unique solution uE for all Here all inequalities are defined pointwise, i.e.
E
> 0.
u
.
(6)
la(x)I ? a > O for x E [O,ll.
W e state a stability inequality in the next theorem where we restrict ourselves to functions u,v with Ru = Rv for simplicity.
Theorem 2 If ( 6 ) holds, then 1 Ilu-vlb,IIIT&u a
-
2 TEvlll Vu,vE C [0,1] with Ru = Rv.
145
STABILITY AND CONSISTENCY ANALYSIS
Here
IlwlL = max{ Iw(x) I : 0 $ x 5 1) and 1 Ilwlll = Ilw(x) Idx. 0
A proof of theorem 2 follows by an estimate of a Green's function, see [51. In the next three theorems we let (7) a(x) > a > O . The constants C, Ci etc may depend on the choice of a, but they are independent of c > O . Theorem 3 k k Assume (7), and let a E C [0,11, b E C ( [ O f 1 I x l R ) . Then
( 8 ) 1~:~) (x)I
Theorem 4 Let the conditions of theorem 3 hold and let cp, function satisfying
- ~ c &p " + a(x)cp; = 0, RQ, = (0,l). Then, for suitable y , E R , w e have uE = g, + and
y,cpp,
be the boundary layer
where
ly
&
I
I g&( * ) (x)I 5 C(l+&-n+le-(l-x)a'c), x E [0,1 , n=0,1, ..., k+l. A similar result was proved in [ I l l for the linear case with a different boundary layer function. The next theorem is needed to prove quadratic convergence for the Mehrste len-scheme below. We let
fE denote the boundary layer function 1 1 ' a(s)ds). f,(x) = exp(- (9)
I
& X
Theorem 5 Let b(x,u) = B(x)u
+
Then, for suitable y , ,
c(x), and assume a,B,cECk+2[0,1] as well as ( 7 ) . 6 & E lR, we have
u & (x) = g & (x) + (ye+6&x)fE(x), x E where l y , l < C , 16 & I I C and lgi") (x)I 5 C(1+&-n+2e-a(1-x)'a)f
[Of
1I
x E [0,1], n=0,1, ...,k.
Theorem 5 is a special case of a result proved in [I31 with methods using singular perturbation theory. 2.
DIFFERENCE FORMULAS Let E > 0, h > 0, h' > 0, a € R , a * 0, and consider the differential operator
J . LOREN2
146
LU =
- tu" + au', u E C2 [-h,h'l.
We list a variety of difference formulas using the three points -h,O,h'. These formulas will be taken in the next section to set up difference schemes for the boundary value problem (2). The formulas will allow for a > O and a < O in order to make an application to turning point problems easier. We note that most of the theoretical results below do not apply to turning point problems, but numerical examples show the usefulness of these formulas also for problems with turning points. We use the abbreviation 2 Ud = (u(-h), u(O), u(h')) for U E C [-h,h'l and set p = ah/(2c), p ' = ah'/(2c). 2a. EQUIDISTANT FORMULAS, h = h' For any U: IR + IR holds -2 a ) + ~ ( - l , O , l ) ) u d= (Lu)( 0 ) (10) lth ~ ( p (-1,2,-1) for the functions u(x) = 1 and u(x) = x, and if (10)is required for u(x) = exp(ax/c), then u ( p ) = u o ( p ) = p cothp. (Compare [3,101.) Other reasonable functions u ( p ) are u,(p) = 1 + I p I , which yields the 2 upwind formula, and u2(p) = 1 + p /(l+lpl), which yields Samarskii's formula [11,18] on the left side' of ( 1 0 ) . For further functions u ( p ) compare [ 19 I. Let p-,p+,po : I R + I R be functions of p with P- + Po + B, Then the identity (11) lth-2pCothp(-1,2,-1) = D-(p)
(Lu)(-h) +
= 1.
a + -(-l,O,l)l 2h
Po(p)
ud
(Lu)( 0 ) + B + ( P ) (Lu)(h)
holds for u(x) = 1, x and exp (ax/&). If (11) is required in addition for u(x) = x2 and u(x) = x exp(ax/e), then
1 ~1 ( c o t h p- -)/(l-~?-~'),
P-(p)
=
~ + ( p )
= e-2p p - ( p ) ,
P
P,(P)
=
1
-
P-(P)
-
~ + ( p ) .
(See [131.) Elementary calculation shows that B - , real analytic functions of p E R satisfying 1 (12) O < B - ( P ) + P + ( P ) B o ( P ) < 1.
Po,
B+ are positive
147
S T A B I L I T Y AND CONSISTENCY A N A L Y S I S
2b. NON-EQUIDISTANT FORMULAS
thus D1ud and D2ud are common finite difference approximations of u' (0)and u" (0)respectively. For any a: R2+ lR the identity (16) {-&o(p,p')D2 + aD1) ud holds for u(x) then u ( p , p ' ) =
=
(Lu)(0)
=
1 and x , and if ( 1 6 ) is required for u(x) where
=
exp(ax/t),
ao(prp')
The upwind formula - ED2 + a6 for a > O , - &D2 + a6' for a < O can also be written in the form - &oD + aD1 with (18) a
=
u1
( p , ~ ' ) =
i
1
+
1
- p'for a < O
p
for a > O
2 As a generalization of Samarskii's function a 2 ( p ) = 1 + p /(l+lpl) we take similarly 1 + p 2/(l+p) for a > O (19) U 2 ( P , P ' ) = I + p12/(1-p') for a < ' ~ In some of the.computations below a different notation of the vectors - &uD2 + aD1 will be convenient. It holds that
i
(20) - &uD2 + aDl = p(-I,l+y,-y) iff
(21) y
=
;,*-,
a u+p' p=E;',,pl
0-P
I
or equivalently (22)
u =
P 2+P
I
2Y y
P
I
The above functions
(25)
y2(p,p1)
=
P
aor
*
a
=
E
1
*
m
-
a l , u 2 thus lead to
(I+(P+P')( l + p )-I
If a < 0 use the relation
y (-p
' ,-p
=
for a > O . l/y(prp') to compute y 1 and
pol p,, p2 are defined with (22). For the exponentially fitted
formula compare [41.
y2.
148
J
. LOREN2
3 . INVERSE MONOTONICITY AND STABILITY For u, vElR n let u I v - u i I v . for i=I,...,n. T:lRn+IRn is called inverse-monotone if Tu 5 Tv implies u 5 v for all u, v E IF?. T is called an M-function [I61 if T is off-diagonally decreasing and inversemonotone. Assume that O: Rn+ Bn is a continuous increasing diagonalfield, and let (26) Tu = Au + BOu, u E XIn, where A E Rnrn is an M-matrix and B E Enfn is a nonnegative matrix. The following result is basic. Compare [I61 for (i) and [ 6 ] for more general related results.
Theorem 6 If B is diagonal, then T is an M-function from lRnonto itself. be a nonnegative diagonal matrix such that all offLet D E Enrn diagonal elements of A + BD are nonpositive. If Ou - OvID(u-v) for all u I v, then again T is an M-function from Enonto itself. In both cases (i), (ii) the stability inequality lu-vl 5 A-1 ITu-TvI V U , V € lRn holds for the operator (26). Here 1wI denotes the vector with components ( I W I =) ~I W ~for I WEIR^. Let the general conditions of section 1 hold: especially, u+b(x,u) is assumed to be increasing. We discuss discretizations of (2) set up with formulas of section 2. First let G = {xor X ~ , . . . , X ~ be + ~ an ~ arbitrary grid with 0 = x 0 < x1 < . . . < x m < Xmtl = 1. Let hi ai
=
=
xi
- xi-l and define
a(xi), p i = aihi/(2&),
pi
=
aihi+l/(2E), (i=l, . . . , m ) .
Using the notation of section 2b we set li
=
-
&oiD2
+
aiDl
=
.. ,m)
pi (-1,l+yir-yi ) , (i=l,.
where ci = o(pi,p;) etc. Finally define TE,,: IRm+2 + lRm by (28) (T&,,u)~ = li(ui-l,ui,ui+l) T + b(xi,ui) (i=l, m) for u E lRm+ 2
...,
and set (F,,,ul
i -
(Tc,,u)
- {Ui
for i=l ,.. . ,m for i=O,m+l
Our discretization of (2) reads ?&,,u=O, u E lRm+2. Of course, the discretization depends on the choice of the function IS.With help of the diagonal-field
149
S T A B I L I T Y AND CONSISTENCY ANALYSIS
(bGu)i = b(xi,ui), i=O,...,m+l, U E I R ~ + ~ the operator ?! (29)
may be written as ErG TErG u = Au + BbGur u E IRm+2 ,
N
...,
where A = A
is a tridiagonal-matrix and B=diag(O,l,l, 1,O). With E rG theorem 6, (i) the following analogue of theorem 1 can be proved. No sign conditions are needed for a(x).
Theorem 7 Let u ( p , p ' ) > p for p t O , p ' > O and let u ( p , p ' ) > - p ' for p < O , p ' < O . Then is an M-field from IRm+2 onto itself for all E > 0 and all ErG grids G. Proof: It is sufficient to show that A is an M-matrix. For
e
=
(l,I,...,l)T
holds A; = (llOr...rO,l)T,and thus A is an M-matrix if
(30) A i r i k l< 0 (i=l,.. . , m ) . (See e.g. [7,Theorem2.2].) Let i be fixed. If a . = 0, then p i = p i = 0, u i > 0. Thus (30) holds by definition of i D2. If a i > O , then p i > O , p ; > O , u > p i , and from (21) follows y i > O , p i > O . If a i < O , then p i < O r p; 0, pi > 0. Since A (30) is proved.
~
=
-P ~
i
ui
>-pir
and again
i i and ~ Ai,i+l ~= -p y the inequality q.e.d.
The functions u I l u 2 defined in section 2b evidently satisfy the condition of theorem 7. Also, uo satisfies it if we set o O ( p r p ' ) = 1 for p p ' = 0. To show this, assume for definiteness p > O r p ' > O . Then y0(p,p')
>Or
p-p'yo(prp')
>Or
and thus (22) yields
~ ~ ( p , p >' p) .
The Mehrstellen-formula of section 2a can be used for a discretization on an equidistant grid G = Gh = IO,h,. . . ,mh,1 1 if a(x) = a E IR. For i=I,...,m the i-th equation reads (31), (epcothp(-1,2,-1) + ~ ~, 1 IUT . 2h( - 1 , 0 , 1 ) 1( U ~ - ~ , U1+ + ( B - ( P ) , O ~ ( P ) ~ B + ( P ) ) (b(Xi-irui-l)r b(xirui)r b ( X i + l r ~ i + l ) )=~ 0 + IRm+2 can where p = ah/ (2c). The corresponding operator Ti:c,h: again be written in the form ( 2 9 ) where B is now tridiagonal. Using
theorem 6, (ii) with D = 11 the following result is obtained [13]: Theorem 8 Let Oib(x,u) - b(x,v) sl(u-v) for all u t v , x E [0,11. If h l i 2 1 a l r then the operator defined with the above Mehrstellen-formula is Erh
150
J
. LOREN2
a n M - f i e l d from Rm+2 o n t o i t s e l f f o r a l l
> 0.-Now
6
assume t h a t ( 6 )
i s s a t i s f i e d . We p r o v e a d i s c r e t e a n a l o g u e o f t h e c o n t i n u o u s s t a b i l i t y i n e q u a l i t y s t a t e d i n theorem 2 , and a l l o w f o r n o n e q u i d i s t a n t g r i d s .
W e r e s t r i c t o u r s e l v e s t o d i f f e r e n c e f o r m u l a s which a r e s e t u p w i t h h e l p o f t h e f u n c t i o n u0 or u 2 a s d e s c r i b e d i n s e c t i o n 2b. F o r d e f i n i t e n e s s w e assume a ( x ) > a > O , and s i n c e t h e boundary l a y e r occurs near t h e r i g h t endpoint w e consider g r i d s G with hi+16hi
(i=l
]...] m ) .
Theorem 9 L e t T c I G : I R m + * + XIm b e g i v e n as i n ( 2 8 ) and l e t t h e f u n c t i o n s
p = po o r y = y 2 , p = p2 b e t a k e n t o d e f i n e li v i a
-(=Yo]
li = p i ( - l , l + Y i l - Y i )
Then
IIu-vII
m
I
Pi = p ( P i , P ; )
Yi
I
= Y(PilP;).
l m 1 h . I (Tcr G -~ Tc , G ~ ) I a j=', 1
I-
f o r a l l u , v E lRm+2 w i t h uo = Proof: a. W e w r i t e
?& ,G u
= Au
"01
%+I
= Vm+l
*
+ BbG u and u s e t h e o r e m 6 1 ( i i i ) . I t i s
s u f f i c i e n t t o show t h a t t h e e l e m e n t s o f t h e j - t h column o f A-l a r e I a l l I z h j I j = 1 I . . ] m . F o r e a c h f i x e d j w e c o n s t r u c t q E lRm+2 w i t h
.
O < g i l a1h j l
(Aq)i~6ij(i=0,...,m+1).
Then t h e j - t h column o f A w i l l b e
.e
1 a 1
5 g 5 -h
b e c a u s e A-'
2 0.
b.
S e t q ( r ) = e2r - i f T c , G i s d e f i n e d w i t h y o ] p o l and s e t q ( r ) = 1 + 2 r + 2 r L if T i n G i s d e f i n e d w i t h y 2 ] p2. F u r t h e r m o r e ] l e t - q(ri), thus O < r i l p i , q . > 1 . We d e f i n e g by r = h i a / ( 2 c ) I qi i - I
1= 1
The i n e - q u a l i t y (Ag) . 2 6
c.
j+l 5 i s m .
-1
+
i s e v i d e n t f o r i=O,i=m+l and f o r 1 i j S i n c e pi > 0 i t remains t o show t h a t
(l+yl)qi
-
y iqiqi+l 2 0 ( i = l l . . . l j - ~ )
-
1 and -h .p' ( 1 l/qj) 2 1 , o r equivalently a 1 ( 3 2 ) qi+l 5 1 + ( l - l / q i ) / y i ~ i = l l . . . l j - l ~ l
For f i x e d i we set
p
= pi]
p'
= p i l r = ril
t = hi+,/hi
=
p'/p.
151
S T A B I L I T Y AND CONSISTENCY A N A L Y S I S
Since ri+l
=
rt the inequality (32) reads
(34) q(rt) I 1 + (l-l/q(r))/y(p,pt), and (33) is equivalent to 1 1 (35) F(l-l/q(r)) >-(l-y(P,pt)t). P
d.
It is sufficient to prove the inequalities (34), (35) for 0 < r s p , 0 < t s 1. Since the function p + y ( p ,pt) decreases, the estimate (34) follows from q(rt) 5 1 - (l-l/q(r))/y(r,rt), and this last estimate is easily proved. To show (35) notice that 1 r+-(1-l/q(r)) is decreasing. Thus it is sufficient to prove (35) for r r = p, and again the estimate l / q ( p )
q.e.d. As an immediate consequence we note the following corollary, where an equidistant grid G = Gh = (O,h,...,mh,.l} is assumed and the norm m IInIIl = h 1 l r l i l , , E n m i= 1 is used. Corollary 1 Let the formula (10) be used to define Tc,h:lRm+2+lRm. If 2 ~(p= ) pcothp or ~ ( p )= 1 + p /(l+Ipl), then (36) Ilu-vl~Ia 1 IITEfhu- TErhvII1 vu,vE lRm+2with uo
=
vo,
-
v~+~.
If a(x) = a = a and the Mehrstellen-formula (31) is used to define T&lh' then again (36) holds as long as O
P
(p)
+p
increases (p > 0 )
the estimate (36) is proved in [131. All functions listed in [191 fulfill (37). 4. CONSISTENCY AND CONVERGENCE Consider an equation (2) under the general conditions of section 1, and let G be a grid 'as above. In section 3 we have described operators
152
J
I R ~ TE ,G: which are T& u = (L = L E IG
. LORENZ
+ x~m , T C l G u = Lu + MbGu, u E discrete analogues of the different a1 operator cu" + a(x)u' + b(x,u) = L u + b(x,u and M = M are (m1m+2)-matrices.) The truncation error is -
E
,G
TE,GUCIG Rm' Now let u & = g, + 1,. We have in mind to split up u in a layer& function l& and a function g with harmless derivatives; see the & theorems 4 and 5. Then 0 = TEIGUClG= LuelG + MbGUEIG = LuclG + M(-L u ) I G f thus 0 = qelG -
&
(38)
rl =
{Lg,lG - M(L,g,) lG)
&
+ (L1,IG - M(Lcl,) lGl.
Therefore the discussion of the truncation error can be split up into two parts, namely an estimate for the harmless part g and an estimate & for a pure layer-function 1 . Such a split up is trivially possible & for linear problems, but, in general, seems to be impossible for nonlinear equations. We indicate the proofs of the following two results [131, where an equidistant grid G = Gh = {O,h,...,mh,ll is assumed. u E Rm+2denotes the discrete solution, and C is a generic constant c,h independent of h and E . Theorem 10 Let a E C2 [0,11, a(x) t a > O , b E C 2 ([O,l]xR). If
1 - a
E
ih
is computed with
IInII1
and thus with Y&I
U
IILV, h "1
38) fol ows
'
using standard arguments. It is just needed that
STABILITY AND CONSISTENCY ANALYSIS
=
&(h-2(-l,2,-l)gd
+
+
g''(xi))
+ a1. (-(-l,O,l)gd 1 2h
153
- gC(xi))
E(O(O~)-I) h -2 (-1t2r-l)gd.
Using standard estimates for difference formulas we find 1 Ig"(s) Ids + &pih-1 J Ig;(s) Ids, liil s T E J Ig;'(s) I ds + piJ & where the integrals are taken over [xi-l,xi+l]. Here p = aih/(2c) , i and summing up we find m 1 1
Now ( 3 9 ) immediately implies I1 5
Ill
5
Ch.
c. The estimate IILcpEIhlll5 C h is trivial if a(x) is constant, but requires care otherwise. We omit the proof here, see [13]. Theorem 10 generalizes results given in [10,11,151 for linear problems. formly in
E ,
For the Mehrstellen-scheme quadratic convergence, unican be stated under special assumptions [131.
Theorem 1 1 Let a(x) = a E I R , a + O , b ( x , u ) = b ( x ) u
-
c(x), 0 , c € C5 [0,1]. If u
is the solution obtained with the Mehrstellen-scheme, then
E
th
A proof follows similar lines as above where again corollary 1 establishes stability uniformly in c . The solution u is spl.it up & using theorem ,5. The boundary layer term does not produce a truncation error here since a(x) is constant. In theorem 10 and 1 1 we stated convergence, uniformly in E , for methods on equidistant grids. The difference formulas were exponentially fitted: this is important for a uniform estimate of the consistency error of the layer-term. (Compare [I41 for conditions which are necessary to obtain uniform convergence on equidistant grids.) We now assume a E C 2 [O,ll, b E C 2 ([O,llxIR), a(x) > a > a ' > O , and consider a nonequidistant grid G with hi+l 5 hi Let TE,G:IRm+2+IRm
(i=l,. . . ,m).
be defined as in (28) with help of the function
u 2 given in (19); thus
li =
-
+ aiD1, p i = aih/(2c). &(1+pi/(l+pi))D2 2
D2 and D 1 are defined in (15). Theorem 12 Under the above conditions holds
154
J . LOREN2
xi
with
= exp(-(l-xi+l)a'/c).
Here C i s a c o n s t a n t i n d e p e n d e n t o f c
and t h e g r i d G. Proof: Theorem 9 i m p l i e s
1
1
-~ u & 1 , ~~ I L1I ,"t
~
hilqiil
i=1 = T c I G u c I Gd e n o t e s t h e t r u n c a t i o n e r r o r of t h e c o n t i n u o u s
if
s o l u t i o n . For f i x e d i E ( 1 , 2 , , ~d = ( u , ( x i - 1 )
-
let
uc ( x i ) r uc ( x i + l ) )
r
Then
I nil
. . ,m}
+
*
aiuk ( x i ) ) I
=
lliud
6
c l u r ] ( x i ) - D2udl + c p i l D j u d l / ( l + p i )
(-ELI:
2
hilu'&'
2
(El) I +
iEhiluz'
5 C
+
(xi)
T
(c3) I}
with
2
Using p i / ( l + p i ) 5 p i 5 C h i / c
Inil < C h i ( l + &- 2 x i )
ck
+ ailDlud - uk(xi)I
l/(l+pi)
5 xi+l.
w e f i n d w i t h theorem 3
and t h e a s s e r t i o n f o l l o w s . q.e.d. F i n a l l y , c o n s i d e r t h e same d i s c r e t i s a t i o n where a s p e c i a l g r i d G w i t h o n l y t w o s t e p - s i z e s H end h i s u s e d : L e t hl = = hL = H 2 h = h L + I = - hm+l.
...
, 0
H
,
x1
.*.
,
H
xL-l
. h .
x~
x
~
Theorem 1 3 3 3 L e t a E C [0,11, b E C ([O,llxIR),
.
h, 1
+xm ~
a ( x ) t a > a ' > O . There e x i s t s C
i n d e p e n d e n t of H , h , and c s u c h t h a t
I I u & I ~ - uc,GILsCH a s l o n g a s h s c n and lc:= $2 & l o g -15 1 &
-
x
L'
Proof:
W e u s e t h e same n o t a t i o n s a s i n t h e p r o o f o f theorem 1 2 and f i n d f o r i = L+l,
...,m:
lqil s C ( c h 2 ( l + ~ - ~ + h ~c-1h2(1+c-2Ai) ) I Ch 2 & - l ( l + & - 2 X i ) .
+
h2(l+c3hi)}
155
S T A B I L I T Y AND CONSISTENCY A N A L Y S I S
(Note that the applied difference formulas are equidistant and make 2 use of pi/(l+pi) ICh2c-2.) This yields
For i=l,...,L-I we have
A.
ie
and in addition holds
= c2
A L < & ea'h/E
In the last inequality we used h/c
DIG.
I
q.e.d. For the simple grid assumed in theorem 13 the number of grid points 1 for & + O , H fixed, necessarily grows like log XLI XL+l'"' & 1 since we required h s &\/Fi, I-x > -2 clog Thus the numerical work L - a' & is not independent of t here, in contrast to the work in the methods converging uniformly in E . But in practice the slow growth like 1 log c is nearly always acceptable.
.
REFERENCES Abrahamsson, L.R., A priori estimates for solutions of singular perturbations with a turning point,. Studies in Applied Math. 56 (1977) 51-69. Abrahamsson, L.R., Keller, H.B., Kreiss, H.O., Difference approximations for singular perturbations of ordinary differential equations, Numer. Math. 22 (1974) 367-391. Allen, D.N. de G., Southwell, R.V., Relaxation methods applied to determine the motion, in two dimensions, of a viscous fluid past a fixed cylinder, Quart. J. Mech. Applied Math. 8 (1955) 129-145. Barrett, K.E., The numerical solution of singular-perturbation boundary-value problems, Quart. J. Mech. Applied Math. 27 (1974) 57-68. Bohl, E., Inverse monotonicity in the study of continuous and discrete singular perturbation problems, in: Hemker, P.W., Miller, J.J.H. (eds.), Numerical analysis of singular perturbation problems (Academic Press, London, 1979). Bohl, E., Lorenz, J., Inverse monotonicity and difference schemes of higher order: A summary for two-point boundary value problems, Aequ. Math. 19 (1979) 1-36. Bramble, J.H., Hubbard, B.E., On a finite difference analogue of an elliptic boundary problem which is neither diagonally domi-
156
J . LOREN2
nant nor of non-negative type, J. Math. and Phys. 4 3 ( 1 9 6 4 ) 117-132.
Flaherty, J.E., O'Malley, R.E., The numerical solution of boundary value problems for stiff differential equations, Math. Comp. 31 ( 1 9 7 7 ) 66-93. Frank, L . S . , Difference singular perturbations - I. A priori estimates, J. Math. Anal. Appl. 7 0 ( 1 9 7 9 ) 180-235. Il'in, A.M., Differencing scheme for a differential equation with a small parameter affecting the highest derivative, Math. Notes Acad. Sci. USSR 6 ( 1 9 6 9 ) 596-602. Kellog, R.B., Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 ( 1 9 7 8 ) 1025-1039. Lorenz, J. Combinations of initial and boundary value methods for a class of singular perturbation problems, in: Hemker, P.W., Miller, J.J.H. (eds.), Numerical analysis of singular perturbation problems (Academic Press, London, 1 9 7 9 ) . Lorenz, J., Zur Theorie und Numerik von Differenzenverfahren fEr singulare Stijrungen (to appear). Miller, J.J.H., Some finite difference schemes for a singular perturbation problem, in: Constructive function theory, Proc. of the internat. conf. on constructive function theory, Blagoevgrad, 1 9 7 7 . Miller, J.J.H., Sufficient conditions for the convergence, uniformly in E , of a three point difference scheme for a singular perturbation problem, in: Springer Lecture Notes in Math. 6 7 9 , 1978.
[ I 6 1 Ortega, J.M., Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables (Academic Press, New York, 1 9 7 0 ) . [ I 7 1 Protter, M.H., Weinberger, H.F., Maximum principles in differen-
tial equations (Prentice-Hall, Englewood Cliffs, New Jersey, 1967).
[ I 8 1 Samarskii, A.A., Introduction to the theory of difference schemes (Nauka, MOSCOW, 1 9 7 1 )
.
[ I 9 1 Stoyan, G . ,
Monotone difference schemes for diffusion-convection problems, ZAMM 59 ( 1 9 7 9 ) 361 - 372.
A N A L Y T I C A L AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N A N A L S S I S S . A x e l s s o n , L . S . Frank, A . van d e r Sluis ( e d s . ) @ N o r t h - H o l l a n d P u b l i s h i n g C o m p a n y , 1981
FINITE ELEMENT GALERKIN METHODS FOR CONVECTION-DIFFUSION AND REACTION-DIFFUSION A. R. Mitchell
* , D.
*,
F. Griffiths
and A. Meiring
**
*Department of Mathematics, University of Dundee, Dundee, Scotland **Department of Applied Mathematics, University of Pretoria, Pretoria, South Africa
Galerkin methods with piecewise polynomial spaces are unsatisfactory for the numerical solution of many time dependent problems. This is particularly so in the case of convectiondiffusion problems for large Peclet numbers, where it is shown that "upwinding" by Petrov Galerkin methods significantly improves matters in one and two space dimensions. In reactiondiffusion problems, Galerkin methods are applied to the Fisher, Nagumo and Fitzhugh-Nagumo equations and solutions involving travelling xaves, pulses and wave trains successfully calculated. I.
INTRODUCTION
Galerkin methods with piecewise polynomial spaces are now being used extensively for the numerical solution of many important partial differential equations of the form Lu - f(u) = 0 where examples of the operator L are
a + a -a
ax
at
11
+
E
- € -
2
a3T ax a-' 2
ax at
(1.1)
v > o
Hyperbo 1ic (1.2) Convection-Diffusion ( I .3) Burgers
v > o
Reaction-Diffusion
(1.4)
E > O
Korteweg de Vries
( I .5)
E > O
Regularised Long Wave
(1.6)
a > O
aL - v - 2 ax aL -vax
I1
,
The non linear function f(u) is zero in all cases except ( 1 . 4 ) . and we include as alternatives the linear and non linear first order operators. Initial and boundary conditions are not specifically mentioned at this stage although they play a major role in determining theoretical and numerical solutions of the above equations. It is also worth emphasizing at the outset that all available theoretical evidence should be used in determining the numerical solution of any particular problem. This is particularly so in problems involving equations like ( 1 . 4 ) and ( 1 . 5 ) where the solutions involve travelling waves and solitons respectively, and the velocity of the wave front or soliton is an important consideration. Equation ( 1 . 1 ) represents the simplest model of the stated types of problem. Further difficulties both theoretical and numerical are present when ( 1 . 1 ) is a system or involves more than one space variable. 157
A.R. MITCHELL et a1
158
2.
THE PETROV GALERKIN FORMULATION
A f i n i t e range of t h e x - a x i s , normalised t o u n i t y , i s e q u a l l y p a r t i t i o n e d so t h a t Nh = 1 , where h i s t h e element l e n g t h . The s o l u t i o n u ( x , t ) o f t h e p a r t i c u l a r d i f f e r e n t i a l e q u a t i o n i s approximated by N
where Qi select
(x) ( i
=
a r e t h e chosen p i e c e w i s e t r i a l f u n c t i o n s .
O,l,--,N)
We
j = O,l,--,N
qj(x)
a s t h e p i e c e w i s e t e s t f u n c t i o n s , l e a d i n g t o t h e Petrov-Galerkin f o r m u l a t i o n ((LU(x,t)
- f(U(x,t)) ,
,
qj(x)) = 0
j = O,l,--,N
(2.2)
where ( , ) d e n o t e s t h e L 2 i n n e r p r o d u c t . The t r i a l and t e s t f u n c t i o n s a r e s e l e c t e d from p i e c e w i s e polynomial s p a c e s S' , of o r d e r r , w i t h v a r y i n g d e g r e e s of c o n t i n u i t y Ck , where r and k a r e i n t e g e r s . The Petrov-Galerkin method (2.2) becomes t h e G a l e r k i n method when t h e t r i a l and t e s t f u n c t i o n s p a c e s co-incide. Spaces i n common u s e a r e
3.
Linear
r = l
Quadratic
r
Cubic
r = 3
=
-1
Discontinuous (Lagrange, S p l i n e )
0
(Lagrange) (Spline)
0
(Lagrange) (Hermite) (Spline)
k = {O
2
= {I
k
=
{1 2
CONTINUOUS I N TIME PETROV GALERKIN METHODS
Each problem l i s t e d i n (1.2) - (1.6) h a s a c e r t a i n minimum c o n t i n u i t y r e q u i r e m e n t of t h e t r i a l and t e s t f u n c t i o n s i f conforming f i n i t e element methods a r e t o be used. I n t e g r a t i o n by p a r t s of t h e i n t e g r a l s a r i s i n g from t h e inner products can g i v e a u s e f u l t r a d e - o f f i n c o n t i n u i t y between t h e t r i a l a n d l g e s t s p a c e s . Examples of t h i s a r e i n c o n v e c t i o n - d i f f u s i o n where t h e i n n e r p r o d u c t (Qy , q j ) can be- r e p l a c e d by - (Qf , $;) , and so Co f u n c t i o n s are s u i t a b l e f o r b o t h t r i a l and t e s t s p a c e s . (Q;', + j ) can be r e p l a c e d by
S i m i l a r l y i n t h e Korteweg d e Vries e q u a t i o n , $;) o r by (4; , $): i f a further
- (4;
i n t e g r a t i o n by p a r t s t a k e s p l a c e and so C ' t r i a l and Co t e s t s p a c e s ( o r v i c e v e r s a ) can b e used. Each i n t e g r a t i o n by p a r t s , of c o u r s e , produces a boundary term which we p r e s e n t l y i g n o r e .
-
I n a l l problems l i s t e d i n (1.2) (1.6) t h e P e t r o v G a l e r k i n method ( 2 . 2 ) produces a f i r s t o r d e r system of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s i n t i m e where t h e m a t r i x c o e f f i c i e n t s a r i s i n g include the following Mass
C(Jli
Convection
C($f
t
Qj)l
, 1$~)1
The non l i n e a r c o n v e c t i o n and r e a c t i o n terms r e q u i r e i n d i v i d u a l c o n s i d e r a t i o n .
F I N I T E ELEMENT GALERKIN METHODS
4.
159
REVIEW OF GALERKIN METHODS
F i n i t e element c o n t i n u o u s i n t i m e G a l e r k i n methods f o r p a r a b o l i c e q u a t i o n s were i n t r o d u c e d by Douglas and Dupont i n t h e i r c l a s s i c p a p e r i n 1970. I n 1973, Wheeler proved optimum r a t e of L 2 convergence ( h +. 0) f o r t h e s e methods f o r a l l s p a c e s Sr For f i r s t o r d e r h p e r b o l i c problems, however, Dupont (1973) showed t h a t G a l e r k i n methods a c h i e v i d t h e optimum r a t e of L 2 convergence O(h4) f o r c u b i c s p l i n e s b u t o n l y O(h3) f o r Hermite c u b i c s . Wahlbin (1974) m o d i f i e d t h e t e s t f u n c t i o n s and showed t h a t t h e d i s s i p a t i v e G a l e r k i n method
.
(Ut + a Ux
, (4
+
h
4 ~ ~ =) ~0 )
j
= 0,1
,--,N
(4.1)
achieved O(h3'5) f o r Hermite c u b i c s . A f u r t h e r s l i g h t m o d i f i c a t i o n by Dendy (1974) o b t a i n e d t h e optimum L 2 convergence O(h4) A f u l l e x p l a n a t i o n of t h e G a l e r k i n method based on Hermite c u b i c s f o r t h e h y p e r b o l i c e q u a t i o n (1.2) can be found i n Hedstrom (1979). A similar s i t u a t i o n e x i s t s f o r order equations l i k e Korteweg de Vries where optimum L 2 convergence o n l y o c c u r s f o r smoothest splines with k 2 1 Wahlbin s u g g e s t s t h e P e t r o v G a l e r k i n method 3 (Ut + a U + v uxXx , 4 + h = o o,,)
.
third
.
......
f o r g e n e r a l p i e c e w i s e polynomial spaces a l t h o u g h t h e m o t i v a t i o n l e a d i n g t o t h e modified t e s t f u n c t i o n s i s n o t too c l e a r . I t a p p e a r s a t t h i s s t a g e t h a t G a l e r k i n methods are o p t i m a l i n L2 f o r second o r d e r problems b u t sub o p t i m a l f o r g e n e r a l s p a c e s . i n f i r s t . a n d t h i r d o r d e r problems. However i n second o r d e r c o n v e c t i o n - d i f f u s i o n problems f o r l a r g e g r i d a h P e c l e t numbers (L = -) t h e r e i s a p r a c t i c a l lower l i m i t f o r t h e element s i z e 2v h , and so a n L2 convergence a n a l y s i s does n o t a p p l y . To i l l u s t r a t e t h e inadequacy of G a l e r k i n methods f o r t h i s t y p e o f problem w e c o n s i d e r t h e s t e a d y s t a t e o f (1.3)
-v
ul'
+ a u' = 0 ,
v > O , a > O
(4.3)
.
where a dash d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o x G a l e r k i n methods based on Lagrange t y p e s p a c e s reduce t o t h e t h r e e p o i n t d i f f e r e n c e e q u a t i o n
-
[ I + dL1 6'Ui + L(Ui+I- Ui-l) = 0 (4.4) 2 dL 6 Ui i s r e f e r r e d t o a s t h e " a r t i f i c i a l v i s c o s i t y " and i s where t h e term i n t r o d u c e d i n a n a t t e m p t t o damp o u t o s c i l l a t i o n s i n t h e s o l u t i o n w h i l e h o p e f u l l y m a i n t a i n i n g r e a s o n a b l e a c c u r a c y o v e r a working r a n g e of L The v a l u e s o f d I 1 1 2 are 0 , 7 L , and + / ( I + -15 L ) f o r l i n e a r , q u a d r a t i c , and c u b i c Lagrange s p a c e s I r e s p e c t i v e l y and t h e s e a r e compared w i t h t h e t h e o r e t i c a l v a l u e c o t h L in
.
-
.
Table 1 f o r a range o f v a l u e s of L For "hard" o u t f l o w (downstream) boundary c o n d i t i o n s t h e s e G a l e r k i n methods l e a d t o e x c e e d i n g l y poor r e s u l t s f o r l a r g e L i n e a r s and c u b i c s underdamp t h e s o l u t i o n and l e a d t o v i o l e n t v a l u e s of L inear ic i l l o f f t h e s o l uT t ihoeno. r e t i c a l o s c i l l a t i o n sLwhereas q u a d r aQuadrat t i c s overdamp and so kCubic
$ ',
0 I 2 3 4 5 10
.
0 0 0 0
0 0 0
(1)
(1) (1) (1) (1) (1) (1)
0
0.33 0.67 1.00 1.33 1.67 3.33
(0)
(0.30) (0.67) (0.75) (0.80) (0.83) (0.91)
0
0.31 0.52 0.63 0.65 0.63 0.43
(0)
(0.37) (0.59) (0.70) (0.76) (0.81) (0.90)
0
0.31 0.53 0.67 0.75 0.80 0.90
160
5.
MITCHELL et al.
A.R.
PETROV GALERKIN METHODS FOR SECOND ORDER PROBLEMS
Having described some of the shortcomings of Galerkin methods we now turn our attention to Petrov Galerkin methods, (Mitchell and Wait r19771, Mitchell and Griffiths [19801) and attempt to find a rationale for matching test functions to particular trial functions for specific problems. If we begin with the convection-diffusion equation -vu"+au'=f,
v > O , a > O
and assume a=-V',
V(x)
potential
( 5 . 1 ) becomes
v
ull
+ V'U' + f
0
=
.
Multiplying this by the integrating factor ev/" and using the Galerkin method, we get v (e"" U')' 4 . dx + eV/' f 4 . dx = 0 %
I
I
J
J
which we can rewrite as the Petrov Galerkin method
where $.
-2Ls eVIV 4 . = e
=
(5.. 2 )
J
j' -2Ls with L the grid Peclet number and s = '/h For large L , e can become indistinguishable from infinity (negative s ) and zero (positive s ) and so we seek a less severe weighting of the trial functions. We assume 1
= 4(s)
$(s)
+
.
a u(s)
a
> 0 parameter
and attempt to find suitable perturbing functions u(s)
.
(a) Linear Trial Functions. The linear trial function is - 1 r s s o
I + s @(s) =
where
s =
-
j
.
I
-
0
Consider a conforming C quadratic test function of the form = 4(s)
$(s)
with the perturbing function u(s) =
(5.3)
0 5 S < + l
+
- 3 s ( I + s ) - 3s ( I - 5 )
Y u(s)
y > 0
- 1 r s r o 0
5
s
5
+
1
(5.4)
(5.5)
The test function satisfies the condition (+I
$(s) ds = 1
,
(5.6)
and has the collective formula qJj
=
(I - 3 a s ) 9
j
This has a similar appearance to the theoretically correct test function given by (5.2). (see Figure I ) .
16 1
F I N I T E ELEMENT GALERKIN METHODS
Figure 1 A l t e r n a t i v e l y we can choose t h e non conforming C with the perturbing function I - I - < s < O
u(s)
-1
l i n e a r t e s t f u n c t i o n (5.4)
+?
=
1 -_ 2
(5.7)
0 5 S < + l
where t h e l a t t e r can b e o b t a i n e d by d i f f e r e n t i a t i o n of t h e l i n e a r t r i a l f u n c t i o n . T h i s t e s t f u n c t i o n s a t i s f i e s (5.6) and a l s o t h e "patch" t e s t s +I I)'(s) d s = 0
i'l
,
I)'(s) sgn ( s ) d s =
-
2
,
(5.8)
-1 -1 and s o can be used f o r second o r d e r problems. Adopting a . f i n i t e d i f f e r e n c e t y p e of n o t a t i o n i t i s e a s i l y shown t h a t a P e t r o v G a l e r k i n method based on (5.3) and (5.4) w i t h p e r t u r b a t i o n f u n c t i o n s g i v e n by (5.5) o r (5.6) l e a d s t o
(U
(U' and
where
(U'
-
AoUj = +1U j + ,
, I).) 3
,
1 6
+ -6
= h(l = (Ao
I)!) 3
=
- 7I Y
- -1 6 2u h
and
Uj-l)
2
-
1
y Ao)Uj
S2)U
(5.9)
j
j
2
6 U. J
=
.
(Uj+l
-
2U. + Uj-]). 3
F u l l upwinding
(d = 1 i n ( 4 . 4 ) ) i s o b t a i n e d i f y = 1 More d e t a i l s o f t h e c o n s t r u c t i o n of t e s t f u n c t i o n s f o r l i n e a r t r i a l f u n c t i o n s c a n b e found i n G r i f f i t h s and M i t c h e l l (1979, 1980)(b) is
A t a n i n t e g e r node t h e q u a d r a t i c t r i a l f u n c t i o n 2 - I < s < O 1 + 3s + 2s (5.10) 2 I -3s+2s O S S < + l
Quadratic T r i a l Functions. O(s) =
A conforming
Co
.
c u b i c t e s t f u n c t i o n i s chosen i n t h e form
a l > o , - I < s < O $(s)
a(s)
= $(6) +
a2
where t h e p e r t u r b i n g f u n c t i o n i s
y
ci
2
> o ,
O < S < + l
(5.11)
A . R . MITCHELL et al.
162
-
- -52 s ( 2 s
+ I)(s + I )
- 5-2 s ( 2 s
-
I)(s
-
- 1 s s < o
(5.12)
I)
O < s < + l
t a h a l f i n t e g e r node, t h e q u a d r a t i c t r i a l f u n c t i o n
=
A conforming
Co
-
.
4s(s + I )
(5.13)
c u b i c test f u n c t i o n i s
(5.14) where
J
and
0
+(s)
(5.15)
.
ds = 2/3
-I
Using t h e t r i a l and t e s t f u n c t i o n s mentioned above t h e evaluated t o give 1
T;(U
inner products a r e
L2
$.I.
I
1 (----
30
1
1 8 0 . € 2 ) 'j+l
1
'j+l
+
4
+
'E
1
180 ('2
+
-
EI))Uj
1
+
-i 'j-1 ?
(--L+--E1 30
1
1
I
0
( Z + -90- E3) Uj + -15u
+ ( i S - - E
j-4
1 )90 U
3
180
1
)U
j-I
j-I
, qJj)*
(U' (-
+
1
7
1 45 E 2 )U j+l
+
2 (7. +
2 E )U 45 2 j + j
-
1
E(E2
2 E 1 )U j+i
+ E I ) U j + (- 2 3 + 45 ( z1 - - - E 1) U
45
4
-I u 3
j+l
- -8 3
u
j+l
+ -14 u 3
- -83 u .J + - 16 u 3
2
j
j-1
-- -83
1
+
j-I
2
1
uj-4 U
+
Tuj-1
j-I
a t t h e i n t e g e r and h a l f i n t e g e r nodes r e s p e c t i v e l y .
F i n a l l y i n t h i s s e c t i o n i t i s remarked t h a t t e s t f u n c t i o n s f o r h i g h e r o r d e r t r i a l f u n c t i o n s c a n be c o n s t r u c t e d i n a manner similar t o t h a t adopted f o r l i n e a r and q u a d r a t i c t r i a l f u n c t i o n s . The i n t e r e s t e d r e a d e r i s r e f e r r e d t o M i t c h e l l and C h r i s t i e C19781. A l s o i n Table 1 ( i n b r a c k e t s ) w e have i n c l u d e d t h e r e v i s e d v a l u e s o f d (formula ( 4 . 4 ) ) when P e t r o v G a l e r k i n methods a r e u s e d w i t h l i n e a r , q u a d r a t i c and c u b i c t r i a l f u n c t i o n s .
163
F I N I T E ELEMENT G A L E R K I N METHODS
6.
TWO DIMENSIONAL UPWIND FINITE ELEMENT SCHEMES
Extending a problem from one t o two s p a c e dimensions o f t e n l e a d s t o d i f f i c u l t i e s a s s o c i a t e d w i t h t h e geometry o f t h e r e g i o n . I n a c o n v e c t i o n - d i f f u s i o n problem, however, a n a d d i t i o n a l d i f f i c u l t y i s t h e d e t e r m i n a t i o n of t h e d i r e c t i o n of upwinding. An a c c u r a t e knowledge of t h i s i s e s s e n t i a l i n o r d e r t o develop upwinding p r o c e d u r e s which p o s s e s s no s p u r i o u s crosswind d i f f u s i o n . I n t h e l i n e a r problem, where t h e v e l o c i t y f i e l d i s g i v e n , t h e d i r e c t i o n of flow i s known i n advance of t h e c a l c u l a t i o n f o r t h e t e m p e r a t u r e and so a r t i f i c i a l d i f f u s i o n can be added t o t h e d i f f e r e n t i a l e q u a t i o n i n t h e d i r e c t i o n of t h e flow. (Hughes [19791, Johnson and Navert [19801, Winze11 [19801). I n t h e non l i n e a r Navier S t o k e s c a s e , however, t h e d i r e c t i o n of f l o w i s n o t known i n advance and can o n l y emerge from t h e c a l c u l a t i o n . We now l o o k a t some methods o f upwinding i n two s p a c e dimensions and a t t e m p t t o a n a l y s e them from t h e p o i n t o f view of e l i m i n a t i n g crosswind d i f f u s i o n . Assume t h a t t h e flow r e g i o n i s covered by a s q u a r e g r i d of s i z e h , where a ( j h , k h ) . Using t h e t r a n s f o r m a t i o n t y p i c a l node h a s c o - o r d i n a t e s
t h e t r i a l f u n c t i o n s can b e e x p r e s s e d as
$j
(X,Y) = $ ( s )
$(t).
The weighted r e s i d u a l f i n i t e e l e m e n t f o r m u l a t i o n o f
-v is
V 2u +
(a. g r a d )
u = 0
v ( g r a d $ , g r a d U)
,
+ h($
v > 0
(6.1)
, 5. g r a d u)
= 0
(6.2)
where t h e d i f f e r e n t i a l and i n t e g r a l o p e r a t o r s are r e f e r r e d t o ( s , t ) co-ordinates, i s t h e t e s t f u n c t i o n . We c o n s i d e r i n i t i a l l y t h e t e n s o r p r o d u c t t e s t and $ ( s , t ) f u n c t i o n ( H e i n r i c h e t a l . c19771) $(s,t) = ($(s)
+
a u(s))($(t)
+
8 u (t))
(6.3)
where a and 8 are upwinding p a r a m e t e r s i n t h e x and y d i r e c t i o n s r e s p e c t i v e l y and t h e p e r t u r b a t i o n f u n c t i o n u i s as o b t a i n e d i n t h e onedimensional c a s e . For l i n e a r t r i a l f u n c t i o n s and t e s t f u n c t i o n s g i v e n by ( 5 . 4 ) and ( 5 . 5 ) , t h e i n d i v i d u a l terms i n (6. I ) g i v e , , i n s t e n c i l n o t a t i o n ,
-
v 2u
=
'
Fl
-1
-'1
-2 2
164
A . R . MITCHELL e t
al.
I f 5 and q denote c o - o r d i n a t e s p a r a l l e l and p e r p e n d i c u l a r t o t h e d i r e c t i o n o f flow, t h e l e a d i n g term i n t h e t r u n c a t i o n e r r o r expansion of t h e combination ( 6 . 4 ) is 2
-1 a 2
h [(a cos 6 +
8
sin 8)
a2,, at
+
aU
(8 COB 8 - a
sin 0) 1 -
am
f.
(6.5)
A r t i f i c i a l d i s s i p a t i o n can a = a cos 8 , a = a s i n 8 and 0 2 8 i Y t h e r e f o r e b e c o n f i n e d t o t h e 5 d i r e c t i o n i f we choose
where
Ba
=
tan
.
e
(6.6)
This c o n t r a s t s w i t h upwind f i n i t e d i f f e r e n c e s where u s i n g t h e s t a n d a r d f i v e - p o i n t replacement of t h e L a p l a c i a n and upwinding i n t h e u s u a l manner w i t h p a r a m e t e r s a and f? i n t h e x - and y d i r e c t i o n s r e s p e c t i v e l y we g e t
-
This has a p r i n c i p a l truncation e r r o r 2 - !-L- [ ( a ax3 + Bay3 ) a U - 2 a a (sax
ac2
a2
X Y
-
Ba
Y
1- a
2
am
+ a a (aax + Ba x Y
)
a 2U 1 7 a0
(6.8)
from which i t can be s e e n t h a t no c h o i c e of a and f3 w i l l a n i h i l a t e t h e a r t i f i c i a l d i s s i p a t i o n i n t h e cross-flow d i r e c t i o n . Numerical r e s u l t s comparing f i n i t e element and f i n i t e d i f f e r e n c e methods a r e g i v e n i n G r i f f i t h s and M i t c h e l l c 19791. I t i s d i f f i c u l t t o g e t s u i t a b l e t e s t problems i n o r d e r t o compare t h e many s t r a t e g i e s t h a t e x i s t f o r s o l v i n g c o n v e c t i o n - d i f f u s i o n problems i n two dimensions. The most commonly used problem, where some s o r t of t h e o r e t i c a l s o l u t i o n i s a v a i l a b l e , i s due t o Raithby c19751. U n f o r t u n a t e l y t h i s problem h a s "weak" o u t f l o w boundary c o n d i t i o n s and so no t h e r m a l boundary l a y e r .
7.
THE REACTION-DIFFUSION PROBLEM
This i s concerned w i t h t h e system of e q u a t i o n s
--
where I! = (u , , u I L , m 2 1 , and D i s a " d i f f u s i o n m a t r i x " w i t h nonn e g a t i v e c o n s t a d t e n t r i e s n a n d i s u s u a l l y d i a g o n a l . The c o e f f i c i e n t s M j ( j = I , -- , m) a r e c o n t i n u o u s m a t r i x valued f u n c t i o n s , and ( x , t ) E R x [ t 2 01 where- R i s a bounded o r unbounded domain i n m-space. Along wyth (7.1) we have the i n i t i a l condition
U(X,O)
=)&u
,
x E
(7.2)
0
and f o r i n i t i a l boundary v a l u e problems we have i n a d d i t i o n t o (7.2) t h e boundary :a condition P - an + Q u = a (7.3)
where t
, and
P
,Q
a an
a r e matrix valued functions of
d e n o t e s normal d e r i v a t i v e .
5 and
t
, 5
depends on
Systems of t h e form ( 7 . 1 )
5 and
a r i s e i n many
f i e l d s of b i o l o g y , chemical p h y s i c s , and f l u i d mechanics. From t h e p r a c t i c a l p o i n t o f view t h e i m p o r t a n t q u e s t i o n s a r e "how do s o l u t i o n s o f (7.1) - (7.3) e v o l v e i n t i m e " and "what i s t h e a s y m p t o t i c b e h a v i o u r of s o l u t i o n s
165
F I N I T E ELEMENT G A L E R K I N METHODS
as t * - ?" It i s known a n a l y t i c a l l y t h a t a l a r g e c l a s s of systems of t h e form (7.1) - (7.3) e x h i b i t " t r a v e l l i n g wave" type s o l u t i o n s and t h e p r i n c i p a l g o a l o f t h i s s t u d y i s t o compare t h e r e s u l t s of n u m e r i c a l e x p e r i m e n t s w i t h some of t h e known a n a l y t i c a l r e s u l t s concerning t r a v e l l i n g waves. I n t h e s p i r i t of t h e p r e v i o u s s e c t i o n s , w e employ f i n i t e element G a l e r k i n t y p e methods i n c o n t r a s t t o p r e v i o u s numerical s t u d i e s where f i n i t e d i f f e r e n c e methods have a l m o s t e x c l u s i v e l y been u s e d .
W e l o o k f i r s t a t a s i m p l e example of (7.1) from t h e f i e l d o f p o p u l a t i o n g e n e t i c s . This i s Fisher's equation
(7.4) with the i n i t i a l condition
and w e c o n s i d e r t h e two c a s e s (i)
- u).
f(u) = u(l f(u) > 0
-
f(u) < 0
in
f o r some
a
f'(0) > 0
and
(0,l).
in
( i i ) f(u) = u(l
Here
u)(u
-
a)
(0,a) E
,
0 < a
b
5
-2 '
and , f ( u ) > 0
,
(0,l)
Here in
f'(0) < 0
(a,l)
, (7.7)
f ( u ) d u > 0.
The most i m p o r t a n t t h e o r e t i c a l r e s u l t s a r e due t o Aronson and Weinberger [I9751 who show t h a t i n c a s e ( i ) ,e i t h e r u ( x , t ) E 0 o r l i m u ( x , t ) = 1 , and i n c a s e ( i i ) , u ( x , t ) Z a o r l i m u ( x , t ) = 0 , l . More complete r e s u l t s c o n c e r n i n g t + m
t h e s e c a s e s can b e found i n F i f e and McLeod c19771, C19791. 8.
SOLUTIONS OF FISHER'S EQUATION
Before a t t e m p t i n g a n u m e r i c a l s o l u t i o n of t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n ( 7 . 4 ) , we e x p l o r e o t h e r p o s s i b i l i t i e s o f s o l u t i o n . We f i r s t i n t r o d u c e t h e moving co-ordina te S = x + c t ,
c c o n s t a n t > 0.
and (7.4) becomes t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n u'l
-
cu' + f ( u ) = 0
.
where a dash d e n o t e s d i f f e r e n t a t i o n w i t h r e s p e c t t o
5
No e x a c t s o l u t i o n i s a v a i l a b l e f o r (8.1) when rewrite (7.6) as t h e system
i s g i v e n by (7.6) and so w e
w = u' w' = cu'
- u2
f(u)
+ u
and t a k e as i n i t i a l c o n d i t i o n s u = 0.99
,
u' = 0.01
at
5
= X
,
A where X i s an a r b i t r a r y p o i n t t o t h e r i g h t o f t h e o r i g i n on t h e c - a x i s . c a l c u l a t i o n i s c a r r i e d o u t s t a r t i n g a t 6 = X and p r o c e e d i n g to t h e l e f t on a g r i d w i t h s p a c i n g h = 0.1. The i n i t i a l p o i n t X i s t h e n moved t o t h e r i g h t and t h e c a l c u l a t i o n r e p e a t e d . T h i s p r o c e s s i s c a r r i e d on u n t i l f u r t h e r m v e m e n t of t h e i n i t i a l p o i n t t o t h e r i g h t does n o t a l t e r t h e s o l u t i o n s i g n i f i c a n t l y . The numerical procedure used i s Hamming's p r e d i c t o r - c o r r e c t o r method (Lambert L1973,
166
A . R . MITCHELL et al.
p . 941) w i t h t h e Runge K u t t a f o u r t h o r d e r method used t o p r o v i d e t h e r e q u i r e d a d d i t i o n a l s t a r t i n g v a l u e s . I n agreement w i t h t h e o r y , s o l u t i o n s are o b t a i n e d f o r c 2 2 4 f ' ( 0 ) = 2. For v a l u e s of c < 2 , o s c i l l a t i o n s o c c u r i n t h e s o l u t i o n . The s o l u t i o n o b t a i n e d f o r c = 4 i s shown i n t h e graph f o r t = 0 i n F i g u r e 2 . We now look a t (8.1) when f ( u ) i s g i v e n by ( 7 . 7 ) . T h i s time a n e x a c t s o l u t i o n g i v e n by Huxley i s u(c) =
where
CI +
exp (- S / ~ ) I - '
- a).
42 (-21
(8.2) 1
O < a S (8.3) 2 T h i s i s a t r a v e l l i n g wave moving t o t h e left w i t h v e l o c i t y c and where u ( - m) = 0 , u(m) = 1. Other e x a c t s o l u t i o n s , which do n o t g i v e t r a v e l l i n g waves have been found by McKean C19701. They a r e c =
u(s) = 3 c j ( m + j where
u(-
~ ( c )= -2 + 1
m)
cash
= u(m) = u(m) = 0
k sn
,
m,
e+ -I-' I + '/a and
J=2),
(L 42 T
1 2
c = ~ , ~ < a - < -
o < k C T 1 ,
c = ~ , a =1 2 '
We now r e t u r n t o t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n (7.4) and s o l v e i t by t h e continuous i n time G a l e r k i n method where t h e weak s o l u t i o n , a f t e r i n t e g r a t i o n by p a r t s gives (8.4)
where a d o t d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o t i m e . The l a s t term is a boundary term which depends on t h e boundary c o n d i t i o n s of t h e problem. E v a l u a t i o n of ( 8 . 4 ) l e a d s t o t h e system of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s M g + S g = F ( g ) + b where
b =
M = C($Jj
, $Jill ,
(bo,O,--,O,bN)
T
.
S =
C($Ji
- 1k =
a =
-2I
$:)I
,
g
= (Uosu,
T, and
The t i m e v a r i a b l e i s now d i s c r e t i s e d a c c o r d i n g t o
t = mk
and we p u t
,
(8.5)
-
m = O,l,2,--
-am)
(CLm+' +
(8.6)
m = 0, I , 2 , - m = O,l,2,--
gm)
i n (8.5). T h i s g i v e s t h e Crank Nicolson G a l e r k i n (C.N.G.) method which i s second o r d e r i n t i m e , and r e q u i r e s t h e s o l u t i o n o f a system of n o n l i n e a r a l g e b r a i c e q u a t i o n s a t e a c h time s t e p . [ F a i r w e a t h e r C19781, Cannon and W i n g C197711. The t r i a l and t e s t f u n c t i o n s a r e t a k e n as piecewise l i n e a r s i n a l l c a l c u l a t i o n s .
-
Equation (8.5) w i t h f ( u ) u(l u) is now s o l v e d u s i n g the C.N.G. method, where t h e i n i t i a l c o n d i t i o n i s t a k e n t o b e t h e s o l u t i o n o f (8.1) when c = 4 , and + and u = 0 as x - + - a . The t h e boundary c o n d i t i o n s a r e u = 1 as x numerical r e s u l t s a r e shown i n F i g u r e 2 where we s e e t h a t t h e t r a v e l l i n g wave w i t h an i n i t i a l speed of 4 converges t o t h e a s y m p t o t i c speed 2 w h i l s t t h e p r o f i l e converges t o t h e shape of t h e s o l u t i o n of (8.1) w i t h c = 2 . I n the c a l c u l a t i o n s h = 0.5 and k = 0.1. T h i s v e r i f i e s n u m e r i c a l l y t h e t h e o r e t i c a l r e s u l t s of Aronson and Weinberger f o r t h i s p a r t i a l d i f f e r e n t i a l e q u a t i o n . F i n a l l y i n t h e c a s e when f ( u ) i s q u a d r a t i c , v a r i o u s examples o f i n i t i a l and boundary d a t a a r e shown t o converge uniformly t o u n i t y w i t h t h e v e l o c i t y of t h e waves converging t o (see Figure 3). 2 from below. -f
F I N I T E ELEMENT G A L E R K I N METHODS
U 1.0
0.8
0.6
0.4
0.2
0.0 0 Figure 2
U h
1.0
-
0.8 -
0.6 -
0.4
-
0.2
-
Figure 3
167
A.R. MITCHELL et al.
168
I
We now turn to ( 8 . 5 ) with f(u) = u ( l - u ) ( u - a) 0< a iand treat a 2 variety of initial and boundary conditions. In all calculations, h = k = 0.5 and I a = - . Choosing a rectangular pulse as shown in Figure 4 for initial condition, the 4pulse either dies away or grows to unity. In the latter case the two travelling waves (one in each direction) obey the Huxley formula (8.2) and (8.3) with regard to shape and velocity of propagation respectively. The threshold curve in Figure 4 shows the values of a and b (shaded area) which produce travelling waves. We now choose a rectangular pulse as a boundary condition at x = 0 as shown in Figure 5 and solve the problem in the quarter plane x z 0 , t 2 0. The solution again either dies away or tends to unity as time increases and the threshold curve is given in Figure 5. 9. NAGUMO'S EQUATION Here we study travelling wave (pulse) solutions of Nagumo's equation -au =at
where 0 < a
5
or as
ax
-21
+
u(l
-
u)(u
and 0 < b
dw = dt
bu
U'"4
cuff+
-
<< 1 .
fU u '
a)
-
b
(9.1)
This is often written as the system
+
-b u
= 0
(9.3)
-
where 5 = x ct , c > 0 and a dash denotes differentiation with respect to 5. Rinzel and Keller [I9731 have solved 69.3) where f is the caricature shown in Figure 6 with the sloping lines at 45 to the u-axis. They obtain the pulse shape and the speed diagram ( c plotted against a) for a range of values of b > 0. They also show that for each a and b in a range of values there are two pulse shaped wave solutions travelling with different velocities and only the faster pulse is stable. In addition to pulse solutions of (9.3), Rinzel and Keller have also demonstrated the existence of periodic wave trains and have shown that for each period two velocities of propagation exist. Once again only the faster wave train is stable. In order to obtain a C.N.G. numerical solution of Nagumo's equation in the form ( 9 . 2 ) we put N
to give
(9.4)
F I N I T E ELEMENT GALERKIN METHODS
169
b 15.0
10.0
5.0
00
0 25
0 5
0.75
10
,
d
Figure 4
and
k
(q+'- $1 4 (U!+'
where In a l l calculations
+ LJ;) ( i = 0,1 ,--,N)
=
WO
=
o
h = k = 0.5
i
=
o,I,--,N.
and t h e i n i t i a l c o n d i t i o n i s
u(x,O) = 0
0
s x
2 L
The boundary c o n d i t i o n s f o r t 5 0 c o n s i s t o f a c o n d i t i o n t o be s p e c i f i e d a t x = 0 t o g e t h e r w i t h one of t h e f o l l o w i n g a t x = L f o r L l a r g e (i)
u = 0
(ii)
aU = 0 ax
(iii) c
au + aU ax a t =
0
( c t h e wave speed)
(9.5)
A.R.
170
MITCHELL et a l .
15c
100
50
0.o
Figure 5
Figure 6
171
F I N I T E ELEMENT GALERKIN METHODS
The s o l u t i o n i s r e q u i r e d i n t h e s e m i - i n f i n i t e s t r i p LO 5 x f o l l o w i n g boundary c o n d i t i o n s a r e c o n s i d e r e d a t x = 0 .
5
L1
x [t 5
01.
The
where I i s c o n s t a n t . A s i n g l e t r a v e l l i n g p u l s e i s o b t a i n e d p r o v i d e d I and T exceed c e r t a i n t h r e s h o l d v a l u e s . I n p a r t i c u l a r f o r b = 0.0025 , I = 1.0 , and T = 15 , a t r a v e l l i n g p u l s e s o l u t i o n i s o b t a i n e d f o r v a l u e s of a i n t h e range 0 2 a 2 0.265. T h i s p u l s e i s shown i n F i g u r e 7 f o r t = 120. The speed curve f o r a range of v a l u e s of a and b f o r t h e t r a v e l l i n g p u l s e i s shown i n F i g u r e 8. u(0,t)
(2)
=
I
t 2 O
A s i n g l e t r a v e l l i n g p u l s e i s o b t a i n e d which t a i l s t o a boundary l a y e r a t
n(Tl
u(0,t) =
(3)
C a r e f u l t u n i n g of
+ T2)
5 t 5
n(T
1 + T2) + T I
n = O,l,2,--
n ( T I + T2) + TI < t < ( n + l ) ( T1 + T 2 ) and
T,
can f i r e a
T2
of t r a v e l l i n g p u l s e s .
example f o r a = 0.075 and b = 0.01 , w e have a t r a i n o f p u l s e s f o r T2 = 110. The f i r s t two p u l s e s are shown i n F i g u r e 9 f o r t = 250.
ax
(4)
1.0
(0,t) =
1 -2 I
x = 0.
For T I = 20
t 2 0
T" a =0.2
0.5 -
0.0 0
I
20
I
40
60
Figure 7
I
80
I
100
x'
,
A . R . MITCHELL et al.
172
C
0.6
0.4
0.2
0.0 0.0
I
I
0.4)
0.2 Figure 8
T h i s Neumann boundary c o n d i t i o n i s a p p l i e d f o r a range o f v a l u e s o f b b u t f a i l s t o produce a wave t r a i n . ( r e p e t i t i v e f i r i n g ) .
10.
FITZHUGH
- NAGUMO
These a r e
2
at
ax
-dw =
b(u
dt
,a
and
EQUATIONS
au a u + u(l = 7
1
I
-
-
u)(u
-
a)
-
w (10.1)
dw)
where 0 < a 5 7 and b d > 0 u s u a l l y small. T h i s system i s a s i m p l i f i e d model of impulse t r a n s m i s s i o n i n a n e r v e axon and c o n t a i n s most of t h e f e a t u r e s of t h e Hodgkin Huxley system b u t i s much easier t o h a n d l e . The changes i n t h e C.N.G. numerical procedure a s compared with Nagumo's e q u a t i o n , are t o r e p l a c e t h e l a s t term i n ( 9 . 4 ) by kb
1 (2+kbd
(10.2)
it0 and (9.5) by
(10.3)
Again i n a l l c a l c u l a t i o n s
h = k = 0.5.
173
F I N I T E ELEMENT GALERKIN METHODS
AuI
1.0
-
-0.5
Figure 9
For Dirichlet boundary conditions ( I ) , (2), and (3) at x = 0 the behavious are the same as in Nagumo's equation. For the Neumann boundary condition (4), however, we are successful in getting repetitive firing. The parameter values which generate such a phenomenon are very critical. Repetitive firing can be obtained by choosing a = 0.139
,
b = 0.008
, d
=
, I
2.54
(10.4)
= 0.6
and the resulting wave train is shown in Figure 10. The relative magnitudes of the parameters a, b, and d in (10.4) are in agreement with the results of Sleeman [19801. Numerical experiments carried out by Rinzel [I9771 suggest that for values of a , b and d for which repetitive firing occurs, I must lie in one or other of two ranges
11.
THE DRIFT TERM
We now look at reaction-diffusion equations involving significant first space derivatives. (drift terms). A s an example consider Fisher's equation in the form
a~
-=--
at
a2y ax
m
a + ~u ( l ax 1
where m (constant) > 0 and 0 < a 5 ct , c > 0 , ( 1 1 . 1 ) righti.e. 5 = x
-
u"
+ (c
- m)
.
u' + u ( l
-
u)(u
-
(11.1)
a)
If we look for waves travelling to the becomes
-
u)(u
This has the Huxley solution provided c - m = & ( - - a1 ) 2
- a)
=
0
.
( I 1.2)
174
A.R.
MITCHELL et al.
and so the velocity of waves travelling to the right is where m > 0 and 0
<
a
5
1 2
.
c
=
42
I
(2
-
a) + m
( I 1.3)
The main problem, however, in handling the drift term arises in the numerical solution of the partial differential equation ( 1 1 . 1 ) for arbitrary initial and boundary conditions. Experience gained in solving ( I 1. I ) without the reaction term has shown that both finite difference and finite element methods can give rise to large oscillations in the numerical solution if m has a significant value in modulus. To illustrate the point we show in Figure I I a C.N.G. solution (h = 0.5 , k = 0.1) of ( 1 1 . 1 ) for an advanced time where m = 20
,
a
=
0.25
, u(0,t)
= 1
, u(100,t)
=
0
.
The offending oscillations are clearly visible and may be removed by "upwinding". 12.
COUPLED REACTION-DIFFUSION SYSTEMS AND HIGHER SPACE DIMENSIONS
So far our numerical experiments have involved no more than one partial differential equation and only one space dimension, although even this limited structure has produced travelling waves in the form of pulses, fronts and periodic trains. Many important problems, however, involve coupled reaction-diffusion equations and higher space dimensions. Examples of the former are pulse evolution on coupled nerve fibres (Eilbeck, Luzader, and Scott C19791) and predator prey equations with diffusion (Murray L19801).
The generalisation to higher space dimensions has been considered by Aronson and Weinberger [I9781 for the equations of population genetics. However, as far as asymptotic behaviour and stability is concerned not very much is known. The paper by Conway, Hoff, and Smoller C19781 gives the most definitive results so far. It is felt that until analytic tools are available to handle systems in higher space dimensions, numerical studies, properly conducted, will be invaluable.
F I N I T E ELEMENT GALERKIN METHODS
175
Figure 10
I
'U 1.5
1.0
0.5
0.0
a
-0.5 Figure 1 1
176
A.R.
MITCHELL et a 1
REFERENCES
[I1 Aronson, D.G. and Weinberger, H.F., Nonlinear diffusion in population
genetics, Combustion and nerve propagation, Partial differential equations and related topics, Lecture Notes in Mathematics vol. 446, Berlin Sprenger-Verlag 1975.
C21 Aronson, D.G. and Weinberger, H.F., Multidemensional nonlinear diffusions arising in population genetics, Advances in Math. 30 (1978) 33-76.
C 31 Cannon, J.R. and Ewing, R.E., Galerkin procedures for systems of parabolic partial differential equations related to the transmission of nerve impulses, Nonlinear Diffusion (eds. Fitzgibbon, W.E. and Walker, H.F.), Research Notes in Mathematics 1 4 , Pitman 1977. C41 Christie, I. and Mitchell, A.R., Upwinding of high order Galerkin methods in conduction-convection problems, Int. J. Num. Meth. Eng. 12 (1978) 1764-1771.
-
C51 Dendy, J.E., Two methods of Galerkin type achieving optimal LL accuracy for first order hyperbolics, SIAM J. Num. Anal. I I (1974) 637-653.
[61 Dupont, T., Galerkin methods for first order hyperbolics : An example, SIAM J. Num. Anal. 10 (1973) 890-899. C71 Eilbeck, J.C., Luzader,S.D. and Scott, A.C., Pulse evolution on coupled nerve fibres (Preprint 1979). [81 Fairweather, G., Finite element Galerkin methods for differential equations, Marcel Dekker, Basel 1978. [9l
Fife, P.C. and McLeod, J.B., The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335-36 1.
ClOl Fife, P.C. and McLeod, J.B., A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, M.R.C. Technical Summary Report 1986 (1979).
[Ill Fisher, R.A., The advance of advantageous genes, Ann of Eugenics 7 (1937) 355-369.
[I21
Fitzhugh, R., Impulses and physiological states in theoretical models of nerve membranes, Biophysical J. 1 (1961) 445-466.
[I31
Griffiths, D.F. and Mitchell, A.R., On generating upwind finite element methods in "Finite elements for convection dominated flows" (ed. Hughes, T.J.R.) AMD V O 34 ~ A.S.M.E. (1979) 91-105.
[I41
Hedstrom, G.W., The Galerkin method based on Hermite cubics, SIAM J. Num. Anal. 16 (1979) 385-393.
CIS1 Hodgkin, A.L. and Huxley, A.F., A qualitative description of membrane current and its application to conduction and excitation in nerves, J. Physiol.117 ( 1 952) 500-544. C161 Heinrich, J.C., Huyakorn, P.S., Mitchell, A.R. and Zienkiewicz, O.C., An
upwind finite element scheme for two dimensional convective transport equations, Int. J. Num. Meth. Eng. 11 (1977) 131-144.
c171 Hughes, T.J.R. and Brooks, A., A multidimensional upwind scheme with no crosswind diffusion in "Finite elements for convection dominated flows" (ed. Hughes, T.J.R.) AMD vol. 34 A.S.M.E. (1979) 19-37.
F I N I T E ELEMENT GALERKIN METHODS
C181
177
Johnson, C . and Ngvert, U . , A n a l y s i s of some f i n i t e element methods f o r a d v e c t i o n - d i f f u s i o n problems, P r i v a t e Communication (1980).
[I91 Lambert, J.D., Computational methods i n o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , Wiley 1973. c201 McKean, H.P.,
Nagumo's e q u a t i o n , Advances i n Math. 4 (1970) 209-223.
1211 M i t c h e l l , A.R. and G r i f f i t h s , D.F.,
Upwinding by Petrov-Galerkin methods i n c o n v e c t i o n - d i f f u s i o n problems, U n i v e r s i t y of Dundee Report NA/36 (1979).
c221 M i t c h e l l , A.R.
and G r i f f i t h s , D.F., The f i n i t e d i f f e r e n c e method i n p a r t i a l d i f f e r e n t i a l e q u a t i o n s , John Wiley, London 1980.
C231
M i t c h e l l , A.R. and Wait, R . , The f i n i t e element method i n p a r t i a l d i f f e r e n t i a l e q u a t i o n s , John Wiley, London 1977.
C241 Murray, J . D . ,
A p a t t e r n f o r m a t i o n mechanism and i t s a p p l i c a t i o n t o mammalian c o a t markings, L e c t u r e Notes i n Bio. Mathematics, Springer-Verlag (1980) ( t o a p p e a r ) .
C251
R i n z e l , J., R e p e t i t i v e nerve impulse p r o p a g a t i o n : numerical r e s u l t s and methods, N o n l i n e a r d i f f u s i o n ( e d s . F i t z g i b b o n , W.E. and Walker, H.F.), Research Notes i n Mathematics 14, Pitman 1977.
C261
R i n z e l , J . and Keller, J . B . , T r a v e l l i n g wave s o l u t i o n s o f a n e r v e c o n d u c t i o n e q u a t i o n , Biophys. J. 13 (1973) 1313-1337.
C271 F a i t h b y , G.D.,
Skew upstream d i f f e r e n c i n g schemes f o r problems i n v o l v i n g f l u i d f l o w , Computer Methods i n Applied Mechanics and E n g i n e e r i n g 9 (1976)
153- 164.
C281 [29]
Sleeman, B . D . ,
349.
F i t z h u g h ' s n e r v e axon e q u a t i o n s , J. Math. B i o l . 2 (1975) 341-
Sleeman, B.D., I n s t a b i l i t y of c e r t a i n t r a v e l l i n g wave s o l u t i o n s t o t h e Fitzhugh-Nagumo nerve axon e q u a t i o n s , J . Math. Anal. A p p l i c s . ( t o a p p e a r )
C301 Wahlbin, L . B . ,
A modified G a l e r k i n p r o c e d u r e w i t h Hermite c u b i c s f o r h y p e r b o l i c problems, Maths. of Comp. 29 (1975) 978-984.
[31] W i n z e l l , B.,
F i n i t e element a n a l y s i s o f d i f f u s i o n - c o n v e c t i o n e q u a t i o n s , Report LiTH-MAT-R-80-07 Linkgping I n s ti t u t e o f Technology, Sweden, ( I 980).
This Page Intentionally Left Blank
ANALYTICAL AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS I N A N A L Y S I S S . A x e i s s o n , L . S . F r a n k , A . van d e r S i u i s ( e d s . ) 0 N o r t h - i l o l l a n d P u b l i s h i n g Company, 1 9 6 1
NUMEBICAL SOLUTION OF SINGULAR PEXTURBATION PROBLEMS AND HYPE3BOLIC SYSTEMS OF CONSERVATION LAWS Stanley &her* Mathematics Dep ax tme nt University o f C a l i f o r n i a Los Angeles, C a l i f o r n i a 90024
U.S.A.
We have r e c e n t l y , with Bjorn Ehgquist and various s t u d e n t s , developed simple upwind f i n i t e d i f f e r e n c e and element methods approximating nonlinear p a r t i a l d i f f e r e n t i a l equations. These methods a r e used t o approximate (1)g e n e r a l systems o f nonlinear hyperbolic conservation laws, ( 2 ) nonlinear s i n g u l a r p e r t u r b a t i o n problems, and ( 3 ) p a r t i c u l a r p h y s i c a l problems involving t h e equations of c m p r e s s i b l e f l u i d dynamics. These methods a r e designed t o have many of t h e following p r o p e r t i e s : 1. High order accuracy i n smooth regions of t h e flow. 2. Shocks which s a t i s f y t h e c o r r e c t jump conditions. 3. Physically c o r r e c t shocks, i.e. those s a t i s e i n g an entropy c o n d i t i o n . 4. Nonlinear s t a b i l i t y . 5. Good shock r e s o l u t i o n minimal overshoot and minimal smearing. (Good. boundary and i n t e r n a l l a y e r r e s o l u t i o n f o r s i n g u l a r p e r t u r b a t i o n problems). 6. (For s t e a d y problems) r a p i d convergence t o s t e a d y s t a t e . 7. (For s t e a d y s i n g u l a r p e r t u r b a t i o n problems) convergence f o r all s u f f i c i e n t l y s m a l l values of E > 0, and f o r all i n i t i a l guesses. 8. Using t h e s m a l l e s t p o s s i b l e number of boundary c o n d i t i o n s .
-
We s h a l l p r e s e n t t h e methods, give d e t a i l e d proofs of sane new theorems, and give computational r e s u l t s . I n s e c t i o n one we p r e s e n t t h e upwind d i f f e r e n c e algorithms f o r a s c a l a r conservat i o n laws i n one space dimensions. Connections with t h e s m a l l d i s t u r b a n c e equation of t r a n s o n i c flow were p o i n t e d out i n L2.1, [ 3 ] , [41. Results proven elsewhere w i l l be discussed here (as i n all s e c t i o n s ) . I n s e c t i o n two we p r e s e n t t h e upwind f i n i t e element algorithm f o r a m u l t i dimensional s c a l a r conservation law. Proofs o f s e v e r a l r e l e v a n t theorems w i l l be given i n Appendix one. I n s e c t i o n t h r e e we p r e s e n t t h e upwind f i n i t e d i f f e r e n c e algorithm f o r a wide c l a s s of s i n g u l a r l y perturbed s c a l a r two p o i n t boundary value problems. I n s e c t i o n four we p r e s e n t t h e upwind f i n i t e element algorithm which a p p l i e s t o a wide c l a s s of s i n g u l a r l y perturbed s c a l a r e l l i p t i c boundary value problems. Proofs of s e v e r a l r e l e v a n t theorems w i l l b e given i n Appendix two. I n s e c t i o n f i v e we p r e s e n t t h e upwind f i n i t e d i f f e r e n c e algorithm f o r g e n e r a l s t r i c t l y hyperbolic systems of nonlinear conservation laws i n one space dimension. I n s e c t i o n six we p r e s e n t t h e d e t a i l e d upwind a l g o r i t h f o r one important p h y s i c a l problem compressible i n v i s c i d gas flow i n Lagrange coordinates -and a l s o d i s c u s s t h e p o t e n t i a l equation of t r a n s o n i c flow and E u l e r ' s equations i n one and two dimensions.
-
* Research
p a r t i a l l y supported by NASA #INCA-2 -0R390-002.
- Ames 179
University consortium interchange
180
S . OSHER
I n s e c t i o n seven we discuss numerical r e s u l t s f o r problems of c m p r e s s i b l e i n v i s c i d gas dynamic.
I. UPWIND FINITE DIFFERENCE APPROXIMATIONS FOR A SCALAR CONSERVATION LAW IN ONE SPACE DIMENSION We consider t h e equation :
with
f
+ f(u)x
ut
(1.1) E
= 0
C2(R1). with - Zn-' ~ i = atp o
x . = jh, t
We s e t up a g r i d
J
+- uj
define t h e d i f f e r e n c e operators To o b t a i n f ' ( u ) 2 0,
=
un
approximating u(x
j
-
? ( u ~ + ~u.).
-
j'
t"), and
J
1 if
f h s t order d i f f e r e n c e approximation, we f i r s t l e t X(u) X(u) 0 i f f ' ( u ) < 0 and t h e n d e f i n e
OUT
f
=J
fJU)
0
f-(u) =
It i s c l e a r t h a t
f;(u)
s i n g l e minimum a t
-2 0,
U
JOU
5
f:(u)
X(s)fl ( s ) d s
(1 0.
- x(s))f~(s)ds Moreover i f
f"(u) 2 0
and
f
has a
u = u, t h e n ( 1 . 2 ) reduces t o : f (u) = f(max(u,u))
+
(1-3 ) We note t h a t
In general
f-(u) = f(min(u,u)).
f;(u) f + and
2 0,
f'-(u) 5 0,
f _ are
f ' ( u ) = fk(u)
and piecewise
C1
second d e r i v a t i v e s only at c r i t i c a l p o i n t s of
C2,
+
f;(u). having p o s s i b l e jumps i n
f.
Our b a s i c upwind d i f f e r e n c i n g i s defined as follows:
(1.4)
f(dx
--f
&+f
-(Uj
1 + A_f+(Uj 11.
For t h e time d i f f e r e n c i n g , we l e t
(1.5) and define
xn
= Atn/&.
The r e s u l t i n g scheme is
-
This scheme is f i r s t order accurate and monotone G i s a nondecreasing f u n c t i o n of i t s arguments i f t h e CFL c o n d i t i o n I h f ' (u)I 5 1 is_v a l i d . !thus, by t h e r e s u l t s of [l], [61, it follows t h a t f o r i n i t i a l d a t a i n L1 n ,L and of bounded
-
181
NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS
v a r i a t i o n , t h e approximate s o l u t i o n converges t o t h e p h y s i c a l l y c o r r e c t s o l u t i o n of (1.1) as
h, atn + 0.
We t u r n now t o higher order upwind approximations of (1.1). Consider a method of l i n e s approach t o
(1.7)
ut = aux
for a w constant
a > 0.
We l e t
uj(t)
approximate
g e n e r a l upwind d i f f e r e n t i a l - d i f f e r e n c e equation:
The scheme is of order of accuracy
u(x.,t) = u(j&,t) J
i f f o r smooth functions
q
via the
q(x),
[ ? I , [41.
Then we have t h e following theorem
Theorem (1.1). If (1.8)i s s t a b l e , t h e n t h e scheme has order of accuracy a t most 2. Moreover t h e r e do e x i s t s t a b l e second order approximations and t h e most compact ( s m a l l e s t p ) i s given by B, = 1, B2 = -1/2 f o r p = 2. Next we c o n s t r u c t a simple second order upwind approximation t o (1.1). Our f i r s t attempt i s t h e simplest p o s s i b l e second order analogue of ( 1 . 4 )
- 1/2 f):n
f(uIx
(1.10)
The r e s u l t i n g scheme w a s shown i n some overshoot for s t e a d y shocks. f(u) =
a(u
- u)’
with
a > 0,
-
(u.) J
+ (A- +
1/2 A2)f - +(u j ) I .
[41, both
a n a l y t i c a l l y and numerically, t o have For t h e important s p e c i a l c a s e
and
<
a r b i t r a r y , we propose i n s t e a d an upwind
second order scheme based on
+
f (u.) +
- +
J
where
This scheme i s fully one sided, uses t h e minimum number of mesh p o i n t s ( t h r e e )
182
S. OSHER
-
away from t h e sonic p o i n t u, and i s of second order accuracy everywhere except a t u = U where it may degenerate t o f i r s t order. We have proven i n [3], [ 4 ] t h a t t h e d i s c r e t e
space norm i s nonincreasing i n L2 time f o r s o l u t i o n s of t h i s d i f f e r e n c e d i f f e r e n t i a l equation. Moreover l i m i t s o l u t i o n s must have t h e c o r r e c t jump conditions and s a t i s f y t h e entropy condition across such jumps.
These r e s u l t s were proven f o r t h e method of l i n e s semi-discrete scheme (l.ll), which we may r e w r i t e as
(1.12) However t h e y axe e a s i l y seen t o go through f o r t h e i m p l i c i t d i f f e r e n c e scheme of Crank-Nicolson type *un+l
At”
( f o r a w value of
- u?) = -l/&
Ln,
J
J
kn).
+
+ “1
(“‘+I
They can a l s o be shown t o be v a l i d for an e x p l i c i t Lax-Wendroff type of d i f f e r e n c i n g with which we experimented s u c c e s s f u l l y i n [41. This method i s l i n e a r l y s t a b l e with CFL number 2, and t h e numerical evidence r e p o r t e d i n [41 i n d i c a t e s t h a t t h i s s t a b i l i t y result i s v a l i d nonlinearly &s well. Next we p r e s e n t a n a l y t i c evidence t h a t our schemes give e x c e l l e n t shock r e s o l u t i o n . For a conservation form approximation t o a s c a l a r conservation l a w
we follow [TI, and d e f i n e a t r a v e l i n g wave s o l u t i o n which moves with speed t o be a s o l u t i o n of
(1.15)
uj -sX
= G(uj+k’
*
.
*
s
’Uj
L R Let t h e s o l u t i o n of (1.15) s a t i s w lhj+-m u = u , limj_tm u j = u , t h e j R jump conditions s(uR u L ) = f ( u ) - f ( u L ) , and O l e i n i k ’ s s t r i c t condition
f ( u ) - f(uR) < f(uL) R
u - u
uL
f(U
- UR
R,
for
u
E
R L ( u ,u 1.
Then we have proven [3], [41 e x i s t e n c e , u n i ueness, monotonicity, and g l o b a l s t a b i l i t y of d i s c r e t e t r a v e l i n g waves f o r ? l . 6 ) , extending r e s u l t s of [ T I t o t h i s n o n - s t r i c t l y monotone scheme. More important, we show t h e e x i s t e n c e of s h a r p p r o f i l e s , which i s t h e content of t h e following: R Theorem ( 1 . 2 ) . ( a ) Let s > 0 and f ’ ( u ) < 0. Then t h e monotone t r a v e l i n g wave s o l u t i o n s t o (1.6) f o r s h r a t i o n a l have t h e p r o p e r t y t h a t t h e r e e x i s t s
uR f o r j >- j,. ( S i m i l a r l y for s < 0 and f’(uL) > 0 ) . L R = 0 f o r convex f with f ’ ( u ) 0 > f’(u ), t h e n t h e r e e x i s t s L - R L j o such t h a t u = u , j < j,, u = u , j > j o + 1 with u E [u,u 1, j,
(b)
with
u
Let
s
j
j
j
j0
E
183
NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS
f - ( u j +1) + f ( u . ) = f - ( u L ) + f + ( u L ). 0 Jo F i n a l l y we a l s o have i n f i n i t e r e s o l u t i o n for s t e a d y shock s o l u t i o n s of t h e second order scheme (1.11)f o r convex f ( u ) .
U
t
j
[uR,u]
s u b j e c t only t o t h e c o n d i t i o n
+
meorem (1.3). The s t e m t r a v e l i n g wave s o l u t i o n s t o (1.11)f o r convex a r e t h e same as t h o s e f o r ( 1 . 6 ) mentioned i n p a r t ( c ) of Theorem ( 1 . 2 ) .
f(u)
These waves look l i k e :
UPWIND FIIYITE ELEMENT APPROXIMATIONS FOR MULTI-DIMENSIONAL
11.
CONSERVATION LAWS
SCALAR
We consider t h e s c a l a r multi-dimensional problem u
(2.1) t > 0,
for f
t
t
x
E
n,
where
C2(n) with values i n
Let
{$}
+ V.f(u)
N R .
is either
or an N
R"
dimensional t o r u s , and
n i n t o closed nonoverlapping polyhedra.
be a decomposition of
assume t h e followirg p r o p e r t y :
u ( x , o ) = o(x)
= 0,
h F(Tj)
if
(resp.
p(T:))
is t h e s m a l l e s t ( r e s p
t h e l a r g e s t ) diameter of t h e b a l l s c o n t a i n i n g ( r e s p . contained i n ) t h e r e e x i s t s a p o s i t i v e constant
j
We approximate
'I? t, hen
such t h a t
K-% 5 i n f p($)
(2.2)
(2.3)
K
We
5 sup ~ ( $ 1 5 a. j
u using t h e decomposition
u h ( x , t ) = ~ $ ( x ) p ~ ( t ) ,with
3
olhJ
$
l?
Ph a p o r t i o n of a hyperplane bounding and we d e f i n e J ,J J t h e u n i t normal. t o P ! p o i n t i n g out of J,l J We t h e n use t h e flux d e c m p o s i t i o n of t h e previous s e c t i o n t o d e f i n e t h e s c a l a r functions
Given
l?. J
t h e c h a r a c t e r i s t i c f u n c t i o n of
l?.
h
v.
J,J'
(2.4)
are, for fixed
t,
piecewise constant functions of
x
having d i s c o n t i n u i t i e s
184
S . OSHER
( i n general) a t t h e boundary of each
5. For J
t
f i x e d and f o r each
P
introduce t h e following piecewise constant f’mctions defined on
j,a’
we
J
h
We d e f i n e we l e t
h
u_ ( x , t ) = BJ. ( t ) ) .
(noting t h a t
vj)-
f(Eh(x,t))
and
f(u -h ( x , t ) ) . v j ) +
i n t h e same fashion.
Finally
be t h e volume of
Ah j
We f i r s t d e f i n e t h e semi-discrete approximation t o
J
with
Bh .(O)
=
J
h un(x)
N e x t we l e t B:(&).
(I T(x)dx)/A;. 8J
For
approximate
n = O,l,.
..,
h
u ( x , d t ) by l e t t i n g Bh
j,n
we l e t
approximate
(2.9) -h
with
h
f3j,o
h = B.(O). J
.1
We impose t h e CFL c o n d i t i o n
The main theorems, s t a t e d i n [lo] and proven i n Appendix 1 below, a r e : h h Suppose u ( x , t ) and v ( x , t ) Theorem (2.1). t h e n have t h e estimate f o r any t 2 0
a r e defined by ( 2 . 3 ) , ( 2 . 8 ) .
W e
185
NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS
Similarly, for n 1 0
h un(x)
that for
defined by (2.3), (2.9), we have f o r any
For a f’unction of the type
Definition (2.1).
where each
h and vn(x)
ThJ
is the centroid o f
x
8J j and
Theorem (2.2).
uh =
cj
c$(x)$
we define
and the sum is taken over all
j,J
such
have a comon plane boundary
!I$
Suppose
Ph Then we have: j,J* i s defined by (2.3), (2.8), or (2.3), (2.9) and t h e
h
u
sequence converges a.e. t o u as h + O . Moreover, suppose B(uh) i s uniformly bounded f o r all h and t . Then u i s a weak solution of ( 2 . 1 ) which s a t i s f i e s the entropy condition. Remark. For N = 1, Richard Sanders [ l ) ] has proven t h a t the assumptions made above are v a l i d for i n i t i a l . d a t a i n BV n L, n L1. 111. UPWIND FINITE DIFFERENCE APPROXIMATION FQR A NONLINm SINGVMLY PEBTURBED TWO POINT BOUNDARY VALUE PROBLFM
We consider the boundary value problem: (3.1)
y“
- a(y)y’ - b(x,y)
(a)
E
(b)
d-1) = A ,
= 0,
-1 L
X
51
Y(l) = B
where 0 < E <<1, A and B are a r b i t r a r y constants, a(y), b(x,y) are C1 functions with b (x,y) 2 6 > 0. The function a(y) can have any number of Y Neverthezeros of a r b i t r a r y order i n t h e i n t e r v a l min(A,B,O) 5 y 5 max(A,B,O). less we present a simple numerical method whose solutions converge t o the correct solution of (3.1) f o r a~ values of E.
We rewrite
a(y)yt = ( f ( y ) ) ’ ,
differencing of the term approximate : (3.2)
(a)
E
uxx
-
dx
f(u)x
for
a(y) = f ’ ( y ) .
Our method i s based on upwind
and uses an a r t i f i c i a l time parameter.
f(y)
- b(x,u)
= ut
t o be solved f o r
(x,t)
E
We
[-1,11 x R+,
with the same time independent boundary conditions. (b)
u ( - l , t ) = A,
U(l,t)
=B
and i n i t i a l condition ( c ) uo(x,o) = uo(x). Our only requirement on
uo(x)
i s t h a t it be bounded and measurable.
Solutions
1a6
S.
OSHER
of our d i s c r e t e approximation t o ( 3 . 2 ) a r e shown t o converge t o s o l u t i o n s of our approximation t o ( 3 . 1 ) independently of our i n i t i a l guess and independently of E. We s e t up a g r i d
x j = jh, t n = dt,
n 2 0 j = O,+-l,.
An e x p l i c i t f i r s t order upwind approximation t o
(3.3)
(a)
. .,+N
with
N h = 1.
(3.1) i s t h u s :
- A t b(x,u;) - x(A-f +(u?.) J + A+f-(uj”))
u?+l = un J J
- 1;
.,+-N
j = O,+),..
n 2 0,
with boundary conditions : (b)
uyN = A,
n
%
= B
and i n i t i a l conditions (c)
0 u j = u (x O
j
1.
We require t h e CFL c o n d i t i o n (d)
h( If1(u)I
E)
< 1 - A t bU(x,u).
+
I n order t o approximate ( 3 . l ) , we seek time independent s o l u t i o n s of ( 3 . 1 ) . Hence we must solve:
(3.4)
(a)
- D-f +(Y j - D+f-(yj) j = O,+-I- ,..., + - N - 1. 0 = -b(x.,y.) J J
( b ) Y-N = A, where, as u s u a l ,
+
D+D-yj
YN = B
D+ = I/&
-
4. -
The main r e s u l t s s t a t e d and proven i n [lo] a r e :
z = ( z )N j j=-N
E
i s t h e l i m i t as
of t h e vectors 0 . Moreover -&At CN 0 t h e i t e r a t e s of (3.3) are such t h a t XN J=-N lun j zJ. 1 -< e j=-Nluj “ 1J. Strong m a x i m u m princ‘iples a r e v a l i d f o r ( 3 . 3 ) and ( 3 . 4 ) and they a r e used t o
u
=
t o ( 3 . 4 ) and furthermore
t h e r e e x i s t s a unique s o l u t i o n
z
n
-+m
computed from (3.3) independently of t h e choice of
-
u
prove a convergence r e s u l t when b(x,O)
0.
piecewise l i n e a r extension of
Then t h e f a m i l y of s o l u t i o n s
z =
(2.3.
J
Let
zhJE(x)
-
be t h e n a t u r a l
zhJE(x)
h, E bounded has a converging subsequence, and l i m i t s o l u t i o n s f o r h + 0 are weak s o l u t i o n s ( s t r o n g i f E > 0 ) of (3.1) f o r t h e corresponding E. All limit s o l u t i o n s have v a r i a t i o n bounded by IBI + IA/. i s precompact i n
L1[-l,ll.
Thus any i n f i n i t e c o l l e c t i o n with
The one drawback of t h e e x p l i c i t time i t e r a t i o n is t h e r e s t r i c t i v e CFL c o n d i t i o n ( 3 . 3 ) ( d ) . Newton‘s method w a s t r i e d on (3.4) numerically with great success. It i s i n t e r e s t i n g t o note t h a t t a k i n g t h e d i f f e r e n t i a l of t h e nonlinear operator i n (3.4) at a s t a t e F = (7.] leads t o a l i n e a r t r i d i a g o n a l problem: J
(3.5)
( a ) F ( x . 1 = -b (x T )Y J Y j’ j j . ,+N 1 j = O,+l,.
.
-
- D - ( f+’ ( T j ) YJ. )
- D+(fL(T- j )Yj
+ E D+D-yj
-
187
NUMERICAL SOLUTION OF S I N G U L A R PERTURBATION PROBLEMS
which has a unique s o l u t i o n s a t i s f y i n g t h e e s t i m a t e
N
c lvjl j =-N
N-1
c
5
IF(xj)l
j =-N+1
y.
independently of E and t h e s t a t e Thus t h e i n v e r s i o n of t h e l i n e a r i z e d problem f o r Newton's method p r e s e n t s no d i f f i c u l t i e s . A suggested procedure i s t o use t h e a r t i f i c i a l time i t e r a t i o n on a coarse g r i d , t h e n use t h e r e s u l t i n g s o l u t i o n s u i t a b l y i n t e r p o l a t e d , on a f i n e g r i d (perhaps r e f i n e d near evident boundary and i n t e r n a l layers) as an i n i t i a l guess f o r Newton' s method. Preliminary c a l c u l a t i o n s i n d i c a t e e x c e l l e n t success with t h i s approach.
IV. UPWINE FINITE ELEMENT AF'PROXlMATIONS FQR NONLINEAR SINGULARLY PERTURBED DIRICHLET PROBLtENS We consider t h e boundary value problem
(4.1) with
UI
= $(x)
f
2
C
- V . f ( u ) - b(x,u)
nu
E
(d). The domain
t h e following p r o p e r t y :
For any
N
R c R
in
= 0
has piecewise
R
nonoverlapping union of closed r e c t a n g u l a r p a r a l l e l i p i p e d s p a r a l l e l t o t h e planes
x . = 0, J
property ( 2 . 2 ) above and t h a t of any p o i n t of
J
.. ,n.
j = 1,.
R CR
h
.
contains a boundary p o i n t of
T!
C2
boundary and has
h > 0 w e can f i n d a domain
Clh
with s i d e s
{<)
We r e q u i r e t h a t
which is t h e
Oh
satisfy
Moreover we r e q u i r e t h a t i f t h e i n t e r i o r Rh,
t h e n it contains a l s o a boundary
R.
We use t h e piecewise constant elements and all t h e other n o t a t i o n of s e c t i o n I1 above t o d e f i n e Ph between j,a
(4.2)
uh(x).
T"j
and
I n a d d i t i o n we approximate
$
by d e f i n i n g :
& aP
(x,)
-
- I(x, - x j )
&I on t h e p l a n a r avhj , a
(Xj) *
vh j,J
I
boundary
S. OSHER
188
(4.3)
- A:
b ( x j , u h (x,)) = 0, h u (x ) with t h e mean value of j i s not empty.
with boundary conditions obtained by r e p l a c i n g on
q(x)
i~
n$
i~
if
n interior
of
ThJ
We a l s o , as i n s e c t i o n 111, have a time dependent extension E
(4.4) with
v1
= q(x)
Av
- V.
- b(x,v)
f(v)
v(x,O) = cp(x)
and
E
Ll(Q)
=
vt
in
n Lm(n),
R
for
t >0
a semidiscrete
appr ox i m a t i o n :
with
h
/
p.(O) = 1 / A . J
J
q ( x ) d x and t h e same boundary condition as i n (4.3), and
Tj
f i n a l l y a f u l l y d i s c r e t e e x p l i c i t approximation
J
with
po ,:
condition:
h
= B.(O), J
J
t h e same boundary c o n d i t i o n
We have t h e following four theorems
BS
i n (4.3), and with t h e CFL
- s t a t e d i n [ll] and proved
i n t h e appendix:
189
NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS
Theorem ( 4 . 1 ) .
There e x i s t s a unique (piecewise c o n s t a n t ) s o l u t i o n t o ( 4 . 3 ) , h u,(x). The s o l u t i o n s have uniformly bounded L1 and L, norms
which we denote aS
h,E + 0.
n
Moreover
is t h e l i m i t as
u,(x)
t
of s o l u t i o n s of ( 4 . 5 )
--tm
and as n --fa, of s o l u t i o n s of (4.6), independently of t h e choice of F i n a l l y , t h e following r a t e of convergence e s t h a t e s a r e v a l i d :
-
k e j g h ~ vhn ( x ) uhE ( x ) ~ 5 Theorem ( 4 . 2 ) .
Suppose
b(x,O)
common p l a n a r boundary with either
(a)
or
(b)
oh
I@(x)
{$,) be t h e s e t of J
l?.Then, f o r f i x e d J
h
5 (1 - at) m a x
j,{Aj
h v n + l ( ~ )j 2 (1
- U&(X)(dX h
Then t h e following maximum p r i n c i p l e f o r
0.
Let
(4.3) is v a l i d .
( 4 . 6 ) , and hence
(4.9)
=
-y
cp(x).
-
Sat)
min
j,
ThCI
which have a
j:
h
3
{Jj3
vn(xe ) 2 0, j h vn(xe ) 5 0, j
Similar a s s e r t i o n s hold for (4.5).
or both.
bounded f o r
0 <
E
<
EO,
o f ( 4 . 5 ) i s such t h a t
[u:(x)]
If t h e family of s o l u t i o n s
0 < h < ho,
h B(uE) i s uniformly
t h e n an e x t e n s i o n of an argument i n [13]
serves t o show t h a t t h e family i s precompact i n
L1(R).
(B(uE)
i s indeed
uniformly bounded i n t h e one dimensional c a s e discussed i n s e c t i o n 11.) Then we have: Theorem (4.3).
Under t h e above assumption, it follows t h a t any c o l l e c t i o n
of s o l u t i o n s of ( 4 . 3 ) ,
-
h
s o l u t i o n of ( 4 . 3 ) .
-
h =
-
E =
0,
then
t e s t function (4.10)
a l i m i t p o i n t of 0
If
uo(x)
q(x)
a,
V
(h,E)
with,
u&x)
hv
[uE ( x ) ) ~ ~has , a converging subsequence i n
k=
0
< F,
pv, . then
E8
u$x)
If h >
-
0,
i s a s o l u t i o n of ( 4 . 1 ) .
i s a we& s o l u t i o n of ( 4 . 1 ) f o r
vanishing near
&,
J,(V X ) * f ( U z ( x ) )
Ll(il)
h t h e n u$x)
is a
If
E = 0, i . e . f o r any
then
- q(x)b(x,uo(x))& 0
to
= 0.
Again, as i n s e c t i o n 111, we l i n e a r i z e ( 4 . 3 ) a t ,uh f o r an unknown l i n e a r equations become
h y .
The
S.
190
OSHER
(4.11)
yh (x.) = 0 i f
with boundary conditions: We have Theorem ( 4 . 4 ) .
For a l l
J
h u (x.)
unique s o l u t i o n sat i s f y i n g
J
&h
n interior
of
?J
i s nonempty.
t h e l i n e a r system o f equations ( 4 . 9 ) has a
V. UFWIM) F I N I T E DIFFERENCE APPROXIMATIONS FOR HYPERBOLIC SYSTEM3 OF CONSERVATION LAWS We consider t h e hyperbolic system of conservation laws Ut
(5.1) 2 d with u : R + R , f : Rd +Rd, r e a l d i s t i n c t eigenvalues X,(u)
+
f(u)x = 0
and t h e Jacobian matrix < h 2 ( u ) < ... < Xd(u).
&(u)
has, for each
u,
We wish t o c o n s t r u c t a vector valued analogue of t h e s c a l a r f l u x s p l i t t i n g (1.2), and t h e approximation t o f ( u ) x defined i n ( 1 . 4 ) . It i s s u r p r i s i n g t h a t we can
-
construct such a scheme s a t i s f y i n g t h e important p r o p e r t y steady d i s c r e t e in t h e same sense as for p a r t ( b ) of Theorem ( 1 . 2 ) shocks a r e resolved e x a c t 1 and f o r Theorem ( 1 . 3 d e o v e r t h e scheme r e s o l v e s c e r t a i n c o n t a c t d i s are continuates even more e x a c t l y no intermediate p o i n t s u . and u j 0+1 J0 needed. P r o p e r t i e s ( 2 ) and ( 3 ) mentioned i n t h e i n t r o d u c t i o n a r e a l s o v a l i d . The scheme i s e x p l i c i t and e a s i l y programmed f o r t h e p h y s i c a l problems mentioned i n t h e next s e c t i o n .
-
I n t h e s c a l a r case, ( 1 . 4 ) can be r e w r i t t e n
I n t h e v e c t o r case, we l e t X(u) be t h e n a t u r a l p r o j e c t i o n onto t h e p o s i t i v e eigenspace of &(u), i . e . i f we l e t T(u) be t h e matrix whose columns a r e t h e eigenvectors of &(u), t h e n
191
NUMERICAL SOLUTION OF S I N G U L A R PERTURBATION PROBLEMS
(5.3)
Then we d e f i n e t h e m a . t r i x
X(u) = T ( u ) d i a g . {1/2
(5.4)
+
1/2 sign(Xk(u))}T-(u)
and
We 'denote t h e p a t h i n Rd connects
u. J
to
connecting
u. ). J +1
The curve
rj
u. t o u. . by r j J -1 J
(and of course
i s decomposed i n t o
d
subcurves
These subcurves a r e r e l a t e d t o r a r e f a c t i o n s o l u t i o n s of (5.1), ( s e e e . g . and axe defined through:
rt
(5.7)
= rk(u(k)),
0
:
rj+l
{ri]:=l
[8]),
-< s 5 s t
(k)
where values
r (u)
are t h e r i g h t eigenvectors of
k ' \ k ( ~ ) . Some s o r t of normalization of
important.
&(u)
corresponding t o t h e eigen-
r k i s needed, b u t i s not very
What i s important i s t h e c o n t i n u i t y and ordering r e l a t i o n s :
U(k-l'(O) U(q&
1
= a (k-l)
=
~ ( ~ ' ( s i ) ,k
= d,d-
l,... , 2 .
= u. J
The e x i s t e n c e of a unique s o l u t i o n t o
(5.7), (5.8), (which
-
i s equivalent t o t h e
d ) , i s guaranteed i f luj u.~ -I 1 is s u f f i c i e n t l y small. This follows from t h e i m p l i c i t f u n c t i o n theorem, a d t h e f a c t t h a t t h e r k ( u ) ,
e x i s t e n c e of
.
k = 1,.. ,d
a r e l i n e a r l y independent, and they depend continuously on
U.
192
S. OSHER
An important consequence of t h i s choice of path i s t h a t t h e scheme "decouples", i.e.,
which follows frm (5.5) and ( 5 . 7 ) . upwind scheme i s ( i n i t s e x p l i c i t time differenced v e r s i o n and dropping
Thus
ou1"
the
n dependence i n
which e x i s t s i f
d,si):
sup lun
5
- unI J 1
i s s u f f i c i e n t l y small.
is i n conservation form (and hence pruperty is convenient t o r e w r i t e it as
To show t h a t t h e scheme
2 of t h e i n t r o d u c t i o n i s v a l i d ) it
(5.11) It i s a l s o c l e a r from t h i s t h a t t h e scheme i s f i r s t order accurate i n smooth regions. The t h r e e theorems r e p o r t e d below have been proved i n [El, .see a l s o [51 f o r more d e t a i l e d r e s u l t s concerning the f u l l p o t e n t i a l equation. Suppose f i r s t t h a t equation ( 5 . 1 ) possesses an entropy function, i . e . a convex s c a l a r valued f u n c t i o n V(U) s a t i s f y i n g V ~ V ( U )&(u) = v ~ ( u ) for some s c a l a r
-
U
f l u x f u n c t i o n G(u). It was shown by Lax [81 t h a t , under c e r t a i n r e s t r i c t i o n s , t h e entropy i n e q u a l i t y f o r weak s o l u t i o n s of ( 5 . 1 ) :
is equivdlent t o t h e g e m e t r i c shock conditions.
.
Suppose
u.(t) J
We then have:
i s a s o l u t i o n of t h e semi-discrete v e r s i o n of
and suppose t h a t as L k - 0 it converges boundedly a.e. t o suppose t h e q u a n t i t y sup I A u (t)l is s u f f i c l e n t l y s m a l l . + j
u ( x , t ) . I n addition Then t h e l i m i t
s o l u t i o n s a t i s f i e s t h e entropy i n e q u a l i t y (5.U). L
Next we l e t u , u' be t h e s t a t e s on t h e l e f t and right (of x = 0, s a y ) f o r a steady k shock s o l u t i o n of ( 5 . 1 ) . This means t h a t both t h e jump condition
NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS
193
L R f(u ) = f(u )
(5.14)
and t h e conditions f o r a zero speed (5.15)
k
xk-l(uL) ik(U
k-t h
We a l s o assume t h a t t h e
shock a r e v a l i d
<0
L
1
R ) < 0 < hk+l(uR)
f i e l d is genuinely nonlinear, i . e .
vu',xk.r k #
(5.16)
0.
We seek s o l u t i o n s of (5.10) which a r e time ( n ) independent and approach u L , uR a t + m. These are c a l l e d s t e a d y d i s c r e t e shocks. We have t h e e x i s t e n c e of steady
shock p r o f i l e s .
-
L R Theorem (5.2). Let lu u I be s u f f i c i e n t l y small. Then a c l a s s of s t e a d y d i s c r e t e shocks e x i s t s and each member i s of t h e following form:
(5.17)
-1
5 jo
u. = u L
for
j
u.
for
j > jo
J
uR
J
u (a), u. ( a ) f o r 0 5 j0 1o+l v e c t o r s s a t i s f y i n g t h e following: and
+
1
a r e a smooth one parameter f m i u of
sCY
L
u ( a ) i s connected t o u = u v i a a curve r J o defined as above b u t j0 j, jo jo-l for which o* rd ,rd-l ,...,:r a r e used; i . e . r jv0 = {uj 1, Y =
...,k
- 1,
= k,k
+ 1,...,d.
1,
j0
rY, Y
and moreover
X,(u(s)) 2 0
( a ) i s connected t o uR via a curve r
uj0+1 which only
j +2
j +2
j0+2
rko ,rk-l ,...,r 1
+ 1,..., d,
0
on each
jo+2
is used, ( i . e .
defined as above b u t f o r j +2
0
-- @jo+13, r v0 , V = j +2
,..., k. The curve r has t h e p r o p e r t y t h a t Xv(u(s)) 2 0 on each r v , j +1 Y = d,d - 1,..., k + 1; X,,(U(S)) 5 0 on each rYo , v = k - 1 , k - 2 ,...,1; j +1 j +1 0 and Xk(u(s)) on rko decreases monotonically for 0 5 s 5 s k , Y
= k
j +1
and moreover
havLng a s i n g l e zero at xk (u) = 0. The v e c t o r equation
(5.14)
s
=!
-s
Xv(u(s)) 5 0
on each
1,2 j +1 0
i n this c l o s e d i n t e r v a l , at which
-
u = u,
194
S . OSHER
i s s a t i s f i e d by every member of t h i s family. We next have t h e i r uniqueness (modulo t h e second (perhaps unnecessary) hypothesis below). Theorem (5.3).
Let
be a s t e a d y d i s c r e t e shock with
(u,]>
s m a l l , and with t h e p r o p e r t y t h a t i f
k (u ) k j of t h e form defined i n t h e previous theorem.
Vl.
5
0, t h e n
16 u 1 +j
sufficiently
Xk(uj+l) 5 0.
Then it i s
THE UPWIND DIFFERENCE ALGORITHM FOR SaME IMPCWTANT EXAMPLES
The equation (5.7) can be i n t e g r a t e d i n closed form f o r many important p h y s i c a l problems. Hence a t h r e e p o i n t scheme ( 5 . 1 0 ) involving various "switches", which depend on t h e s i g n of Xk(u), can be constructed and t e s t e d . This was done, with very s u c c e s s f u l numerical r e s u l t s , sane of which we r e p o r t i n t h e next s e c t i o n . The algorithms mentioned here a r e taken fTm [12] and [51. We begin with one dimensional gas flow for a non i s e n t r o p i c gas i n Lagrange coordinates. The equations a r e w r i t t e n i n t h e form
(6.1)
7
-vx=0
vt
+ p,
Et Here
Let
+
for
= 0
(PV),
= 0,
z i s s p e c i f i c volume, v i s v e l o c i t y , u = (T,v,E) T , f ( u ) = (-v,p,pv), then
0
and t h e eigenvalues of
k(u)
p = (y
y
Tt;
E
e -?
- 1) -
1.k
is energy,
-1
p
is p r e s s u r e .
O
1
are
(6.3)
f o r a l l values of u, t h e r e a r e no switches, and t h e 2 3 scheme becomes f a F r l y simple, once t h e i n t e g r a t i o n i n (5.7) i s performed exactly Since
h
1
< 0 e x < h,
NUMERICAL SOLUTION OF S I N G U L A R PERTURBATION PROBLEMS
- see
[121 f o r t h e d e t a i l s .
In p a r t i c u l a r
3 2
Tn
j -1
with
(6.5)
dJ. = 2 ( J T n ) ' d m
and (using t h e above d e f i n i t i o n s )
195
196
S.
OSHER
with
(6.7)
' k e n m e r i c a l approximation is now obtained by using (6.1), (6.4) and
(6.6).
I n [121 t h e algorithms were worked out for, among o t h e r s , t h e one dimensional i s e n t r o p i c gas dynamics equations i n E u l e r i a n coordinates, and a l s o t h e two dimensional v e r s i o n of t h e system (with t h e help of a dimensional s p l i t t i n g algorithm). For l a c k of space, we anit t h e d e t a i l s h e r e b u t p r e s e n t sane computational r e s u l t s i n t h e next s e c t i o n . One f u r t h e r example, from [51, i s t h e p o t e n t i a l f l a w equation i n one space dimension w r i t t e n &s a hyperbolic system
(6.8) with
c = c(p) =
= AypYm1 for
dP
m e eigenvalues of
&
-e
vectors
r1,2 =
(*.').
A > 0,
axe
=
X1,2
1
T
= u
5 3. c
The equation (5.7) here becomes
(6.9) .
with s o l u t i o n s
(6.10)
w i t h corresponding eigen-
u
+_
2 c Y - 1
.
= constant.
197
NUMERICAL SOLUTION OF S I N G U L A R PERTURBATION PROBLEMS
The paths of i n t e g r a t i o n
rj
=
r:
U
ri
a r e j u s t piecewise l i n e a r i n t h e
c, u
plane. The algorithm, complete with switches and p o s i t i v e numerical evidence, i s given i n [51. VII.
"WEXAMPLES
I n all cases we t a k e f o r a Riemann problem
at/&
= .1,Ax = .05, A t = .005. We a l s o t a k e i n i t i a l d a t a
- two constant
states.
1. 1 Dimensional Lagrange Equations
We d i s p l a y t h e r e s u l t s f o r a s t e a d y contact d i s c o n t i n u i t y , (Figure 1). The 7 component i s p l o t t e d a t t = 0 and a t i t s e v a l u a t i o n a f t e r 20 time s t e p s . The i n i t i a l d a t a of two constant s t a t e s proves t o be an exact s t e a d y s o l u t i o n of t h e d i f f e r e n c e appr ox h a t ion. 2.
1 Dimensional Euler Equations
-
( a ) steady shock (Figure 2 ) and a s t e a d y c o n t a c t d i s We d i s p l a y two cases c o n t i n u i t y (Figure 3). The d e n s i t y component i s p l o t t e d a t t = 0 and t = .1. Again zero speed c o n t a c t s are r e s o l v e d e x a c t l y , while it takes two p o i n t s t o r e s o l v e zero speed shocks as i n t h e s c a l a r case.
3.
2 Dimensional f i l e r Equations
We d i s p l a y a zero speed shock having a 45" angle of o r i e n t a t i o n obtained via a s p l i t t i n g algorithm (Figure 4). The p component is p l o t t e d a t t = 0 and t = .l, along t h e l i n e x = y. The s t a t i o n a r y shock again y i e l d s a sharp profile. (Figure 1) Lagrange Eqtns I D Nonisentropic 0-speed c o n t a c t d i s c o n t i n u i t y ( I n i t i a l Data is s t e a d y s o l u t i o n ) 7 specific volume 1.0
-
(
)
-
h d
-
h "
-
A -
-
A
- - ---
-
-
-
Ax
-
-
= .05
A t = .005 speed of c o n t a c t = 0 0.0
.1
.2
.3
4 X
.5
6
.7
8
.9
198
S . OSHER
(Figure 2 )
Euler E q t n s
-
0-speed shock
I D Nonisentropic
2.0
QX = .05 A t = .005 shock speed = 0 0.0
.1
.2
.3
4
.5
6
.7
8
.9
.5
6
.7
8
*9
X
(Figure 3 ) Euler Eqtns I D Nonisentropic 0 speed c o n t a c t d i s c o n t i n u i t y ( I n i t i a l Data i s steady s o l u t i o n )
-
(deis it,)
1.0
v s - - t -0-%0.0
= 0
t =.l
-1
.2
*3
4 X
-
(Figure 4) Euler Equations 2D I s e n t r o p i c 0 speed shock a t 45" angle P r o f i l e of p(x,y) d o n g l i n e . x = y . (deGity)
1.0
. 5 T-7- 7--c- 7-T-
Ax =
+t
-*-
nt
= O t = .1
0. 0
.1
.2
.3
4 X
.5
.6
-
=
.05
.a05
shock speed = 0
.7
.8
-9
199
NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS
The r e s u l t s for moving shocks and c o n t a c t d i s c o n t i n u i t y were not as good- t h e r e was some smearing. The c l o s e r t o zero speed, t h e b e t t e r t h e shock r e s o l u t i o n i n a l l o m calculations. APPENDIX I
-Proof
of Theorem (2.1):
d
(A 1.1)
Here each uh uh
- vh - vh
h
B,
h
d dt
is a connected union of
< 0 adjacent t o €Ih. < 0 in and u’
Each
- vh
Th
such t h a t
h
h
h u
- vh
2 0
T?J
- h ’ B V i s a connected union of
2 0
-h
adjacent t o
Bv.
in
0;
and
such t h a t
Using (2.8) we have
h 2 However ing function,
2( f ( and u)
e
2
V)
-
i n t h e s e i n t e g r a l s and (f(u) * v)+ is an i n c r e a s i s decreasing. Thus t h i s expression is non p o s i t i v e .
A similar r e s u l t follows f o r
lq
$vh
- uh)dx.
Summing over
d e s i r e d r e s u l t f o r t h e semi-discrete case.
For t h e e x p l i c i t t h e d i s c r e t i z a t i o n , we have f o r
x
E
T?: .J
v
gives us t h e
200
S . OSHER
where t h e to
Th
j‘
BB
3’
a r e t h e piecewise constant values t a k e n on by
h un(x)
adjacent
i s a nondecreasing h I f we d e f i n e another s o l u t i o n v n ( x ) = CY (x)?)jl,.,,
Thus, by t h e CFL condition, t h e f u n c t i o n G ( j )
function of i t s arguments. t h e n we have
j
and (dropping t h e
We sum over
j,
h
j
superscript):
t h e r i g h t s i d e telescopes., and we have:
(A 1.6)
t h e theorem follows e a s i l y . Proof of Theorem (2.1) We multiply ( 2 . 8 ) by
cp(x
j’
t ) , where
x
J
is t h e c e n t r o i d of
with support i n t > 0, and t h e n sum over a l l
Now
j.
’8j ’
This gives us :
and
m
cp E Co,
NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS
( A 1.8)
J
n
201
W p - f(uh)dx
J
1.7).
S u b t r a c t ( A 1.8) from (A
=
q
We have:
cp(x,t)(f(B$t))
*
v::’+dx
jar?
J
f
rJ (cp(x,t) j a T hJ
=q j
- dXj’t))(f(B$t))
dXj,t)(f(2(t))
. v;)
‘
dx
2 . J cp(x,t)(f(?(t)).vJ)-
Thus t h e r i g h t hand s i d e of (A 1 . 9 ) can be r e w r i t t e n as
r,/
(cp(x,t)
dx
<
The f i r s t sum on t h e r i g h t above may be r e w r i t t e n as
( A 1.10)
.;I
-
dXj,t))[(f($(t)).
-
vJh)-
Jd“
(f(?(t)).
vhj ) - l d x
j&$
Next we i n t e g r a t e with r e s p e c t t o t and l e t h --f 0. Since B(uh) i s uniformly bounded, i t i s easy t o s e e *can ( A l.g), (A 1 - 1 0 ) , and t h e Lebesgue dominated convergence theorem, t h a t :
j] dxdt((Ptu
( A 1.11)
+
OCP.
f ( u ) ) = 0.
A s i m i l a r proof works f o r t h e fully d i s c r e t e case.
It remains f o r us t o prove t h a t t h e entropy c o n d i t i o n i s v a l i d , i . e . f o r any r e a l c
and any nonnegative
(A 1.12)
cp
E
m
Co
j] dtdx(cpt/u -
t > 0 we must show
with support i n c(
+ Wp.
sgn(u
-
c)(f(u)
-
f ( c ) ) = 0.
We s h a l l do t h i s f o r t h e filly d i s c r e t e problem by u s i n g an argument of Crandall and Majda [11. With z v w = m a ( z , w ) , z A w = min(z,w), a d i r e c t a p p l i c a t i o n of t h e i r argument gives us, f o r s o l u t i o n s of ( 2 . 3 ) , (2.9),
S. OSHER
202
We a l s o have
(A 1 . 1 4 )
-
W . sgn(uh- c ) ( f ( u h ) - f ( c ) ) d x = x j 8 V - (cp sgn(uh c ) ( f ( u h ) - f ( c ) ) d x n j j
Multiply (A(1.13))by
cp(x t ) , sum over
5'
j,
t h e n s u b t r a c t (A(1.14)) from it.
The
r e s u l t follows as i n t h e f i r s t p a r t of t h e proof of t h i s theorem. APPENDM I1
Proof of Theorems ( 4 . 1 ) and (4.21. We begin with t h e following i n e q u a l i t y
(A 2 . 1 ) which is v a l i d f o r any two s o l u t i o n s of (4.5) having d i f f e r e n t i n i t i a l d a t a . "his follows as i n t h e proof of Theorem (3.1) above. A similar e s t i m a t e a l s o follows as above f o r s o l u t i o n s of (4.6) We must now merely o b t a i n t h e be any smooth f u n c t i o n i n
Ll, Lm and maximum p r i n c i p l e r e s u l t s . Let w cp on b. Let wh i n Th be
agreeing with
5
and l e t wh t a k e on t h e boundary values on t h e mean value of w over !I$, h Define $(u ) t o be t h e operation defined by (4.3). Then
(A 2 . 2 )
q u h )
- $(Wh)
anh.
h 1.
= A-$"
It i s easy t o modify our familiar argument i n t h e proof of Thecrrem ( 2 . 1 ) ( s e e a l s o [lo], Theorem 5), t o show t h a t t h i s implies
The r e s u l t i s immediate f o r t h e
L1 norm.
Next suppose uh (x ) i s a p o s i t i v e l o c a l maximum among t h e values n j neighboring 7$ Then
.
(A 2.4)
un(xa h ) j
far
'-j
J
which is a nondecreasing f u n c t i o n of a l l i t s arguments, by t h e CF'L reiuirement (4.7). Then it follows t h a t
NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS
h
5 'j,n
(1
203
- at) - A t b ( xj ' 0).
I n t h e s t e a d y case, t h i s gives us (A
h
2.6)
8. J
5 E1 b ( xj ' 0).
A similar argument works f o r a l o c a l n e g a t i v e minimum, t h u s t h e L , uniformbound i s immediate. h F i n a l l y from (A(2.4)) we l e t Bk,,n be t h e nonnegative maximum value of B
v, n
i n t h e c l a s s of values on which 'G(j)
depends.
We now use (A 2 . 4 ) , and an argument similar t o t h a t of ( A 2 . 5 ) gives us (4.9)(a). A similar argument gives us ( 4 . 9 ) ( b ) . Proof of Thearem ( 4 . 3 ) . I n view of t h e proof of Theorem ( 2 . 2 ) , we need only show f o r any t e s t f u n c t i o n 7 , that
We extend a l l t h e f a c e s of each
l?J
h u n t i l t h e y r e a c h &l, t h i s gives us a new
collection {T?, 3 and a;. We keep t h e same set o f values, i . e . h The sum ( A 2 . 7 ) i s i n v a r i a n t a f t e r t h i s extension. = uh(x i f T,! c J j J J Hence f o r s i m p l i c i t y we do not change n o t a t i o n and s t i l l consider ( A 2 . 7 ) .
3
u (x., )
Th.
Now consider a f a c e quantities
v
(where
with
and
ph
j,J
1
separating
%
l?J
and
$.
i s a u n i t vector i n t h e d i r e c t i b n from xJ
N
x
In t h e sum above, t h e
involve a term
to x
j
p o i n t i n g towards
on t h e l i n e between x and xJ. Now we sum over j and J j,J h j use t h e hypothesis t h a t B(uE) i s uniformly bounded. The r e s u l t ( A 2 . 7 )
and
follows immediately. Proof of Theorem ( 4 . 4 ) . Follows (as d i d t h e L1 extimate i n t h e proof of Theorem ( 4 . 1 ) ) from a s l i g h t e x t e n s i o n of t h e proof of Theorem ( 2 . 1 ) . See a l s o [lo], Thearem 7.
204
S.
OSHER
BIBLIOGRAPHY [ l ] &andall, M. and Majda, A . , Monotone d i f f e r e n c e approximations f o r s c a l a r conservation laws, Math. Camp. 34 (1980), 1-22. 121 Engquist, B. and Osher, S . , S t a b l e and entropy s a t i s f y i n g approximations f o r transonic flow c a l c u l a t i o n s , Math. Comp. 34 (1980), 45-75.
[3] Engquist, B. and Osher, S . , One-sided d i f f e r e n c e schemes and t r a n s o n i c flow, Proc. Nat. Acad. S c i . U.S.A., (1980), t o appear.
[41 Engquist, B. and Osher, S . , One-sided d i f f e r e n c e approximations f o r nonlinear conservation laws, Math. Comp., (submitted).
151 Engquist, B. and Osher, S . , Upwind d i f f e r e n c e schemes f o r systems of conserv a t i o n laws t h e p o t e n t i a l f l o w equations, ( i n p r e p a r a t i o n ) .
-
161 Harten, A., w a n , J. M. and Lax, P. D., On f i n i t e d i f f e r e n c e approximations and entropy conditions f o r shocks, Comm. Pure Appl. Math. 29 (1976), 297-322.
[TI
Jennings, G., D i s c r e t e shocks, Comm. Pure Appl. Math. 27
(1974), 25-37.
[81 Lax, P. D . , Shock waves and entropy, i n : Zarontonello, E. H . , (ed. ), Contribu t i o n s t o Nonlinear Functional Analysis (Academic p r e s s , New York, 1971).
[gl Osher, S . , Approximation par e'le'mentes f i n i s avec de'centrage pour des l o i s de conservation non l i n e a i r e s multi-d.hensionelles, Comptes Rendues. Acad. S c i . P a r i s (1980) t o appear. [lo] @her, S., Nonlinear s i n g u l a r p e r t u r b a t i o n problems and one-sided d i f f e r e n c e schemes, S W , t o appear.
[11] Osher, S . , Approximation par e'le'mentes f i n i s avec de'centrage pour des problemes de p e r t u r b a t i o n singulaFre non-lineaires multi-dimensionelles, Comptes Rendues. Acad. S c i . P a r i s , t o appear. [121 Osher, S . and Solomon, F . , Upwind d i f f e r e n c e schemes f o r c m p r e s s i b l e gas dynamics, ( i n p r e p a r a t i o n ) .
[131 Sanders, R . , Convergence of t h r e e p o i n t monotone schemes, ( i n p r e p a r a t i o n ) .
A N A L Y T I C A L AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS I N A N A L Y S I S S. A x e i s s o n , L . S . F r a n k , A . v a n d e r S i u i s ( e d s . ) 0North-Holland P u b l i s h i n g Company, 1981
NONSTATIONARY FILTRATION I N PARTIALLY SATURATED M E D I A
PI. B e r t s c h and L.A.
Peletier
1. INTRODUCTION Consider a homogeneous, i s o t r o p i c and r i g i d porous medium f i l l e d w i t h a fluid.
Let
q
denote t h e macroscopic v e l o c i t y o f t h e f l u i d and
c
the
v o l u m e t r i c m o i s t u r e c o n t e n t . Then t h e f l o w i s governed by t h e c o n t i n u i t y e q u a t i o n
% +div q at
= 0
and D a r c y ' s l a w q = -K(c) grad where
K
i s t h e h y d r a u l i c c o n d u c t i v i t y and
( c f . BEAR C l l , p . 4 8 8 ) .
@
o
, the t o t a l potential
I f a b s o r p t i o n and chemical, osmotic and thermal e f f e c t s
a r e i g n o r e d , we may w r i t e
(3)
@ = * + 2 ,
where
$
i s t h e h y d r o s t a t i c p o t e n t i a l due t o c a p i l l a r y s u c t i o n and
z the
g r a v i t a t i o n a l p o t e n t i a l , t h e z - c o o r d i n a t e chosen v e r t i c a l , p o s i t i v e p o i n t i n g upwards. Elimination o f
@
and
a c($) at
q
from ( 1 ) - ( 3 ) y i e l d s t h e e q u a t i o n
= divIK(c)grad $1 +
a K(c)
.
F o r values o f c , which a r e small compared t o t h e s a t u r a t i o n v a l u e a reasonable c h o i c e f o r JI = $ ( c ) and K = K ( c ) i s
205
(4)
s
206
where
M . BERTSCH and L . A . P E L E T I E R
q ~ ~ , K ~ , and p n
a r e p o s i t i v e c o n s t a n t s such t h a t
n
>
ptl
.
S u b s t i t u t i o n i n t o (4) leads t o the equation Ct
=
A(cm) t ( c ' ) ~
m = n - p ,
(5)
where t h e c o n s t a n t s have been absorbed i n t h e independent v a r i a b l e s . I f c
is
z , ( 5 ) r e s u l t s i n t h e "porous media e q u a t i o n "
independent o f
c t = A(cm) On t h e o t h e r hand, i f
Because m
>
1
c
o n l y depends on
, equation
z
and
t
, ( 5 ) reduces t o
( 5 ) i s o f degenerate p a r a b o l i c type, and i t s s o l u t i o n s
may e x h i b i t sharp i n t e r f a c e s between r e g i o n s which a r e d r y (c=O) and r e g i o n s I n r e c e n t y e a r s e q u a t i o n ( 5 a ) has been s t u d i e d v e r y which a r e w e t (00) e x t e n s i v e l y C81. F o r e q u a t i o n ( 5 b ) we r e f e r t o C61.
.
I n t h i s paper, however, we s h a l l be more concerned w i t h f l o w s i n which t h e moisture content
c
i s close t o the s a t u r a t i o n value
reasonable c h o i c e f o r t h e r e l a t i o n between c
and J,
c . For t h i s
purpose, a
is:
Equation ( 4 ) can now be w r i t t e n as
I
c J,t
t where
D = Koc
a
= d i v I D ( $ ) g r a d J , ) taz D($)
0 = A$
.
Thus
if
-1
< J, <
0
$ > O ,
if
s a t i s f i e s a parabolic equation i n the unsaturated
JI
r e g i o n and an e l 1 i ; p t i c e q u a t i o n i n t h e s a t u r a t e d r e g i o n . F i n a l l y , across t h e boundaries between these r e q i o n s one r e q u i r e s JI
and
g r a d J,
c
and
continuous.
q
t o be continuous, i . e .
207
NONSTATIONARY FILTRATION
I n t h i s paper we s h a l l s k e t c h a few o f t h e r e s u l t s which have been o b t a i n e d about t h e f r e e boundary problem d e r i v e d above. S p e c i f i c a l l y , we s h a l l d i s c u s s
two cases:
I.
K(c)
11.
K(c)
1 ;
f
increasing w i t h
c
.
I n b o t h cases we s h a l l c o n s i d e r one-dimensional f l o w o n l y . F o r convenience we shall set
c
.
= 1
2. CASE I . We c o n s i d e r t h e C a u c h y - D i r i c h l e t problem = uxx
(C(U)), u(0,t)
=
-1
, u(1,t)
= t1
C(U(X,O)) = v o ( x ) where
T > 0
i n Case
. Equation
QT = ( 0 , l )
on
(0,Tl
on
C0,11
x
(0,TI
,
(8) r e s u l t s from e q u a t i o n ( 6 ) under t h e assumptions made so designed t h a t n e a r x = 0 ,
I; and t h e boundary c o n d i t i o n s ( 9 ) a r e
t h e f l o w i s u n s a t u r a t e d and n e a r function
in
vo
x = 1
, the
f l o w i s s a t u r a t e d . The i n i t i a l 1 i s so chosen t h a t t h e r e e x i s t s a f u n c t i o n uo E C ( [ O , l l ) w i t h
the properties: ( a ) uo(0) = -1
, uo(l)
( c ) c ( u o ( x ) ) = vo(x)
= t1
on
, (b) uo(x)
C0,ll
2
-1 on
C 0 , l l and
.
One approach t o t h e s t u d y o f Problem I i s t o assume t h e e x i s t e n c e o f an i n t e r f a c e x = c ( t ) , s e p a r a t i n g t h e u n s a t u r a t e d and t h e s a t u r a t e d f l o w r e g i o n s , t o s o l v e e q u a t i o n (8) on b o t h s i d e s o f i t and t h e n t o p a t c h t h e s o l u t i o n s together a t t h i s i n t e r f a c e using the c o n t i n u i t y o f
c
a c o n d i t i o n f r o m which t h e f u n c t i o n
u
and ux
. This
leads t o
can be determined C51.
Another approach i s t o d e f i n e a c l a s s o f weak s o l u t i o n s on t h e e n t i r e cylinder , t o e s t a b l i s h t h e e x i s t e n c e o f one such s o l u t i o n , and t o show t h a t
oT
i t has p r o p e r t i e s which a r e t o be expected on p h y s i c a l grounds. Here we s h a l l
a d o p t t h e second apnroach. Thus, we i n t r o d u c e t h e f o l l o w i n g not-ion o f a weak s o l u t i o n . DEFINITION. A f u n c t i o n
u(x,t)
s u l u t i ~ no f Problem I i f ( i )
, d e f i n e d a.e. i n 0, , w i l l be c a l l e d a weak c(u)
E
C(OT)
, where
u
i s p o s s i b l y r e d e f i n e d on a
2 08
M . BERTSCH and L . A .
s e t of measure z e r o i n QT , ( i i ) u and ( i i i ) u s a t i s f i e s t h e i d e n t i t y
for all
4
E
C1(oT)
which v a n i s h f o r
Note t h a t t h e f u n c t i o n
u(x)
-
-u
E
PELETIER
1 L 2 (O,T;HO(O,l))
x = 0,l
and
t =
T
I
where
i(X) =
2X
- 1
.
i s t h e unique e q u i l i b r i u m s o l u t i o n o f Problem I
1 be such t h a t there e x i s t s a finnetion uo E C ( C 0 , l l ) , v0 which s a t i s f i e s the p r o p e r t i e s ( a ) - ( c ) . Then there e x i s t s one, and o n l y one weak THEOREM 1 C31. Let
solution o f Problem I . The e x i s t e n c e o f a weak s o l u t i o n i s proved by approximat ing
c
and
uo
by sequences o f smooth f u n c t i o n s { c n l and {uOn1 such t h a t c n' 2 l / n f o r a l l n 2 1 The equat i o n ( c ~ ( u ) =) ~uxx i s now u n i f o r m l y p a r a b o l i c and has a unique smooth s o l u t i o n un , which s a t i s f i e s t h e boundary c o n d i t i o n s ( 9 ) and t h e i n i t i a l
.
.
I t i s t hen shown t h a t a subsequence c o n d i t i o n u(x,O) = uOn(x) , 0 5 x 5 1 2 1 { u 1 o f f u n ] converges weakly i n L (0,T;H ( 0 , l ) ) t o an element u i n t h i s P space, which possesses a l l t h e p r o p e r t i e s r e q u i r e d o f a weak s o l u t i o n .
The uniqueness o f a weak s o l u t i o n i s proved by s u b t r a c t i n g t h e i n t e g r a l i d e n t i t i e s (10) f o r two s o l u t i o n s , and choosing a s u i t a b l e t e s t f u n c t i o n
+.
As may be expected, t h e s o l u t i o n s o f Problem I a r e ordered according t o t h e
i n i t i a l c onc ent ra t i o n s
vo
. Thus we have:
THEOREM 2 C31. Let vO1 and vO2 s u t i s f y the asswnptions of Theorem 1, and l e t u1 and u2 be the weak s o l u t i o n s of Problem I wh.ich correspond t o r e s p e c t i v e l y vO1 and
vO2
. Then
vol
2
vo2
As t o t h e r e g u l a r i t y o f
on
uo
c0,ll
=)
c ( u ~ )2 c(u,)
a.e. i n
OT
*
one can show:
THEOREM 3 131. Let the hypotheses of Theorem 1 be s a t i s f i e d , arid l e t 2 2 weak s o l u t i o n of Problem I . Then u E L (0,T;H ( 0 , l ) ) ,
u be t h e
To d e s c r i b e t h e p r o p e r t i e s o f t h e i n t e r f a c e , we d e f i n e t h e f u n c t i o n
NONSTATIONARY FILTRATION
v = c ( u ) a.e.,
where
v
.*
, and t h e s e t s
C(0,)
D
209
{(x,t)cQT:V(x,t)
=
P = { ( X , ~ ) E Q ~ : V ( X ~. ~ ) = ~ I Thus i n D
P
t h e f l o w i s u n s a t u r a t e d and i n
i t i s s a t u r a t e d . By t h e c h o i c e o f
boundary c o n d i t i o n s , and t h e r e g u l a r i t y o f t h e s o l u t i o n
, there exists a
u
6
0
such t h a t
(0,6) Define f o r
x
(0,Tl
D
c
and
(1-6,l)
x
(0,Tl
c
.
P
(11)
t c TOYTI :
L - ( t ) = sup
{x E ( o , l ) : ( E , t ) l b y V F ~ ~ O , X ) }
5t ( t ) = i n f
I'1x,(o,l):(~,t)i7j,vgc(x,l1
I.
Then i n view of ( 1 1 ) 6 s c-(t)
t
5
Suppose t h a t i n i t i a l l y , r,-(O) such t h a t vo(x)
<
1 on
(t) =
1-6
for
~'(0)
, i.e.
<
[O,R) , v o ( x )
=
0 < t
5
T
.
t h e r e e x i s t s a number
1 on
Then one may wonder whether c - ( t ) = r ; + ( t ) f o r a l l t i s o l a t e d pockets o f ( u n ) s a t u r a t e d f l o w may emerge f o r
R
(0,l)
[R,ll (0,TI , o r whether t > 0 The c o n t e n t of t h e
t
.
f o l l o w i n g theorem i s t h e a s s e r t i o n , t h a t no such pockets w i l l come i n t o e x i s t e n c e . THEOREM 4 [31. L e t t h e hypotheses of Theorem 1 be s a t i s f i e d , and l e t
<-(O)
= ~'(0)
. Then
< - ( t ) = r,'(t)
I n what f o l l o w s we s h a l l w r i t e
f o r a22
t
E
[O,T1
.
210
M . BERTSCH and L.A.
0 =
COROLLARY 5.
PELETIER
{ (x,t):O<x<<(t),O
Recently i t has been shown by VAN DUYN C41 t h a t t h e i n t e r f a c e i s continuous: THEOREM 6. L e t t h e hypotheses of Theorem 4 be s a t i s f i e d , and l e t i n addition 2ta ( C 0 , l l ) ( a c ( 0 , l I ) such t h a t ug(0) = ug(1) = 0 Then 5 c C(C0,TI) u0 c C
.
0
D and P , where
I t f o l l o w s from Theorem 6 t h a t t h e s e t s
a r e open. I t can be shown by s t a n d a r d t h e o r y [71 t h a t t h e s o l u t i o n Problem I s a t i s f i e s t h e e q u a t i o n ut = uxx Let
$
coo
E
Co(P)
. Then
because
in
c(u) = 1
D
u
of
. 0
a.e. i n P
, the
i n t e g r a l i d e n t i t v (101
becomes 0 =
11
{bxux-$t}
dxdt =
E
2
L (0,T;H
11
Gxuxdxdt
!
QT Because, b y Theorem 3, u
2
(0,l))
t h i s implies t h a t
$uxxdxdt = 0
,
0
P
i.e. t h a t uxx = 0
a.e. i n
0
P
.
A f u r t h e r r e g u l a r i t y r e s u l t enables one t o extend t h i s e q u a l i t y t o t h e whole of 0 P Since u ( - , t ) c C(CO,l]) f o r each t c (0,1] , i t f o l l o w s t h a t
.
u(x,t)
=
x-r;(t) l-<(t)
.
r;(t)
5
x c3
NONSTATIONARY FILTRATION
f o r a l l t E (0,TI , F i n a l l y , because uX(-,t)
t
C([0,11)
u(.,t)
c H
2
(0,l)
a.e.
on
21 1
COYTI , i t f o l l o w s t h a t
on COYTI , and i n p a r t i c u l a r across t h e i n t e r f a c e .
a.e.
We conclude t h i s s e c t i o n w i t h a r e s u l t a b o u t t h e l a r g e t i m e b e h a v i o u r o f s o l u t i ons. THEOREM 7 C31. Let t h e conditions of Theorem 1 be s a t i s f i e d . , and l e t
u be the
weak s o l u t i o n of Problem I . Then
Ic(u(x,t))-c(G(x))l
C
where
c e -71 2 t / 3
5
i s a constant, which only depends on
v0
.
3. CASE 11.
We c o n s i d e r t h e C a u c h y - D i r i c h l e t problem
where
D
T
= Koc
(iii)
>
0
, and K'(r)
, and
t h e assumptions about
we assume a b o u t t
0
for all
r
K E
that ( i )
C0,ll
.
vo
K
a r e t h e same as i n Case I. I n ( 1 2 ) , 2 E C (C0,lI) , ( i i ) K(0) > 0 ,
The d e f i n i t i o n of a weak s o l u t i o n u o f Problem I 1 i s t h e same as w i t h Problem I, e x c e p t t h a t t h e i n t e g r a l i d e n t i t y (10) now becomes
I t i s p o s s i b l e "21 t o d e r i v e e s s e n t i a l l y t h e same r e s u l t s as i s done f o r
Problem I,i . e . t h e e x i s t e n c e , uniqueness and r e g u l a r i t y o f a weak s o l u t i o n u , and t h e e x i s t e n c e and c o n t i n u i t y o f t h e i n t e r f a c e . On s e v e r a l p o i n t s t h e methods w i c h a r e used a r e d i f f e r e n t f r o m t h e ones developed i n C31. P a r t i c u l a r l y t h e
M. BERTSCH and L . A . P E L E T I E R
212
uniqueness p r o o f i s based on [61. Furthermore one can show t h a t t h e r e e x i s t s a unique s t eady -s t a t e s o l u t i o n
Q(x)
and t h a t
Ic(u(x,t))-c(Q(x))l
for a p o s i t i v e constant
C
, which
5
C e -a2t / 4K(0)
o n l y depends on
vo
C21.
REFERENCES: [11 BEAR, J . , Dynamics o f f l u i d s i n porous media, New York: American E l z e v i e r
P u b l i s h i n g Company I n c . 1972. r 2 1 BERTSCH, M., To appear. I31 DUYN, C.J. van, and L.A. PELETIER, N o n s t at ionary f i l t r a t i o n i n p a r t i a l l y s a t u r a t e d porous media. To appear i n Arch. Rat. Mech. Anal. 141 DUYN, C.J. van, N o n s t a t i o n a r y f i l t r a t i o n i n p a r t i a l l y s a t u r a t e d porous media: c o n t i n u i t y o f t h e f r e e boundary. To appear i n Arch. Rat. Flech. Anal. 151 FASANO, A. and M. P R I M I C E R I O , L i q u i d f l o w i n p a r t i a l l y s a t u r a t e d media. To appear. [ 6 1 GILDING, B.H. and L.A. PELETIER, The Cauchy Problem f o r an e q u a t i o n i n t h e
t h e o r y o f i n f i l t r a t i o n , Arch. Rat. Mech. Anal.
, 61(1976)
127-140.
[71 LADYZHENSKAJA, O.A.,
V . A . SOLONNIKOV and N.N. URAL'CEVA, L i n e a r and Q uasil i n e a r Equations o f P a r a b o l i c Type, T r a n s l a t i o n s o f Mathematical Monographs Volume 23, Porvidence, R . I . : American Mathematical S o c i e t y 1968.
C81 PELETIER, L.A. , The porous media e q u a t i o n , Proceedings of t h e conference on " B i f u r c a t i o n Theory; A p p l i c a t i o n o f N onlinear Analysis i n t h e Physical Sciences", B i e l e f e l d 1979.
A N A L Y T I C A L AND NL'MEKICAL APPKDACHES TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S . A x e l s s o n , L.S. F r d n k , A . v a n del Sluis ( t , d s . ) @ N o r t h - I I o l l J n d P u b l i s h i n g Company, 1 9 8 1
A-POSTERIORI ERROR ESTIMATES AND ADAPTIVE FINITE ELEMENT COMPUTATIONS FOR SINGULARLY PERTURBED ONE SPACE DIMENSIONAL PARABOLIC EQUATIONS Hans-JLirgen R e i n h a r d t Fachbereich Mathematik
J. W . Goe the-Universi t l t Frankfurt Fed. Rep. Germany
For t h e Crank-Nicolson-Galerkin-method of s o l v i n g a s i n g u l a r l y p e r t u r b e d p a r a b o l i c i n i t i a l v a l u e problem, upper and lower a-pos t e r i o r i error estimates are established. As a principal tool a-posteriori err o r e s t i m a t e s i n [14] f o r s i n g u l a r l y p e r t u r b e d boundary v a l u e problems i n o r d i n a r y d i f f e r e n t i a l e q u a t i o n s a r e extended t o a p p r o x i m a t e l y given inhomogeneous right-hand s i d e s c o n t a i n i n g boundary l a y e r c o n t r i b u t i o n s . Herewith, a - p o s t e r i o r i e r r o r e s t i m a t e s f o r t h e f i n i t e element s o l u t i o n o f t h e given p a r a b o l i c problem a r e d e r i v e d on e v e r y d i s c r e t e time l e v e l which p r o v i d e a b a s i s f o r an a d a p t i v e mesh r e f i n e m e n t . The e f f i c i e n c y of o u r method i s demonstrated by a numerical example.
INTRODUCTION
Compared w i t h t h e number of p a p e r s concerned w i t h t h e a s y m p t o t i c s o l u t i o n o f s i n g u l a r p e r t u r b a t i o n problems, t h e r e a r e few which s t u d y t h e i r numerical s o l u t i o n , and even among t h e s e very few develop a d a p t i v e methods f o r such problems ( c f . Brandt [6], L e n t i n i - P e r e y r a Pearson [12], R e i n h a r d t [ 1 4 ] ) . There seems t o be no a d a p t i v e numerical s o l u t i o n f o r s i n g u l a r l y p e r t u r b e d p a r a b o l i c problems. I n t h e p r e s e n t p a p e r , f o r t h e Crank-Nicolson-Galerkin-method of s o l v i n g a p a r a b o l i c i n i t i a l v a l u e problem w i t h a s i n g u l a r l y p e r t u r b e d one-dimensional e l l i p t i c p a r t , a - p o s t e r i o r i e r r o r e s t i m a t e s and an a d a p t i v e mesh r e f i n e m e n t on e v e r y d i s c r e t e t i m e l e v e l a r e developed.
[Ill,
A s an i m p o r t a n t t o o l f o r t h e d e r i v a t i o n of a - p o s t e r i o r i e r r o r e s t i m a t e s f o r t h e Crank-Nicolson-Galerkin-method, i n S e c t i o n 1 t h e r e s u l t s of [ I 4 1 a r e extended i n
two d i r e c t i o n s which a r e a l s o i n t e r e s t i n g i n themselves. F i r s t , we a l l o w boundary l a y e r c o n t r i b u t i o n s i n v o l v e d i n t h e inhomogeneous right-hand s i d e of t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n where lower and upper a - p o s t e r i o r i e r r o r e s t i m a t e s f o r t h e f i n i t e element s o l u t i o n a r e a l s o e s t a b l i s h e d . Secondly, we c o n s i d e r t h e extended s i t u a t i o n t h a t t h e right-hand s i d e s of t h e f i n i t e element e q u a t i o n s a r e o n l y g i v e n approximately by l i n e a r f u n c t i o n a l s . In* t h i s c a s e , t h e l o c a l energy-norms of t h e e r r o r w i l l a g a i n be r e a l i s t i c a l l y e s t i m a t e d by s u i t a b l e approximate e r r o r i n d i c a t o r s . We s a y t h a t bounds a r e " r e a l i s t i c " i f they do n o t roughly o v e r e v t i m a t e t h e e r r o r . I n t h e second s e c t i o n , we a r e a b l e t o a p p l y t h e above r e s u l t s by a s s o c i a t i n g a s y s tem of e l l i p t i c e q u a t i o n s w i t h t h e given s i n g u l a r l y p e r t u r b e d p a r a b o l i c problem. The p r e s e n t approach i s s i m i l a r t o R o t h e ' s method ( c f . Kazur-Wawruch [g] , K a r t s a t o s Z i g l e r [lo]). By time d i s c r e t i z a t i o n ( b e f o r e t h e p a r t i t i o n o f t h e s p a c e v a r i a b l e ) , a system of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s i s a s s o c i a t e d w i t h t h e p a r a b o l i c problem. Then t h e Crank-Nicolson-Galerkin-method i s c o n s i d e r e d as a f i n i t e element method f o r t h e numerical s o l u t i o n of t h i s system w i t h a p p r o x i m a t e l y g i v e n r i g h t hand s i d e s . We a r e t h u s l e d t o a s i t u a t i o n where t h e r e s u l t s o f S e c t i o n i can be a p p l i e d and y i e l d lower and upper a - p o s t e r i o r i e r r o r e s t i m a t e s on e v e r y d i s c r e t e time l e v e l . They p r o v i d e a b a s i s f o r t h e d e s i g n of a d a p t i v e f i n i t e element s o l v e r s . For an i n t e r e s t i n g example, e f f i c i e n t computations w i l l f i n a l l y b e p r e s e n t e d . 213
214
H.-J.
REINHARDT
In t h e r e s u l t s of [ 1 4 ] and t h e f i r s t s e c t i o n t h e r e a r e involved i d e a s of r e c e n t papers of Babugka-Rheinboldt [ I ] - [4] concerning t h e numerical s o l u t i o n of one- and two-dimensional boundary- and eigenvalue-problems by t h e f i n i t e element method. However, t h e r e s u l t s of [ I ] - [ 4 ] cannot be a p p l i e d t o s i n g u l a r p e r t u r b a t i o n problems because t h e e r r o r estimates would n o t b e r e a l i s t i c . The p r e s e n t approach f o r t h e development of a - p o s t e r i o r i e r r o r e s t i m a t e s on e v e r y d i s c r e t e time l e v e l f o r t h e numerical s o l u t i o n o f p a r a b o l i c problems i s n o t r e s t r i c t e d t o s i n g u l a r l y p e r t u r b e d d i f f e r e n t i a l e q u a t i o n s . Babugka-Rheinboldt [ 2 ] d e r i v e a - p o s t e r i o r i e r r o r e s t i m a t e s f o r ( n o t s i n g u l a r l y p e r t u r b e d ) p a r a b o l i c problems. T h i s approach d i f f e r s from o u r s and i s s i m i l a r t o t h e "method o f l i n e s " where, by d i s c r e t i z a t i o n of t h e space v a r i a b l e , a system of i n i t i a l v a l u e problems i s a s s o c i a t e d w i t h t h e parabolic equation. Numerical experiments show t h a t o u r a d a p t i v e methods a l s o e f f i c i e n t l y s o l v e problems w i t h i n t e r n a l l a y e r s . Asymptotic methods f o r t h e s o l u t i o n of p a r a b o l i c problems i n v o l v i n g i n t e r n a l l a y e r s a r e c o n s i d e r e d i n e . g . Bobisud [5], Polk [13]. I n a forthcoming paper w e s h a l l e s t a b l i s h a - p o s t e r i o r i e r r o r e s t i m a t e s and a d a p t i v e f i n i t e element s o l v e r s f o r such problems i n o r d i n a r y a s w e l l a s p a r a b o l i c d i f f e r e n t i a l equations. 1 . THE FINITE ELEMENT SOLUTION OF ONE-DIMENSIONAL ELLIPTIC EQUATIONS WITH
RIGHT-HAND
SIDES C O N T A I N I N G BOUNDARY LAYER CONTRIBUTIONS
This f i r s t s e c t i o n d e a l s w i t h t h e f i n i t e element s o l u t i o n of s i n g u l a r l y p e r t u r b e d o r d i n a r y d i f f e r e n t i a l e q u a t i o n s which o c c u r a s t h e e l l i p t i c p a r t s of t h e p a r a b o l i c e q u a t i o n s c o n s i d e r e d i n t h e f o l l o w i n g s e c t i o n . The r e s u l t s of R e i n h a r d t [I41 a r e extended t o e q u a t i o n s w i t h inhomogeneous right-hand s i d e s c o n t a i n i n g boundary l a y e r c o n t r i b u t i o n s . This extended s i t u a t i o n w i l l be t r e a t e d a n a l o g o u s l y t o [ 1 4 ] , and upper as w e l l a s lower a - p o s t e r i o r i e r r o r e s t i m a t e s w i l l be proved ( c f . Thm. 1 . 1 ) .
F i n a l l y , a more g e n e r a l problem w i l l be c o n s i d e r e d where t h e right-hand
s i d e s of t h e f i n i t e element e q u a t i o n s a r e only given approximately i n form of l i n ,ear f u n c t i o n a l s . By v i r t u e of s u i t a b l e s t a b i l i t y p r o p e r t i e s and t h e r e s u l t s i n d i c a t e d above, we a r e a g a i n a b l e t o e s t i m a t e t h e e r r o r w i t h r e s p e c t t o l o c a l energy-
norms by computable, now approximate, e r r o r i n d i c a t o r s ( c f . Thm. 1 . 2 ) . These ext e n s i o n s of [I41 a r e needed t o d e r i v e a - p o s t e r i o r i e r r o r e s t i m a t e s f o r t h e parab o l i c e q u a t i o n s c o n s i d e r e d i n S e c t i o n Z . See [I51 f o r a survey of t h e r e s u l t s of [ 1 4 ] . 1 . I . ASYMPTOTIC APPROXIMATION Let us c o n s i d e r t h e s i n g u l a r l y p e r t u r b e d boundary v a l u e problem (1.1)
-E
2
u'l
2
+ b u = fE
i n (O,l)y
u(0) = u ( l ) = 0
i n i t s v a r i a t i o n a l formulation
c o n t a i n i n g a small parameter
E
in O < E < E
. Here
B E ( . , . ) denotes the associated
sesquilinearform
The c o r r e s p o n d i n g energy-norm is denoted by
1
llullE = B E ( ~ , ~ ) l ' 2u, , v r H ( 0 , l ) .
A-POSTERIORI
215
ERROR ESTIMATES
Analogously t o [ 1 4 ] , we assume t h a t t h e f u n c t i o n b
is continuously d i f f e r e n t i a b l e ,
t h a t t h e d e r i v a t i v e b ' i s uniformly bounded w i t h r e s p e c t t o 0 < Bo6bE(x)
< 4 ' < m , xe[O,I],
w i t h some p o s i t i v e c o n s t a n t s 6
6; independent of
0'
E,
and t h a t
O < E \ < E ~, E.
Our s t u d i e s i n [I41 a r e
extended i n such a way t h a t t h e inhomogeneous r i g h t - h a n d s i d e f boundary l a y e r c o n t r i b u t i o n s which means t h a t f
=
may c o n t a i n
F, + 4, + YE where F
E'
a r e continuous functions s a ti s f y i n g yE = o(Ps(,)
4 € ( x ) = O(Pr(f) e x p ( - c x / E ) ) ,
(1.3)
uniformly i n [0, I]. resp.,
is
c
In t h e
case
set
m.
c =
of
boundary
no
t h a t b o ( x ) 2 Bo,
e x p ( -c( ] - X ) / E ) ) ( E + o )
d e n o t e polynomials of d e g r e e r >I 0 and s a 0 ,
Here Pr and P
a p o s i t i v e c o n s t a n t , and O(.)
layer
d e n o t e s t h e well-known Landau symbol.
contributions
let there exist
Furthermore
1 -x
@E*
we
have
4
E
continuous functions
I +'
bo and
E
:0
Fa
and such
and, w i t h numbers v I , v 2 > 0 , l e t t h e f o l l o w i n g conver-
x E [0, I ] ,
gence r e l a t i o n s b e s a t i s f i e d
(1.4)
IFE
-
FoI0,2 = O ( E ' ~ )
,
2 IbE -
2
=
0(cv2) ( ~ ' 0 ) .
For t h e norms and s p a c e s we use t h e n o t a t i o n s of C i a r l e t [ 7 ] .
A s i n [ 1 4 ] , a l s o i n t h e p r e s e n t extended s i t u a t i o n t h e s o l u t i o n u
of ( I . 2 ) can be
a s y m p t o t i c a l l y r e p r e s e n t e d by t h e s o l u t i o n of t h e reduced problem p l u s approximat i o n s of t h e boundary l a y e r s o l u t i o n s . Let u s d e n o t e t h e s o l u t i o n of t h e reduced
prob Zem (1.5)
bLu = F
2 2 1 by uo (= F o / b o ) , and l e t u s assume t h a t u E H ( 0 , l ) and b o c C [0,1].
Let
+
and JI
) = - Uo(l),
respectively. $
and JI,
a r e approximations o f t h e boundary l a y e r s o l u t i o n s a t x = O
and x = I , r e s p . . T h e i r r e p r e s e n t a t i o n by means of t h e Green's f u n c t i o n t o g e t h e r w i t h t h e a s y m p t o t i c b e h a v i o u r of @ - and Y - y i e l d t h e r e l a t i o n s
where c o = m i n ( c , b o ( 0 ) ) , c1 = m i n ( c , b o ( l ) ) i n t h e c a s e of a and c~ = b o ( 0 ) , c 1 = b ( I )
finite
c > 0,
i n t h e c a s e of c = - . I n o r d e r t o t r e a t b o t h c a s e s s i -
m u l t a n e o u s l y , we a l l o w r , s = - I
i n (1.8) which r e p r e s e n t s t h e c a s e of no boundary
l a y e r c o n t r i b u t i o n s (and c o = b (O), c1 = b ( I ) ) .
Here and i n t h e f o l l o w i n g , p o l y -
nomials w i t h t h e same s u b s c r i p t s need n o t b e t h e same. Defining v
=
uo +
+ JI,,
O < E Q E ~ , and a p p l y i n g t h e s t a b i l i t y i n e q u a l i t y
216
H. - J .
(1.3)] w i t h u - v
[14,
E
(1.9)
llUE
-
one o b t a i n s +
=o(E"'
VEIIE
This means t h a t v
E'
REINHARDT
Eu2
-t E3/')
(E'O).
i s an asymptotic approximation f o r u
with r e s p e c t t o t h e e n e r -
gy-norm. The o r d e r of convergence i n ( 1 . 9 ) with r e s p e c t t o t h e maximum-norm i s r e duced by t h e power 1 / 2 ( c f . [14, 1.2.
(1.4)]),
I n t h e f o l l o w i n g , we assume t h a t u1,u2>3/2.
SOME PRELIMINARY ESTIMATES FOR THE FINITE ELEMENT SOLUTION some p r e l i m i n a r y e s t i m a t e s
For t h e f i n i t e element s o l u t i o n a s s o c i a t e d w i t h (1.2),
a n a l o g o u s l y t o [14].
w i l l now be d e r i v e d . The e r r o r w i l l be s p l i t
The i m p o r t a n t
e s t i m a t e s of Lemma 1 . 1 modify c o r r e s p o n d i n g r e s u l t s o f [I41 f o r t h e extended s i t uation considered here. Let A denote a p a r t i t i o n o f [0, I ] ,
A: 0 = x
<
xI <
. . . < xJ-I
<
xJ = I , w i t h n o t nec-
e s s a r i l y e q u i d i s t a n t g r i d p o i n t s . Let u s use t h e n o t a t i o n s
With t h e s p a c e M of a l l c o n t i n u o u s piecewise l i n e a r f u n c t i o n s over t h e g r i d , we define M
1 = M n H ( 0 , l ) . Now t h e
f i n i t e element s o l u t i o n u A , € e Mo a s s o c i a t e d w i t h
( 1 . 2 ) i s t h e unique s o l u t i o n of t h e e q u a t i o n
,
B E ( ~ A , E ,=~ () f c , v )
(1.10)
vEMo
.
The f i n i t e element method (1.10) i s s t a b l e i n such a s e n s e t h a t u bounded by t h e s o l u t i o n u
of ( 1 . 2 )
(with respect t o
i s uniformly
A,c
and A). I n f a c t one h a s
E
I n t h e proof of t h e f o l l o w i n g e r r o r e s t i m a t e s we need two assumptions on t h e g r i d . With p o s i t i v e c o n s t a n t s p o , p l independent of € , A and m > m ' : m a x ( r , s ) , (1.11)
p0 i h j / h j - l
s
p1
,
j=2
assume t h a t
,...,J ,
and
>
c'h
(1.12)
PI
where c ' = m i n ( c o , c l ) . Assumption ( 1 . 1 1 )
mE(-InE)
,
i s t h e same as [ 1 4 , (3.9)].
It is a l s o re-
q u i r e d ' i n Babugka-Rheinboldt
[I]
tions".
a r e a l s o c a l l e d " l o c a l l y quasi-uniform".
Grids of type ( 1 . 1 1 )
i n connection with so-called "admissible p a r t i -
( 1 . 1 2 ) i s s a t i s f i e d provided t h a t
In o r d e r t o derive e r r o r e s t i m a t e s we s p l i t the e r r o r e =
+
(1.13)
PA,€ (cf. pa,,
[14, (3. I ) ,
= uE
-
nASE ,
(3.1411) where
PA,€
Condition
i s s m a l l compared w i t h h .
E
=
nASE- u A , €
, +
A,E
= u
E
= nAu0 +
1
- u
0,
A,E +
in e
qE
A,€
=
.
Ila v denotes t h e l i n e a r i n t e r p o l a n t of a f u n c t i o n V E H ( 0 , l ) . Furthermore we s p l i t pa,,
a s follows
A-POSTERIORI
(1.14)
pa,,
-
= (uE
ERROR E S T I M A T E S
v c ) + ( u o - IIAuo) + (?
t
-
217
.
II G ) B E
The f i r s t c o n t r i b u t i o n can be e s t i m a t e d by means of t h e convergence r e l a t i o n ( 1 . 9 ) f o r t h e a s y m p t o t i c approximation v
.
For t h e second c o n t r i b u t i o n i n ( 1 . 1 4 ) ,
the
standard interpolation estimates y i e l d
(1.15)
/lug -
2
nA~o/IE,I
I 2,2,1, ,
j=l,...,J , J The t h i r d c o n t r i b u t i o n w i l l b e e s t i m a t e d i n t h e f o l l o w i n g l e m m a 6 C h . ( l +C.)/u
J
j
J
where C = E/h j j' which e x t e n d s a c o r r e s p o n d i n g e s t i m a t e i n [ 1 4 , Lemma 3. I ] .
LEMMA I . I . Under the asswnption
of
( I . 12) there e x i s t s a constant C > 0 independent
and the p a r t i t i o n A such t h a t f o r a l l 0 <
E
E G E
and a21 j = 2 , .
. . ,J-I
the fol-
lowing estimates hold
PROOF: Without r e s t r i c t i o n o f g e n e r a l i t y suppose t h a t
6exp(-l/p
E
I
) . Hence by
means of (1.12) one h a s h 2 ~ ( r + I ) / c and t h e f u n c t i o n s (x/E)'exp(-c
. ., r + l , a r e
w=O,.
0'
monotonically decreasing i n [x,, I ] .
x/E),
Therefore the following
estimates hold
=
where
p,,(t)
i ~ x j - I / ~ ~ w e x ~ ~ - c o x j - l / ~/E)exp(-c ~ p u ~ x .o xj - 1 / E l , =
4-1
1:1 =o (
j = 2 , . . ,J,
U=O,.
.,r+~,
I / c ~ + ~ ) - p~) t~W ~- ' . ~ By ( u means of t h e a s y m p t o t i c r e p r e s e n t a -
t i o n of $ ' and i n t e r p o l a t i o n e s t i m a t e s , one o b t a i n s
2
2 -nA'~11,2,1
I'E
' j
2 C E
2 1 ' ~ 1 1 , 2 , 1 .< J
CE
/
/P,+~
2
.
~ X ~ ( - ~ C ~ X ~ j =-2 ~, . / ,J, E ) ,
w i t h some polynomial P r + l of degree l e s s o r e q u a l r + l . The same argument a s above implies that
Hence f o r t h e LL-norm
2 2 2 I $ E - n A $ E 1 0 , 2 , 1 6 l $ E ( x j - l ) l h j 6 C hJ . l P , + ] ( ~ ~ - ~ / c exp(-2c )l ox j-1 j
/E),
j=2
,...,J ,
and f o r t h e energy-norm
- nA$E
IIE,I
j
< c (E
2 + hJ. ) I P r + l ( X ~ - ~ /[ Ee)x p ( - 2 c o x j - l / ~ ) , j = 2 , .
..,J.
Assumption (1.12) i s e q u i v a l e n t t o t h e f o l l o w i n g i n e q u a l i t y e x p ( - c ' h . / ( p E ) ) < J I j=l, J , which,combined w i t h ( l . l l ) , i m p l i e s t h a t h. h -u L Y (?)'exp(-c o x j - 1 / E ) < (+)" e x p ( - c o h j - l / c ) \< p I ( € ) exp(-c Oh .J / ( p I E ) )
...,
-v,m-v j
Q P] E
-u,m-v
(hj/E)mexp(-c'h./(pl~)) 6 p l J
E~
h
m j
,
u=O
,...,m,
j=2
E
,...,J .
m,
.
218
H - J . REINHARDT
We have t h e r e f o r e proved t h a t
For Ji
one concludes a n a l o g o u s l y t h a t
The r e p r e s e n t a t i o n ir tion ( l . l 6 ) . 0 Combining ( l . 9 ) ,
E
- nA ir €
=
-
$E
IlA$€ + JiE
-
llA IJJ E f i n a l l y y i e l d s t h e a s s e r -
( I . l 5 ) , and ( l . l 6 ) , one r e a d i l y s e e s t h a t
The l a s t two c o n t r i b u t i o n s i n t h e s p l i t t i n g (1.14) v a n i s h a t t h e g r i d p o i n t s . Cond i t i o n ( 1 . 9 ) modified f o r t h e maximum-norm t h e r e f o r e y i e l d s
(1.18)
l p A , E ( x j ) l = O(E)
,
j=O
,...,J.
1.3. A-POSTERIORI ERROR ESTIMATES a-posteriori estiTaking i n t o account t h e e s t i m a t e s ( 1 . 1 7 ) and (1.18) f o r p A,€' mates f o r t h e e r r o r of t h e f i n i t e element method w i l l be d e r i v e d q u i t e a n a l o g o u s l y t o [14, S e c t . 41 e
A,E
= u
E
-
u
A,E
.
Besides t h e s p l i t t i n g (1.13) we a d d i t i o n a l l y s p l i t t h e e r r o r
as f o l l o w s
( i ,)€ , i = 1 , 2 The f u n c t i o n s e A
(more p r e c i s e l y : t h e i r r e s t r i c t i o n s t o any s u b i n t e r v a l
I , ) , s a t i s f y t h e f o l l o w i n g a u x i l i a r y problems
-E
2 (2lIl e A,E
+ b 2E e(A2 ), € = r j
i n Ii, j = 1 ,
...,J.
The inhomogeneous right-hand s i d e of (1.21) i s given by t h e residual (1.22)
r
i
= (fE
-
2 b E u A , c ) I I i , j=1,
...,J.
The s o l u t i o n s of (1.20) and (1.21) s a t i s f y t h e b a s i c r e l a t i o n
According t o t h e l a s t r e l a t i o n i t i s now o u r aim t o d e r i v e r e a l i s t i c e s t i m a t e s f o r t h e two c o n t r i b u t i o n s on t h e right-hand s i d e of ( 1 . 2 3 ) . By v i r t u e of s t a b i l i t y i n e q u a l i t i e s from t h e t h e o r y of s i n g u l a r l y p e r t u r b e d o r d i n a r y d i f f e r e n t i a l equat i o n s , t h e l o c a l energy-norms of e ( l ) are bounded a s f o l l o w s ( c f . A.E
[14,
(4.9)])
A-POSTERIORI
219
ERROR ESTIMATES
= 1 3 i 1 1 2 ( 1 + O ( h . ) ) ( l + 0 ( E ! I 2 ) ) and B ' = max{bE(x), ~ ~ 1 . The 1 . variational j J J j J f o r m u l a t i o n of ( 1 . 2 1 ) and t h e i n e q u a l i t y of P o i n c a r g - F r i e d r i c h s immediately y i e l d the following a-posteriori e r r o r estimates f o r e ( 2 ) A.E '
where y
where 8. = minIb (x), J
indicator
X E
rl
I.). L e t u s c a l l t h e right-hand s i d e of (1.25) error J
j' I t i s now i m p o r t a n t t o d e r i v e a r e l a t i o n between t h e energy-norms o f e A ( 1,)E and
e(')
For n o t s i n g u l a r l y p e r t u r b e d problems such a r e l a t i o n i s proved i n Babuska-
A,€'
Rheinboldt [ 2 , S e c t . 3.21 i n t h e form I l e ( l ) l l A,€
E
=O(h')lleL:lllE.
In t h i s case i t
s u f f i c e s t o e s t i m a t e t h e energy-norm of t h e second c o n t r i b u t i o n e ( 2 ) of t h e s p l i t t i n g ( 1 . 1 9 ) i n o r d e r t o o b t a i n an e s t i m a t e f o r t h e whole
error e
.
A,E
A,€
because
lleA,EII, = ( I + O ( h ' ) ) / l e A ( 2,)E I I E For t h e p r e s e n t s i n g u l a r p e r t u r b a t i o n problem nega-
t i v e powers of
E
would a p p e a r i n t h e bounds which t h e n become t o o p e s s i m i s t i c .
However, i n t h e c a s e o f a s m a l l
E
t h e f o l l o w i n g theorem s t a t e s t h a t t h e l o c a l
energy-norms of t h e e r r o r can be e s t i m a t e d e s s e n t i a l l y by t h e e r r o r i n d i c a t o r s . Moreover, a lower a - p o s t e r i o r i e r r o r e s t i m a t e i s p r e s e n t e d which shows t h a t t h e e r r o r i n d i c a t o r s indeed a r e r e a l i s t i c bounds.
Under t h e r e g u l a r i t y assumptions s t a t e d in Section 1 . 1 there e x i s t s a
THEOREM 1 . 1 .
constant c
>O
j = l , ...,J, 0
such t h a t the following lower a-posteriori error estimates hold,
< E < E
.
s u f f i c i e n t l y small 0
Tf one assumes the conditions ( 1 . 1 1 ) and ( I . l Z ) , then f o r a l l the following upper a-posteriori error estimates are
< E 6
valid,
PROOF: ( i ) . For t h e lower e s t i m a t e we o n l y show t h a t (1.26) i s a consequence of According t o t h e uniform boundedness o f b ' one has B ! / B . = 1 + O ( h . ) . 2 2 -1 "2 J- JJ = B . ( I + ~ 1 ~ 2 ~ 8 . ' immedi) Thus m u l t i p l i c a t i o n of [ 1 4 , (4.15)] by (B2 + TI 2 . ) j J J J a t e l y y i e l d s the a s s e r t i o n (1.26). [14,
(4.15)].
( i i ) . Analogously t o 1114, Thm. 4 . 1 1 ,
E
2
C. + h 3 + ~j
(l,l7),
E
(l.l8),
( 1 . 2 4 ) , and (1.25) imply
2(m-(m'+l)) 2 m ' + l h
j
J
,
j = 2 , ...,J- I , O <
E < E
1'
220
H . 4 .
REINHARDT
Again u s i n g ( l . l 8 ) , t h e whole e r r o r a t t h e mesh p o i n t s s a t i s f i e s (leA,E(xj-l)
1)-
I+
leA,E(xj)
7
\<
C(E + 1 0 A , E ( x j - l )
1
C(E’ + h - ’ Qj
IcA,€(xj)
+
Combining t h e l a s t e s t i m a t e , ( 1 . 2 4 ) and ( 1 . 2 5 ) ,
j = 2 , ...,.II, 0 <E<
l i e n , € 11’E , I .i
2
<
2
j=2
( I + c e . ) n f + c E ( E +K.), J J J
,...,J - 1 ,
o
2
+Uj)’
one has
< E <<El.
2
A s I + O ( f . ) = ( I + O ( e . ) ) , t h e a s s e r t i o n (1.27a) i s proved. The i n e q u a l i t i e s J J (1.27b) and ( 1 . 2 7 ~ ) f o l l o w q u i t e s i m i l a r l y ( c f . [ 1 4 , Thm. 4 . 1 1 ) . 0
1.5. A-POSTERIORI ERROR ESTIMATES WITH APPROXIMATE ERROR INDICATORS I n the following Section 2 the
situation
arises
that
the
inhomogeneous
right-hand s i d e o f t h e f i n i t e element e q u a t i o n (1.10) i s only given approximately i n form of a l i n e a r f u n c t i o n a l . For t h i s more g e n e r a l c a s e w e s h a l l a g a i n d e r i v e a-posteriori error estimates. of ( 1 . 2 ) d e f i n e s a l i n e a r f u n c t i o n a l u
The right-hand s i d e
1
(1.28)
uE(v) = (fE,v), V G H (0,l).
It s a t i s f i e s the e s t i m a t e s
‘
lfE10,21v/0,2
‘
l f E 1 0 , 2 IIVllE
where, a c c o r d i n g t o (1.3) and ( l . 4 ) , E.
1
on H ( O , l ) ,
Suppose t h a t f
functional C
.
If,lo,2
i s uniformly bounded w i t h r e s p e c t t o
(and u ) i s n o t known b u t given approximately by a l i n e a r
Consequently, one cannot compute u
element approximation 6
A,€
(1.29)
d e f i n e d by
A,€
B E ( G A , € , v )= a E ( v ) , V E M
( c f . (1.10)) b u t an f i n i t e
.
It i s readily seen t h a t the following s t a b i l i t y inequality
(1.30)
IIaA,EIIE
‘ lIdEII*
and t h e e r r o r e s t i m a t e (1.31)
IIuA,E
- “,EllE
‘
IIUE
-
CEIl
I
1 1 ~ 1 =1
h o l d . A s u s u a l t h e norm o f a f u n c t i o n a l u on H ( 0 , l ) i s d e f i n e d by I v e H (0,l)). = s u p ~ l u ( v ) I /lIvIIE: 0 During t h e computations t h e e r r o r i n d i c a t o r s r-
(cf. (1.25)) a r e not available. In j o r d e r t o d e f i n e and compute approximate e r r o r i n d i c a t o r s , we s h a l l f i r s t i n t r o d u c e r e s i d u a l s f o r t h e p r e s e n t c a s e where a f u n c t i o n a l a p p e a r s on t h e right-hand s i d e of ( 1 . 2 9 ) .
I n t h e s p l i t t i n g (1.19) of e
i n g v a r i a t i o n a l e q u a t i o n s ( c f . (1.21)) (1.32)
where B
B
(2)
.(eA,€,$) =
€ 9 3
BE($)
A,E
the function
APE
2 - (bEuA,€,$),$EHA(I.), j = l , J
s a t i s f i e s t h e follow-
...,J,
1 .(.,.) d e n o t e s t h e r e s t r i c t i o n of B (.,.) on H (0,l)xH ( I . ) . I n s e r t i n g
EYJ
1
O
J
A-POSTERIORI
where t h e residual functional p (1.34)
PE(v) = d (v)
-
221
ERROR E S T I M A T E S
i s d e f i n e d by
2(b u A , E , v )
I vEHo(O,l).
,
For c l a r i t y , t h e norms o f t h e r e s t r i c t e d f u n c t i o n a l s a r e w r i t t e n w i t h a c o r r e s ponding s u b s c r i p t , i . e . l l u I l H ~ ( 1 ~ =) ~S U P { I O ( V ) IIIvIIE,I, / : 0 J
1
f
VEH~(I~)]*
110
of P E i s uniformly bounded with r e s p e c t t o
The norm
E provided t h a t - SEl1 i s uniformly bounded. I n f a c t , then t h e uniform boundedness o f I / u E l li m p l i e s t h a t
2
of I l d E I I and, moreover, t h e L -norm of ii
A,E
and A .
i s uniformly bounded w i t h r e s p e c t t o
E
Let u s now c a l l
i s d e f i n e d e s p e c i a l l y by an element g E L L ( O , l ) , I i . e . BE(v) = ( g E , v ) , v r H (O,]), then by means of t h e i n e q u a l i t y of P o i n c a r i -
approximate error i n d i c a t o r s . I f d Friedrichs
We a r e now a b l e t o e s t a b l i s h a - p o s t e r i o r i e r r o r e s t i m a t e s where t h e approximate e r r o r i n d i c a t o r s a r e i n v o l v e d . For b r e v i t y , l e t us d e n o t e
terms of magnitude O ( X . + 6 . ) J J have t o be added on t h e right-hand s i d e s . A s a consequence of ( 1 . 2 6 ) , ( 1 . 2 7 ) , I f one i n s e r t s ii
j
i n s t e a d of q . i n ( 1 . 2 6 ) and (1.27), J
( 1 . 3 1 ) , and (1.33) t h e whole e r r o r S = u - ii s a t i s f i e s lower and upper A,€ E A,E a - p o s t e r i o r i e r r o r e s t i m a t e s s t a t e d i n t h e f o l l o w i n g theorem. THEOREM 1 . 2 . Under the r e g u l a r i t y asswrrptions stated i n Section I . 1 there e x i s t s a
constant C > 0 such t h a t the following lower a-posteriori error estimates hold,
j=l,
...,J, O < E $ E . The
Assume ( 1 . 1 1 )
A . defined b y ( 1 . 3 6 ) f u l f i l l the r e l a t i o n s J
numbers 6 .
and ( l . l 2 ) ,
J'
then f o r a l l s u f f i c i e n t l y small O < E < E ~the following
upper a-posteriori estimates are v a l i d , ( I ' 39a)
1 1 'A,E
IIE, 1
j
\< ( 1
+
ce.)fi. J
J
+
O ( E I / ~ ( E + K . )
J
+ 6 . + A.), j = 2 ,...,J - I , J
J
222
REINHARDT
H.-J.
(1.39~)
where the
~ ~ ~ A , E ~ ~,< E'i,, I J K.
J
ceJfiJ-,
+
+ 6, + A,)
o(E'/2(E+KJ)
are defined in Theorem 1 . 1 .
2 . THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PARABOLIC I N I T I A L VALUE PROBLEM BY THE CRANK-NICOLSON-GALERKIN-METHOD
I n t h i s s e c t i o n we e s t a b l i s h a - p o s t e r i o r i e r r o r e s t i m a t e s f o r t h e Crank-NicolsonGalerkin-method of s o l v i n g p a r a b o l i c i n i t i a l v a l u e problems with t h e s i n g u l a r l y p e r t u r b e d e l l i p t i c p a r t s t u d i e d i n S e c t i o n I . We a r e a b l e t o apply t h e e r r o r e s t i mates of t h e p r e c e d i n g s e c t i o n by a s s o c i a t i n g a system of e l l i p t i c e q u a t i o n s w i t h t h e given p a r a b o l i c e q u a t i o n . For e v e r y d i s c r e t e t i m e l e v e l t h e e r r o r between t h e f i n i t e element s o l u t i o n and t h e s o l u t i o n of t h e e l l i p t i c e q u a t i o n s a t i s f i e s lower and upper a - p o s t e r i o r i e r r o r e s t i m a t e s where t h e a s s o c i a t e d s e s q u i l i n e a r forms and norms have t o be a d j u s t e d ( c f . Thm. 2 . 2 ) . For t h e d i f f e r e n c e between t h e s o l u t i o n of t h e e l l i p t i c e q u a t i o n s and t h e p a r a b o l i c problem, a s t a n d a r d a n a l y s i s of t h e
Crank-Nicolson-Galerkin-method
y i e l d s an e r r o r e s t i m a t e i n terms of t h e s t e p width
i n t h e t - d i r e c t i o n ( c f . Thm. 2 . 1 ) . The p r e s e n t approach i s similar t o t h e method of Rothe ( c f . Kagur-Wawruch
[9], K a r t s a t o s - Z i g l e r
[lo])
where, b e f o r e t h e p a r t i -
t i o n of t h e space v a r i a b l e , a d i s c r e t i z a t i o n of t h e time i n t e r v a l i s performed. To conclude t h i s s e c t i o n , a d a p t i v e computations a r e p r e s e n t e d . According t o our l o c a l a - p o s t e r i o r i e r r o r e s t i m a t e s , t h e a u t o m a t i c mesh refinement i s implemented such t h a t t h e mesh t e n d s t o a s t a t e which i s c a l l e d a s y m p t o t i c a l l y e q u i d i s t r i b u t e d . This p r o c e s s c a u s e s a d e s i r a b l e mesh d e s i g n where t h e boundary l a y e r s a r e d e t e c t e d and r e s o l v e d . 2.1.
STABILITY OF THE CRANK-NICOLSON-GALERKIN-METHOD
Let us c o n s i d e r s i n g u l a r l y p e r t u r b e d p a r a b o l i c d i f f e r e n t i a l e q u a t i o n s of t h e form
(2. la)
ut =
E
2
u
xx
-
b%u + wE
,
(x,t)
E
(0, I)x(O,T]
,
0<
E
Q
E ~ ,
w i t h homogeneous D i r i c h l e t boundary c o n d i t i o n s (2.Ib)
u(0,t) = u(l,t) = 0
,
t E(O,T],
and an i n i t i a l c o n d i t i o n (2.Ic)
u(x,O) = g ( x )
,
X E
(0,l).
For t h e sake of s i m p l i c i t y , w e assume t h a t t h e continuous i n i t i a l f u n c t i o n g i s independent of
E,
Furthermore l e t w
and t h a t t h e c o m p a t i b i l i t y c o n d i t i o n g ( 0 ) = g ( 1 ) = 0 h o l d s . be of t h e form
(2.2)
WE(X,t) = Wo(X,t)
where Wo,
To, T I , Q E , I
+
T O ( t ) o E (XI + T l ( t ) Y E ( X ) , ( x , t )
E
[O,I]x[O,T],
a r e continuous f u n c t i o n s . For t h e p r e s e n t p a r a b o l i c prob-
A'POSTERIORI
223
ERROR ESTIMATES
lem we a l s o a l l o w boundary l a y e r c o n t r i b u t i o n s i n t h e inhomogeneous t e r m w a r e r e p r e s e n t e d by 0
I
E
E'
which
s a t i s f y i n g ( 1 . 3 ) . L a t e r on we s h a l l a d d i t i o n a l l y r e q u i r e
some r e g u l a r i t y of g and w
.
1
L e t bEE C [ O , I ]
be a f u n c t i o n o f t h e s p a c e v a r i a b l e
only which, t o g e t h e r w i t h i t s d e r i v a t i v e , i s uniformly bounded w i t h r e s p e c t t o F i n a l l y , w i t h a f u n c t i o n b E C 1 [0, I]
E.
we assume t h a t lb2 E - bol0,2 = o ( ~ ~ / ~ ) .
W e s t a r t w i t h t h e f o l l o w i n g v a r i a t i o n a l f o r m u l a t i o n of ( 2 . 1 ) (2.3a)
(ut,v) +
E
(2.3b)
u(.,O)
g
=
2
2
+ (b u , v )
(u',v')
1
(wE,v), v r H ( O , l ) ,
=
O
,
and assume t h a t t h e r e e x i s t s a unique s o l u t i o n u
of ( 2 . 3 ) . I n o r d e r t o a p p r o x i -
mate ( 2 . 3 ) by a f i n i t e element method, we s u b d i v i d e t h e time i n t e r v a l [O,T] e q u i d i s t a n t s u b i n t e r v a l s , t n = n k , n=0, i n the t-direction.
For e v e r y n=O,
...,N
...,N ,
in
where k = T / N d e n o t e s t h e s t e p width of [0,1]
l e t a partition A
of n o t neces-
s a r i l y e q u i d i s t a n t mesh p o i n t s be g i v e n . L e t us emphazise t h a t t h e p a r t i t i o n A and a l s o t h e f i n i t e dimensional s p a c e s M and M
a r e dependent on t h e t - l e v e l and
may be d i f f e r e n t f o r d i f f e r e n t n . For c l a r i t y , we t h e r e f o r e write h ' and hn f o r t h e maximal and minimal s t e p w i d t h , r e s p . , on t h e t - l e v e l . The s t a n d a r d Crank-Nicolson-Galerkin-(CNG-)method ui,,.
Mo,
n=0,. . . , N ,
-
i s now one of f i n d i n g f u n c t i o n s
which s a t i s f y t h e e q u a t i o n s 2 ( ( u ~+ ~( ( u ~" ) ) '' , v ' 1 A,€
k1 ( ~n+l A , E unA , E , v ) + ~ (2.4)
G
1
n+l
2
E
-(w
n
+ wE,v)
,
+
1 2 ( un+l -(b 2 E A,E
...,N-I.
+
un A,E)' v ,
v e M o , n=O,
For b r e v i t y , we w i l l be s u i t a b l y chosen l a t e r . n n n-l ) / k , v n t 1 l 2 = (vnt1+vn)/2 w r i t e un(x) = u ( x , t ), w ( x ) = w E ( x , t n ) , a n d a v = (v - v E . n2 E t f o r f u n c t i o n s v l a L ( O , l ) , i = n - l , n , n + l . Then ( 2 . 4 ) can be r e w r i t t e n i n t h e form The i n i t i a l a p p r o x i m a t i o n uo
(2.5)
n+ 1 n+1/2 , v ) , vcMo, 06nSN-1. ( a t u A , E , v ) + E ~ ( ( u ~ ~ " ~ ) ' , +v '( b) E2 uAn, + € 1 / 2 ,v ) = (wE A.E
I n t h e f o l l o w i n g lemma we s h a l l e s t a b l i s h t h e s t a b i l i t y ( i n t h e t - d i r e c t i o n ) o f t h e CNG-method.
I t s proof i s d e r i v e d from e s t i m a t e s of Wheeler [I81 ( c f . a l s o
F a i r w e a t h e r [E,
14.3, 14.41, ThomGe-Wahlbin
[17])
where, i n t h e p r e s e n t i n e q u a l i -
t i e s , t h e energy-norm c o n t a i n i n g t h e s m a l l parameter LEMMA 2.1. For t h e s o l u t i o n
1
i s involved.
of t h e CNG-method t h e foZlowing estimates are
valid,
PROOF: For e v e r y z n E H ( O , l ) ,
E
n=O
,...,N ,
the relation
224
REINHARDT
H . -J.
i s s a t i s f i e d . I n s e r t i n g v = a un+] i n ( 2 . 5 ) and z t
A,E
n
= (u
i n the l a s t equation, it follows t h a t
T
n A,E
) I
a s w e l l a s z n = b un E A,€
= 1/4 yields
which i m p l i e s
Thus (2.6) i s proved; (2.7) f o l l o w s by summation.U
In t h e s e n s e of t h e Lax-Richtmyer-theory,Lemma 2 . 1 s t a t e s t h a t t h e CNG-method i s a s t a b l e method f o r t h e numerical s o l u t i o n of t h e i n i t i a l v a l u e problem (2.1) where in
the
stability
inequality,
in contrast
2
to
t h e c l a s s i c a l t h e o r y , t h e energy-
norm i s used f o r t h e s o l u t i o n and t h e L -norm f o r t h e inhomogeneous right-hand s i d e . Stability
holds
without
r e s t r i c t i o n on t h e s t e p w i d t h s i n t h e t- and x-di-
any
r e c t i o n . Such a method i s c a l l e d " u n c o n d i t i o n a l l y s t a b l e " . One can a l s o e s t a b l i s h t h e s t a b i l i t y of t h e CNG-method with r e s p e c t t o t h e maximum-norm ( o v e r t h e g r i d p o i n t s ) which, by t h e way, r e q u i r e s a r e s t r i c t i o n on t h e s t e p w i d t h s . A s t h i s i s n o t r e l e v a n t f o r o u r subsequent s t u d i e s , we s h a l l n o t go i n t o f u r t h e r d e t a i l s . We a d d i t i o n a l l y r e w r i t e t h e CNG-method ( 2 . 5 ) i n t h e f o l l o w i n g form which w i l l be
a l s o used f o r computational purposes, E
(2.8)
2k n+l ~ ( ( u )',v') A,€
=
-E
+ ({I
2k
7 ( ( unA , € ) ' , v ' )
k 2 n+I +--b }u 2 E A,E'~)
k 2 n + ( { I --b }u
2
E
A,E
( c f . Section I . ] ) ,
I n a d d i t i o n t o BE(.,,)
, v ) + k ( w n + 1 / 2 , v ) , V E Mo,
n=O,. , . , N - I .
we denote t h e s e s q u i l i n e a r form o c c u r i n g
on t h e l e f t - h a n d s i d e of ( 2 . 8 ) by (2.9)
2
D (u,v) = L ( u ' , v ' )
,
+ (d:u,v)
~m
1
u , v ~ H(0,1),
2
and d: = 1 t ( k / 2 ) b E . D E ( . , . ) o b v i o u s l y d e f i n e s a s c a l a r product where Z = on H I ( 0 , l ) ; t h e a s s o c i a t e d (energy-)norm i s denoted by llu llD = D E ( u , u ) 1 12 2.2.
.
THE ASSOCIATED SYSTEM OF ELLIPTIC EQUATIONS
Let us now a s s o c i a t e a system of N e l l i p t i c e q u a t i o n s w i t h t h e p a r a b o l i c problem
.
( 2 . 3 ) . For e v e r y n=O,. , N - l
l e t v:+l
6
I
H o ( O , I ) be t h e s o l u t i o n of
A-POSTERIORI
(2.10)
n+ I , v ) = (d-vn,v) DE(vE E
E
-
225
ERROR E S T I M A T E S
? 2 ( ( vnE ) ' , v ' ) + k(wE n+ 112 , v )
,
v ~ HI ~ ( 0 , l ) .
For n = O we s e t vo = g . The p r e s e n t system has t h e same form a s t h e CNG-method (2.8) b u t c o n t r a r y t o (2.8) t h e s e s q u i l i n e a r forms a r e c o n s i d e r e d on H 1 ( 0 , I ) and n o t on 2 Ma. Assuming g E H ( 0 , l ) and d e f i n i n g (2.11)
f:+'
=
n+ 112
i2(v:)"
+ d-vn E E + kwE
,
n=O,.
. ., N - I ,
then (2.10) i s e q u i v a l e n t t o (2.12)
n+ 1
The s o l u t i o n u
n+ 1
.
n+ 1
DE(vE .v) = ( f E
n+ I A,€
,v)
,
1
n=O, ...,N - I .
veH (O,l),
of t h e CNG-method ( 2 . 8 ) i s n o t t h e f i n i t e element p r o j e c t i o n of
I n f a c t t h e right-hand
s i d e s of ( 2 . 8 ) a r e n o t d e f i n e d by f n + l b u t t h e y a r e vE given by l i n e a r f u n c t i o n a l s d e f i n e d by t h e f i n i t e element s o l u t i o n un of t h e preceding t-level.
AYE
Hence we have t h e s i t u a t i o n t r e a t e d i n S e c t i o n 1.5 where t h e
inhomogeneous right-hand
s i d e s of t h e f i n i t e element e q u a t i o n s a r e o n l y g i v e n n n a p p r o x i m a t e l y . The e r r o r v - u w i l l be s t u d i e d i n t h e f o l l o w i n g s e c t i o n . T h i s
A,€
E
s e c t i o n d e a l s w i t h t h e d i f f e r e n c e vn - un E
E
and
the
asymptotic
b e h a v i o u r of vn.
Q u i t e a n a l o g o u s l y t o Lemma 2 . 1 , one o b t a i n s t h e f o l l d w i n g s t a b i l i t y i n e q u a l i t y f o r t h e s o l u t i o n s of ( 2 . 1 2 ) ,
An a p p r a i s a l f o r t h e e r r o r e n = un E
w i t h e n i n s t e a d o f vn
E'
E
-
vn w i l l now be d e r i v e d by a p p l y i n g ( 2 . 1 3 ) E
by e s t a b l i s h i n g t h e s y s t e m of d i f f e r e n t i a l e q u a t i o n s f o r en
E'
and by e s t i m a t i n g t h e a s s o c i a t e d right-hand s i d e s . This i s t h e s u b j e c t of t h e f o l 2 2 lowing theorem. A s u s u a l L ( L ) d e n o t e s t h e space of a l l f u n c t i o n s z ( x , t ) such t h a t T I 2 2 llzlIL2(L2) = Iz(x,t)I d x d t < 0 0 3 3 2 2 THEOREM 2 . 1 . Let us asswne t h a t ( a u E / a t ) E L (L ) f o r the s o l u t i o n u of ( 2 . 3 ) .
.
Then the error e n = un - vn s a t i s f i e s the equations E
n+ 1
(2.14) ( a t e E
.v)
+ E
2
E
E
n+1/2 2 n+l/2 n+ I 1 ((eE ) ' , v ' ) + (bEeE , v ) = ( p E , v ) , veH (O,l), 06nCN-I,
where
, n=O,...,N - I ,
and t h e following error estimate holds,
,
n=O
,...,N ,
0
< E < E
.
226
H .-J. REINHARDT
Hereby t h e f o l l o w i n g e s t i m a t e s can immediately be i n f e r r e d ,
N o t i c i n g eo E 0 , t h e a s s e r t i o n (2.15) f o l l o w s from t h e s t a b i l i t y i n e q u a l i t y (2.13) with e n an:
p
n+ 1
i n s t e a d of vn and w:+'/~.O
To conclude t h i s s e c t i o n we s t u d y t h e a s y m p t o t i c behaviour of t h e s o l u t i o n v
n+ 1
of (2.12). According t o t h e d e f i n i t i o n (2.2) of w - . f! can be w r i t t e n i n E 1 1 I t h e form ( c f . ( 2 . 1 1 ) ) f = F + k.r @ + k . r ; l E t where L T~1 = (Ti(to) + T i ( t 1 ) ) / 2 , E E O E 2 1 2 i = 1 , 2 , F = Z g" + d i g t kWA'2. I f one a d d i t i o n a l l y d e f i n e s d i = I k ( k / 2 ) b o and
kW:l2,
F 1 = d-gE+ 0
0
then t h e convergence
h o l d s . Hence, f o r n = O i n (2.12), we have t h e s i t u a t i o n o f S e c t i o n I where t h e s e s q u i l i n e a r form B (.,.) the solution v
I
E
h a s t o be r e p l a c e d by D E ( . , . ) .
According t o S e c t i o n 1 . 1 ,
has the asymptotic representation Vl
(2.16)
11. ( I n- )
(with respect t o
voI = Foldo I +
,
= v;
+
$ EI
+
*J
+ O ( E 3/2)
where co = m i n ( @ / m ,
c )
,
and c > O i s t h e c o n s t a n t o c c u r i n g i n ( 1 . 3 ) . I f k < 2 / c 2 , one o b v i o u s l y has c
i = 1 , 2 , because d+ > I .
$I
I f no boundary l a y e r c o n t r i b u t i o n i s c o n t a i n e d i n w
and $ ' s a t i s f ; $,(x) 1
, $,(XI I
= O ( k exp(-@xlE))
= O ( k exp(-@(l-x)/Z))
T h i s c a s e can a g a i n b e t r e a t e d s i m u l t a n e o u s l y by s e t t i n g c = hence co =
G / m ,c 1 =
,
c1 = m i n ( e / m , c )
m,
r,s = -1,
i
E'
= c3
then
. and
elm.
For any n ) 1 one observes t h a t t h e right-hand s i d e of (2.12) c o n t a i n s t h e boundary l a y e r c o n t r i b u t i o n s of t h e s o l u t i o n of t h e preceding t - l e v e l .
The d e g r e e of t h e
c o r r e s p o n d i n g polynomials i n c r e a s e s b y one once t h e n e x t t - l e v e l i s reached. For n+ 1 every n 2 0 the solution v of (2.12) can t h e r e f o r e b e r e p r e s e n t e d i n t h e form (2.18)
where
n+l
n+l
A-POSTERIORI
n+ I
Analogously t o S e c t i o n 1 . 1 ,
227
ERROR ESTIMATES
we d e n o t e by F
n+ I
t h e p a r t of f
n o t c o n t a i n ng t h e
boundary l a y e r c o n t r i b u t i o n s which i s given by (2.20)
Fntl = E2(v:)"
- n
+ dEvo + kWnc1/2, n=O
,...,N - I .
Thus t h e r e s u l t s of S e c t i o n I can be a p p l i e d f o r any t - l e v e l where B ( . , . ) and b
2
have t o be r e p l a c e d by D (.,.) and d l , r e s p . . 2.3. A-POSTERIORI ERROR ESTIMATES A-posteriori
e r r o r e s t i m a t e s and, i n t h e f i n a l s e c t i o n , an a d a p t i v e mesh s e l e c t i o n
on e v e r y d i s c r e t e t - l e v e l w i l l now be e s t a b l i s h e d by v i r t u e of t h e r e s u l t s of
S e c t i o n 1 . 5 . Before a p p l y i n g t h e s e r e s u l t s , we s h a l l a d j u s t t h e assumptions of Section I t o f i t t h e present s i t u a t i o n , e s p e c i a l l y (1.12). ( 2 . 8 ) and ( 2 . 12), B c ( . , . )
and b!
Corresponding
have t o r e p l a c e d by D (.,.)
and d:,
to
r e s p . . d+ is
o b v i o u s l y bounded from above and from below. T h e r e f o r e i n p a r a b o l i c problems of t h e form ( 2 . 1 )
one needs o n l y t o r e q u i r e t h e uniform boundedness of b E , and hence
the heat equation with b
E 0 i s i n c l u d e d i n o u r c o n s i d e r a t i o n s . According t o t h e
convergence assumption on b of F
2
E
n+ 1
E'
condition (1.4) i s s a t i s f i e d f o r F
and b E , r e s p . , and w i t h v l = v 2 = 3 / 2 . For p a r t i t i o n s A
mains unchanged where, f o r s i m p l i c i t y , we assume t h a t p
n'
and d l i n s t e a d
condition ( 1 . 1 1 )
re-
and p I a r e a l s o independ-
e n t of n. I n s t e a d of c o n d i t i o n (1.12) we s h a l l now r e q u i r e (2.21)
c'h,
2 p I m ~ ( - l n ~, ) n=O
,...,N ,
f o r some m > N + m ' , w h e r e m' = m a x ( r , s ) , c ' = m i n ( c o , c l ) , because t h e d e g r e e s of t h e polynomials c o n t a i n e d i n $n and JI:
a r e l e s s o r e q u a l n + r and n + s , r e s p . .
c o n d i t i o n i s f u l f i l l e d even f o r k = O ( E v ) and u i n O C V < I p r o v i d e d t h a t compared w i t h h I -v = O(E (- In€))
.
I n f a c t k = 0 ( E v ) i m p l i e s t h a t m=O(E-')
+
This i s small
E
and p m E ( - l n E ) 1
=
0 ( ~ ' 0 ) .
t h e CNG-method i s e q u i v a l e n t t o ( c f . ( 2 . 8 ) ) (2.23)
n+ 1 n+ I DE(uA,€,v)= 3 (v)
,
VE
M
0'
n=O,..., N - I . n+ I
The r i g h t - h a n d s i d e of e q u a t i o n ( 2 . 1 2 ) o b v i o u s l y s a t i s f i e s ( f E 1
,v)
=U
n+ 1
(v),
vCH (0,l). The r e s i d u a l f u n c t i o n a l a s s o c i a t e d w i t h ( 2 . 2 3 ) w i l l be e s t i m a t e d i n t h e f o l l o w i n g lemma. Note t h a t t h e norm of t h e r e s i d u a l f u n c t i o n a l i s d e f i n e d w i t h r e s p e c t t o
II.lI D'
220
H.-J.
REINHARDT
LEMMA 2.2. The residual functional associated with (2.23) i s given by
and s a t i s f i e s
where _n+l
(2.26)
r
j
--
n+l
(atuA,E
+
2 ntl/2 bEuA,€
-
wn+l/2 ) \ I j , j = l , ..., J , n=O E
,...,N - I .
PROOF: For v e H o1 ( I . ) one h a s BE , j ( u i , E , v ) = ( b Z u A , E , v ) and hence J -n+1 k 2 n n+ 112 pE (v) = - y ( b E ~ A , E ,+ ~ ()~ ; , ~ , v + ) k(WE ,v)
- [(un+l,v) A,€
t
k(b2untl,v)] 2 E A,E
=
- k (i-?+',v), J
j=l,..
.,J.
The i n e q u a l i t y of P o i n c a r g - F r i e d r i c h s y i e l d s
A p p l i c a t i o n of Schwarz' i n e q u a l i t y f i n a l l y proves
Let us c a l l
the
n+ 1 ( 2 . 2 5 ) t h e error indicator 71 for l e v e l j o u r e s t i m a t e s i n S e c t i o n 1.5, i t remains t o show t h a t t h e
right-hand
side
t According t o n+l' f u n c t i o n a l s d e f i n e d by ( 2 . 2 2 )
(2.9),
of
a r e s u f f i c i e n t l y c l o s e t o g e t h e r . By t h e d e f i n i t i o n s
( 2 . 2 2 ) , a n d Schwarz' and Hijlder's i n e q u a l i t i e s t h e f o l l o w i n g e s t i m a t e s can
e a s i l y be d e r i v e d ,
The i n i t i a l approximation uo
A,E
of
(2.28)
i s not y e t d e f i n e d . We choose uo
B E ( ~ i , E . =~ )( g , v )
,
as t h e s o l u t i o n
vEMo.
T h i s i s an e l l i p t i c problem o f t h e form c o n s i d e r e d i n [14].
As g ( 0 ) = g(1) = 0,
no boundary l a y e r s occur. Theorem 1 . 2 combined w i t h Lemma 2 . 2 and (2.27) now y i e l d s t h e f o l l o w i n g theorem -level a r e essenn+ I n+ 1 t i a l l y bounded from above and from below by t h e e r r o r i n d i c a t o r s 71 provided
s t a t i n g t h a t t h e l o c a l energy-norms of t h e e r r o r e n + ' on t h e t t h a t en note
A,€
j
i s s u f f i c i e n t l y s m a l l on t h e p r e c e d i n g t - l e v e l . For b r e v i t y l e t us
de-
A-POSTERIORI
and
229
ERROR ESTIMATES
8:
3
,
Ilvn - u i , E l l D , I
=
j = l , . . , J , n=O ,.., N .
j
THEOREM 2 . 2 . L e t us adopt the r e g u l a r i t y assumptions of S e c t i o n 2 . 1 and geH'(0, I ) .
Then there e x i s t s a constant C > O independent of
t
such t h a t t h e foZZowing Lower
a-posteriori error e s t i m a t e s hold, (2.30) Fj:"$ J
(1 + (1
j = l , ...,.I, n=O,
k 2 - 1 k 2-2 - 1 / 2 +-4.) --TI E.) 2
2 J
...,N - I ,
J
n+l 3/4+&n+l n ( l + C h . ) ( l + C < 1 / 2 +) C ( 2 +e.), I I e A , ~ D,I j~
11
j
J
6n+1 s a t i s f y ( 2 . 2 9 ) and j' j
where t h e numbers 8n 6
~~=,(e~)2.
1 where t h e
K
+
n+l
(sJ
+
e nJ ) ,
OsncN-I,
. are defined i n Theorem I . 2 .
1
PROOF: I n S e c t i o n 1.5 one h a s t o s u b s t i t u t e un
where vn i s t h e s o l u t i o n of A,€ DE(v;,€,v)
=
0:(v)
I I u ~ , -~
The numbers A n a r e d e f i n e d by 6; = j According t o ( 1 . 3 1 ) and ( 2 . 2 7 ) , one h a s
The numbers X
j
and vn
A,€
, vn A,E
A,E
V E
f o r ii
A,€
and u ~ , ~r e ,s p . ,
Mo.
1 1 D,I,
,
j=l,
...,J,
n=l,
...,N
3
i n S e c t i o n 1 . 5 have t o be r e p l a c e d by A:
1 0:
- 5:llHi(I.),
which, O J J. Thus Theorem 1 . 2 immediately
...,
J
=
n n-I , j=l, 6 8 j j proves t h e a s s e r t i o n s . Note t h a t t h e K remain unchanged because t h e a s y m p t o t i c j n+ 1 n+ 1 a r e w r i t t e n i n term o f E and n o t i n terms of ? representationsof 9 and q E according t o ( 2 . 2 7 ) ,
satisfy X
(cf. (2.19)).0
2.4.
COMPUTATIONAL RESULTS
In t h i s f i n a l section
we
present
f i n i t e element computations f o r an example
which i s e s p e c i a l l y i n t e r e s t i n g because t h e r e does n o t a p p e a r any boundary l a y e r a t x = O b u t o n l y a t x = 1 . The e r r o r i n d i c a t o r s n o t i c e t h i s very w e l l and c o r r e spondingly t h e a u t o m a t i c mesh r e f i n e m e n t i s e x e c u t e d o n l y towards x = I . Hence o u r e r r o r i n d i c a t o r s d e t e c t t h e boundary l a y e r s . The s t r a t e g y of t h e mesh r e f i n e m e n t
on e v e r y d i s c r e t e t - l e v e l i s t h e same a s t h e one i n [14,
S e c t . 51. Those s u b i n t e r -
v a l s a r e always h a l v e d where t h e c o r r e s p o n d i n g e r r o r i n d i c a t o r s a r e of t h e same magnitude a s t h e maximal one. A s i n [I41 t h e e r r o r w i t h r e s p e c t t o t h e energy-norm
.
230
H -J.
REINHARDT
tends t o a minimum i f t h e e r r o r i n d i c a t o r s become approximately e q u a l f o r a l l subi n t e r v a l s . According t o Theorem 2 . 2 ,
i n t h e l a t t e r c a s e t h e mesh i s a s y m p t o t i c a l l y
e q u i d i s t r i b u t e d with r e s p e c t t o t h e e r r o r i n d i c a t o r s a s w e l l a s w i t h r e s p e c t t o t h e l o c a l energy-norms on every t - l e v e l
( c f . Lentini-Pereyra
[I I],
Russell-
C h r i s t i a n s e n [I61 f o r t h e d e f i n i t i o n o f “ a s y m p t o t i c a l l y e q u i d i s t r i b u t e d ” ) . Let u s emphazise t h a t a d a p t i v e mesh refinement towards t h e boundary l a y e r s c a u s e s a cons i d e r a b l e r e d u c t i o n of t h e e r r o r even when t h e mesh p o i n t s l i e o u t s i d e t h e bounda r y l a y e r . T h i s demonstrates t h a t o u r method e f f i c i e n t l y s o l v e s t h e s i n g u l a r pert u r b a t i o n problem and n o t only t h e reduced e q u a t i o n . Let us c o n s i d e r t h e d i f f e r e n t i a l e q u a t i o n (2.32)
ut
= E
u xx - u + w E .
(x,t)
E ( 0 , I)x(O,
11,
w i t h homogeneous boundary c o n d i t i o n s and an i n i t i a l f u n c t i o n g neous f u n c t i o n w
i s chosen such t h a t t h e s o l u t i o n u
u E ( x , t ) = t { x + (exp(-(l + x ) / E )
-
SO.
The imhomoge-
of (2.32) h a s t h e form
exp(-(l - x ) / E ) ) ( l
-
exp(-2/c))-I}.
Obviously, t h e r e does n o t a p p e a r a boundary l a y e r a t x = O , and t h e boundary l a y e r
a t x = l shows a v e r t i c a l i n c r e a s e f o r i n c r e a s i n g t . For t h e s o l u t i o n of t h e p i e c e w i s e l i n e a r CNG-method, t h e i n t e g r a l s i n t h e s t i f f n e s s m a t r i x and i n t h e right-hand s i d e s a r e e v a l u a t e d approximately by means o f Simpson‘s q u a d r a t u r e . The e r r o r w i t h r e s p e c t t o t h e L‘-
and energy-norm i s com-
puted by a h i g h l y a c c u r a t e Gaussian q u a d r a t u r e formula. A f t e r some c a l c u l a t i o n s t h e e q u a t i o n s f o r v? = uo ( v . b 2( c . ) J
(2.33)
E
J
-
1 2 2 . w.)v0 J J j-1 + ( w j b2e ( c j t l )
+ (v.b2(c.) + (v. + u . ) b2 E(xj) J E J J J =
( x . ) assume t h e form
J
+
3
-
1 2 C j t l wj)vqtl
w j b2E ( c j + ] )
+
2 4 w2 . ) 0~ . J J
Z v . g ( c . ) + ( v . + w . ) g ( x . ) + 2 u j g ( c j t l ) , j = l , ...,.J-I , J J J J J
where
v
= 1 + hj/hjtI, w = 1 + hj+l/hj, 2 w = b w j j j j j ’ For n = O t h e e r r o r i n d i c a t o r s a r e computed by
(2.34)
2 0
c
j
=
(xj-l + x . ) / 2 .
where to = g - b E u A I E .Note t h a t t h e d i s t i n c t i o n t o [ 1 4 , ( 5 . 4 ) , ( 5 . 5 ) ]
j t h e e v a l u a t i o n of t h e i n t e g r a l s by Simpson’s q u a d r a t u r e .
s t e m from
n+l ntl = u (x.) can be w r i t t e n a s f o l l o w s , j A,€ J n+ 1 n+ I ( v . B + ( c . ) - 1 2 tJ. k wJ . )j-1 ~ + ( wJ. B + ( c j t l ) - 12 b j + l k w j ) v j t l J J + ( v . { B + ( c . ) + B + ( x . ) } + w . { B t ( c j t l ) + B + ( x . ) } + 2 4 k w2. ) v n+l J J J J J ~j = ( u . B ( c . ) + 1 2 C . k ~)vn + (ujB-(cjtl) + 1 2 2 j + l k w . ) ~ n + J - J J j j-1 J JtI
For n > O t h e e q u a t i o n s f o r v
(2.36)
J
A-POSTERIORI
where, a d d i t i o n a l l y t o (2.34), B+(x) = 2 t k b ' ( x ) . t
-
a r e computed by
231
ERROR E S T I M A T E S
The e r r o r i n d i c a t o r s f o r l e v e l
fiy = k [ l + k2 ( Bj 2 + ~ 2J E ? ) ] - 1 / 2 T If."(, ( J j-1 ) I 2 + l f ~ ( x j ) 1 2 ) 1 f 2n=I, ,
(2.37)
...,N ,
where t h e r e s i d u a l f n i s d e f i n e d by (2.26).
j
The computations a r e performed on e v e r y d i s c r e t e t - l e v e l a c c o r d i n g t o t h e s t r a t e g y d e s c r i b e d above. The a d a p t i v e mesh r e f i n e m e n t i s r e s t r i c t e d t o a maximal number of 30 s u b i n t e r v a l s . I n Table I ,
f o r every t = O . 1 n
e r r o r a r e p r e s e n t e d which a r e
,
n = l , ... , l o , v a r i o u s norms of t h e
o b t a i n e d a t t h e f i n a l mesh d i s t r i b u t i o n (of l e s s o r
e q u a l 30 s u b i n t e r v a l s ) . The mesh d i s t r i b u t i o n i n t h e l a s t column of Table 1 i s des c r i b e d a s f o l l o w s . S t a r t i n g w i t h h = 0 . 1 , t h e number b e f o r e t h e o b l i q u e s t r o k e
haZving number x of t h e c o r r e s p o n d i n g s u b i n t e r v a l d e f i n e d by indicates the j The one f o l l o w i n g t h e o b l i q u e s t r o k e g i v e s t h e number of s u b i n t e r v a l s h . = h/2'j. J with t h i s associated h a l v i n g number. For example 0 / 4 , l f l , 2 / 3 , 3 / 2 means t h a t t h e r e a r e 10 s u b i n t e r v a l s i n a sequence from l e f t t o r i g h t with t h e f o l l o w i n g widths
I.-I,
I.-I,
l.-l,
l.-l,
5.-2, 2.5-2, 2.5-2, 2.5-2,
1.25-2, 1.25-2
.
The t o t a l number of s u b i n t e r v a l s i s t h e r e f o r e g i v e n by t h e sum of t h e numbers following
the oblique strokes.
II'
t
IleA,~ D
0.0 0.1
0.2 0.3 0.4
0.5 0.6 0.7 0.
a
0.9 I
.o
n I
IeA,s
n I
0,2 leA,c 0,m
0.00000
0.00000
0.000
1.36699-4 I .94aa6-4 2.7 1908-4 3.56520-4 4.41857-4 5.29460-4 6.16732-4 7.05091-4 a. 2642 1-4 9. I 1585-4
5.31839-5 9.23475-5 1.34963-4 1.a3306-4 2.29007-4 2.77720-4 3.24828-4 3.73627-4 4.97725-4 5.19505-4
5.447-4 8.326-4 7.450-4 1.580-3 1.924-3 2.272-3 2.604-3 2.937-3 3.232-3 3.610-3 Table I
For d e m o n s t r a t i o n p u r p o s e s , a t l e v e l t = 1.0 t h e second t a b l e i n d i c a t e s how t h e a d a p t i v e mesh s e l e c t i o n r e d u c e s t h e e r r o r s t e p by s t e p . The f i n a l row i n Table 2 shows t h e e r r o r and t h e a s s o c i a t e d mesh d i s t r i b u t i o n a l r e a d y p r e s e n t e d i n Table I . i n d i c a t e s where t h e maximal e r r o r i n d i c a t o r i s l o c a t e d . The jmax = fi" i t s e l f is given i n the t h i r d c o l u m . maximal e r r o r i n d i c a t o r maX jmax
A t each s t e p ,
file
232
H . 4 .
ii 10
10
jmax
IleA,JD
10 12
2.340-2 1.586-2
13 14 15 14 14
1.018-2
1.41087-1 7.49768-2 3.05361-2 1.04045-2 4.75615-3 3.84864-3 2.46533-3 9.11585-4
lo
leA,E
REINHARDT
I 0,2
mesh d i s t r i b u t i o n
J
I
1
1
2 3 4 5 6 7
18
6.507-3 3.901-3 2.005-3 1.066-3 5.227-4
1.37056-1 7.21868-2 2.84840-2 8.71083-3 3.48466-3 2.94337-3 1.99033-3 5.19505-4
10
12 13 14 I5 16 20 30
0110 018, 018, 018, 018, 018, 018, 018,
14 13,212 13,2/1,3/2 /3,2/1,311,412 /3,211,311,4/1,5/2 13,211,4/2,5/2,6/4 12,2/2,3/2,5/4,6/4,7/8
For comparison we have computed t h e f i n i t e element s o l u t i o n w i t h e q u i d i s t a n t s t e p widths h = 1/30 on e v e r y t - l e v e l w i t h o u t mesh a d a p t a t i o n . A t t = 1.0, t h e e r r o r with r e s p e c t t o t h e energy-norm t u r n e d o u t t o be t i v e computation we achieved
9.11585-4
4.14661-2
whereas by an adap-
( c f . Table I and 2 ) .
One observes t h a t with t h e r e s t r i c t i o n o f l e s s o r e q u a l 30 s u b i n t e r v a l s t h e meshes a r e n o t y e t a s y m p t o t i c a l l y e q u i d i s t r i b u t e d . This can be b e t t e r achieved and t h e e r r o r can be f u r t h e r reduced i f one a l l o w s more s u b i n t e r v a l s . However, t h e smalle s t s t e p w i d t h i s then c o n s i d e r a b l y s m a l l e r than t h e t h i c k n e s s of t h e boundary l a y e r something which has t o be regarded a s a t y p i c a l f o r t h e numerical s o l u t i o n of s i n g u l a r p e r t u r b a t i o n problems. F u r t h e r computations of o t h e r examples, a l s o of t h e h e a t e q u a t i o n , and w i t h s m a l l e r and l a r g e r
E
and k show t h a t t h e mesh a d a p t a t i o n a c c o r d i n g t o o u r e r r o r
i n d i c a t o r s p r o v i d e s a d e s i r a b l e d e s i g n of t h e mesh. A l l computations were p e r formed on a DEC-Computer PDP 1 1 / 0 3 . REFERENCES
[I]
Babugka, I . , R h e i n b o l d t , W.C., E r r o r e s t i m a t e s f o r a d a p t i v e f i n i t e element computations, SIAM J . Numer. Anal. 15 ( 1 9 7 8 ) , 7 3 6 - 7 5 4 .
[ 2 ] Babuska, I . , R h e i n b o l d t , W.C., A - p o s t e r i o r i e r r o r e s t i m a t e s f o r t h e f i n i t e element method, I n t e r n a t . J. Numer. Methods Eng. 12 (1978), 1597-1615. [ 3 ] Babugka, I . , R h e i n b o l d t , W . C . , A n a l y s i s of o p t i m a l f i n i t e element meshes i n R1, Math. Camp. 3 3 ( 1 9 7 9 ) , 4 3 5 - 4 6 3 . [ 4 ] Babugka, I . , R h e i n b o l d t , W . C . , R e l i a b l e e r r o r e s t i m a t i o n and mesh a d a p t a t i o n f o r t h e f i n i t e element method, Techn. Note BN-910, I n s t . f o r P h y s i c a l S c i e n c e & Technology, Univ. Maryland, C o l l e g e Park ( A p r i l 1 9 7 9 ) . [ 5 ] Bobisud, L . E . , P a r a b o l i c e q u a t i o n s w i t h a s m a l l parameter and d i s c o n t i n u o u s d a t a , J. Math. Anal. Appl. 26 ( 1 9 6 9 ) , 208-220. [6] Brandt, A . , M u l t i - l e v e l a d a p t i v e t e c h n i q u e s f o r s i n g u l a r - p e r t u r b a t i o n problems, i n : Hemker, P.W. and M i l l e r , J . J . H . ( e d s . ) , Numerical a n a l y s i s of s i n g u l a r p e r t u r b a t i o n problems ( P r o c . Conf. Nijmegen, 1 9 7 8 ) , pp. 53-142 (Academic P r e s s , London-New York-San F r a n c i s c o , 1 9 7 9 ) . [ 7 ] C i a r l e t , P . G . , The f i n i t e element method f o r e l l i p t i c problems (North-Holland, Amsterdam-New York-Oxford, 1 9 7 8 ) .
A-POSTERIORI
ERROR ESTIMATES
233
[8] F a i r w e a t h e r , G . , F i n i t e e l e m e n t G a l e r k i n methods f o r d i f f e r e n t i a l e q u a t i o n s , L e c t u r e N o t e s i n P u r e a n d A p p l . M a t h . , V o l . 34 ( D e k k e r , New Y o r k - B a s e l , 1 9 7 8 ) . [g] KaEur, J . , Wawruch, A . , On t h e a p p r o x i m a t e s o l u t i o n f o r q u a s i l i n e a r p a r a b o l i c e q u a t i o n s , C z e c h o s l o v a k Math. J . 27 ( 1 0 2 ) ( 1 9 7 7 ) , 220-241.
[lo]
K a r t s a t o s , A . G . , Z i g l e r , W . R . , K o t h e ’ s method a n d weak s o l u t i o n s o f p e r t u r b e d e v o l u t i o n e q u a t i o n s i n r e f l e x i v e Banach s p a c e s , Math. Ann. 219 ( 1 9 7 6 ) , 1 5 6 - 1 6 6 .
[I I]
L e n t i n i , M . , P e r e y r a , V . , An a d a p t i v e f i n i t e d i f f e r e n c e s o l v e r f o r n o n l i n e a r two-point boundary problems w i t h m i l d boundary l a y e r s , SIAM J. Numer. A n a l . I4 ( 1 9 7 7 ) , 91-111.
[I21 P e a r s o n , C . E . , On a d i f f e r e n t i a l e q u a t i o n o f t h e b o u n d a r y l a y e r t y p e , J . Math. and P h y s . 4 7 ( 1 9 6 8 ) , 134-154.
[I31 P o l k , J . F . ,
Asymptotic approximations t o t h e s o l u t i o n of t h e h e a t e q u a t i o n , Rocky Mountain J . Math. 6 ( 1 9 7 6 ) , 697-708.
[I41 K e i n h a r d t , H . - J . , A-posteriori e r r o r e s t i m a t e s f o r t h e f i n i t e element solut i o n of a s i n g u l a r l y perturbed l i n e a r ordinary d i f f e r e n t i a l equation, s u b m i t t e d t o SIAM J . Numer. A n a l .
[I51 R e i n h a r d t , H . - J . , On t h e f i n i t e e l e m e n t s o l u t i o n o f a o n e - d i m e n s i o n a l s i n g u l a r p e r t u r b a t i o n p r o b l e m , t o a p p e a r i n Z . Angew. Math. Mech. 61 ( 1 9 8 1 ) . [I61 R u s s e l l , K . D . , C h r i s t i a n s e n , J . , A d a p t i v e mesh s e l e c t i o n s t r a t e g i e s f o r s o l v i n g b o u n d a r y v a l u e p r o b l e m s , SIAM J . Numer. A n a l . 15 ( 1 9 7 8 ) , 59-80. [I71 Thomke, V . , W a h l b i n , L . , On G a l e r k i n methods i f l s e m i l i n e a r p a r a b o l i c p r o b l e m s , SIAM J. Numer. A n a l . 12 ( 1 9 7 5 ) , 378-389. [I81 W h e e l e r , M.F., A p r i o r i L 2 e r r o r e s t i m a t e s f o r G a l e r k i n a p p r o x i m a t i o n s t o p a r a b o l i c e q u a t i o n s . SIAM J. Numer. A n a l . 10 ( 1 9 7 3 ) , 723-759.
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ANALYTICAL AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS I N ANALYSI S S . A x e l s s o n , L . S . Frank, A . v a n tier S l u i s ( e d s . ) @ N o r t h - H o l l a n d P u b l i s h i n g C o m i j u u , 1981
DIFFRACTION OF WAVES BY CONES AND POLYHEDRA Michael E . T a y l o r Rice University Houston, Texas, U.S.A.
This l e c t u r e i s a r e p o r t on j o i n t work w i t h J e f f Cheeger; d e t a i l s a r e given i n 131. A cone w i t h v e r t e x a t t h e o r i g i n i n lRm+', o r i t s complementarv cone, i s d e s c r i b a b l e as t h e cone over a subdomain R of t h e u n i t s p h e r e Sm i n n m + ' , More g e n e r a l l y , i f N i s any compact Riemannian rnaqifold of dimension m , p o s s i b l v w i t h houndary, t h e cone over N , C(N), i s t h e space IR' x N t o g e t h e r w i t h the'Riemannian m e t r i c dr
(1)
2 + r 2g
where g i s t h e m e t r i c t e n s o r on N . Our purpose i s t o understand s o l u t i o n s t o t h e wave e q u a t i o n ( - - a2
(2)
at2
A )u = 0
where A i s t h e Laplace o p e r a t o r , a n e g a t i v e s e l f a d j o i n t o p e r a t o r , on C(N). I f aN # @ , we impose D i r i c h l e t boundarv c o n d i t i o n s on X ( N ) , though manv o t h e r boundarv c o n d i t i o n s could be e q u a l l v e a s i l y t r e a t e d . One g o a l i s t o understand t h e l o c a t i o n of s i n g u l a r i t i e s of s o l u t i o n s t o ( 2 ) , p a r t i c u l a r l v t h e fundamental s o l u t i o n
s i n t(-A)'
(3)
1
7
and t o g i v e a d e s c r i p t i o n o f t h e " d i f f r a c t e d wave" produced from t h e v e r t e x r = 0,which belongs t o t h e m e t r i c completion of C(N). We do t h i s when N i s a compact manifold w i t h smooth boundarv, and i n some c a s e s when t h e boundarv of N has c o r n e r s , i n enough g e n e r a l i t v t o d e s c r i b e t h e d i f f r a c t i o n of waves i n R n bv p o l y h e d r a , The i n i t i a l s t e p i n s o l v i n g (2) i s t o use t h e method o f s e p a r a t i o n o f v a r i a b l e s , w r i t i n g A on C(N) i n t h e form
where
'
A = - + ! ! ! -a'+ - A
a i 2 r ar r 2 ar i s t h e Laplace o p e r a t o r on t h e base N .
t h e e i g e n f u n c t i o n s and e i g e n v a l u e s of where a = (l-m)/2.
If 235
-
A,
L-l
Let
and s e t
y
I-(j
, q j ( x ) denote
3
= (p.
J
+ z 2 )% ,
236
M.E. TAYLOR
w i t h g . ( r > w e l l behaved, and i f we d e f i n e t h e second J Lp bv 3' m a M L p g ( r > = ( 7+ - - + ;3 ) g ( r > (5 1
order operator
ar
ar
one s e e s t h a t
T h i s s u g g e s t s making u s e of t h e Hankel t r a n s f o r m , d e f i n e d f o r v EIR' b y (9)
J;m J"(Ar)
H y ( g ) ( ~ )=
rdr.
The Hankel i n v e r s i o n formula ( s e e [ 6 1 ) s a y s (10)
g ( r ) = HJH"(gI)(r)*
Moreover, t h e r e i s a P l a n c h e r e l formula I:lg(r)12
r dr
I ; I H V ( g ) ( A ) l 2 AdA.
=
I n view o f ( 5 ) and ( 9 ) , (11)
H V ( r- a L,,g)
= =
JiL,,
- (11)
g r" dr
(r'J,(Ar))
- A 2 JZg r " J v ( A r ) rm d r
= -1'
Now from ( 6 )
we have
H~ ( r - ' g ) .
i t f o l l o w s t h a t t h e map
"pc g i v e n bv
-a
H, ( r g,), . . . > 2 1 p r o v i d e s a n i s o m e t r v of L (C(N)) w i t h L2 (R + , 3 d A , A 2 ), s u c h t h a t A i s c a r r i e d i n t o m u l t i p l i c a t i o n bv - A2 Thus (12) p r o v i d e s a s p e c t r a l r e p r e s e n t a t i o n o f A. C o n s e q u e n t l y , f o r well-behaved f u n c t i o n s f , w e have
(12)
g = (HV (r-'g0), 0
.
(13) f ( - A ) g ( r , x )
=
r'
C ~ ~ f ( X Z ) J w . ( X r ) X ~ ~ s l -(As) ' J " g (s)ds
j j Now w e c a n i n t e r p r e t (13) i n t h e f o l l o w i n g f a s h i o n . o p e r a t o r v on N bv
(14)
-
J
j
2 112
~ = ( - A + p l )
.
dAw.(x). J
Define t h e
237
D I F F R A C T I O N OF WAVES BY CONES A N D POLYHEDRA
Thus vyj = \,)j y j . I d e n t i f v i n g o p e r a t o r s w i t h t h e i r d i s t r i b u t i o n a l k e r n e l s , we can d e s c r i b e t h e k e r n e l of f ( - b ) a s a f u n c t i o n on @xIR* t a k i n g v a l u e s i n o p e r a t o r s on N , bv t h e formula 0
(15)
f ( - * ) = (rlr2)'
f('h2)J ( a r l ) J v
'?
( k 2 ) Jd2
= K(r1,r2,;),
s i n c e t h e volume element on C(N) i s r m d r dx. This a n a l v s i s a p p l i e s t o f ( 2 2 ) = e 2-1 s i n 2 t f o r any E > o , and one g e t s f o r t h e fundamental s o l u t i o n t o t h e wave e q u a t i o n ( 2 ) , 0
3-
l i m ( r 1 r 2 ) d Zm (Qe-(E+it)' J\,o r l ) Jv(2r2)Ad 3 f. d o 2 2 = - ( 1 / v ) ( r 1 r 2 ) a - k l< i\ LmO I m Q v- L I ((rl+r2+(z+it)2)/~) where t h e l a s t i d e n t i t v i s t h e Lipschitz-Hankel i n t e -
(16) ( - \ ) - + s i n t ( - A ) l = with S = 2rlr2,
-
g r a l f o r t h e Legendre f u n c t i o n Qv-15; s e e Lebedev 161. i n t e g r a l formulas f o r Q,;-$, in particular
Qv-k
(cosh) ) =
\
m
'I
Using o t h e r
z
( 2 cosh s
-
2 coshv) )-+e-'g
ds
where t h e p a t h of i n t e g r a t i o n i s a s u i t a b l e p a t h from
'7
to
+ w
i n the
complex p l a n e , one o b t a i n s t h e f o l l o w i n g a l t e r n a t e i n t e g r a l r e p r e s e n The k e r n e l i s e q u a l t o
t a t i o n of ( - a ) - $ s i n t ( - A ) % .
if
a0
(19) where
(l/n)(rlr2)'
p1
cos v v
(
h
lrl-r2/4 t
< r1+r 2
[rl+r2+2rlr2cosh 2 2
s -t2J-'% e-'"
ds
if t>rl+r2 = c o s - l ( ( r l2+ r 22- t 2 ) / 2 r l r 2 ) and p2 = c o s h - ' ( ( t 2 - r 2l - r 22) / 2 r l r 2 ) .
Formulas ( 1 7 ) - ( 1 9 ) w i l l provide t h e b a s i s f o r t h e a n a l v s i s of t h e fundamental s o l u t i o n t o ( 2 ) . s k e t c h e d above.
They were d e r i v e d i n
121,
by t h e method
It i s a p p a r e n t t h a t an u n d e r s t a n d i n g of t h e s o l u t i o n o p e r a t o r c o s vs t o t h e wave e q u a t i o n on t h e base N f o r s 6 [O,vl, i s a c r u c i a l i n g r e d i e n t i n u n d e r s t a n d i n g ( 1 8 ) , ( 1 9 ) . One a l s o needs t o understand e - S V , t h e P o i s s o n semigroup on N , b u t t h i s i s a s i m p l e r problem. In from a d e s c r i p t i o n of t h e s i n g u l a r i t i e s o f c o s s l 1 6 x2 one c a n d e s c r i b e t h e s i n g u l a r s u p p o r t of and of e-"bx ( - A > - k si n t )%6 !r x > ' a ( g e n e r a l i z e d ) f u n c t i o n o f ( r l , x l ) , a s f o l l o w s . We c o n s i d i i ?he f o l l o w i n g t h r e e r e g i o n s ,
particular,
(-3
238
M.E.
TAYLOR
Region I: 0 < t < d i s t ( ( r l , x l ) , ( r x ) ) 2’ 2 Region 11: d i s t ( ( r x ), ( r 2 , x 2 ) ) < t 4 rl+r2 1’ 1 Region 111: t 7 r1+r2. In r e g i o n 1 o u r fundamental s o l u t i o n v a n i s h e s , s o i t has no s i n g u l a r i t i e s t h e r e , I n r e g i o n 11, t h e i n f l u e n c e o f t h e v e r t e x h a s n o t v e t been f e l t , s o p r o p a g a t i o n o f s i n g u l a r i t i e s r e s u l t s €or manif o l d s w i t h smooth boundarv ( s e e [8]) a p p l y , v i a a s i m p l e f i n i t e p r o p a g a t i o n speed argument. We c a n a l s o r e a d o f f t h e s i n g u l a r s u p p o r t from ( 1 8 ) . Indeed, c o n s i d e r t h e o p e r a t o r on N g i v e n by (20)
\:‘(2
where 0 4 3’ 1 n e a r
[-pi,
pll,
(21)
-
cos s
2
COS
p,)-k
c o s bs d s
p,
n . L e t y ( s ) be a smooth f u n c t i o n on [O,f,] which i s and v a n i s h e s n e a r 0 . Extend y a s an even f u n c t i o n on and write (20) a s
/,
\ Py ( s ) ( 2 c o s 0
s
-
2 cos p,)-&os
+ 4 (‘i(l-p(s))(2cos
’Js’ds s
-
2 cosp,)-’
cos vs d s .
I n t e g r a t i o n by p a r t s shows t h a t t h e second term i s a smoothing o p e r a t o r . I f we make t h e change o f v a r i a b l e s = /,-u, t h e f i r s t i n t e g r a l i n (21) becomes -PI
where g ( u , PI) i s smooth and v a n i s h e s f o r u i s c l e a r l v o f t h e form (23)
f (v)cos 11
PI” + g
PI
(v)sin
‘pi.
The i n t e g r a l ( 2 2 )
PI
(24) i n t h e s e n s e t h a t t h e remainder term a f t e r t h e sum o f the f i r s t k terms maps 9 = Is ( v s ) t o OB s+k+3’2 f o r e a c h s 6 .’R I n d e e d , i f one defines 8 t o be t h e d u a l o f 16 f o r s 4 0, t h i s mapping p r o p e r t y h o l d s f o r a l l r e a l s . ( I f a N = $ , a s = HS(N); i f 2 N i s smooth, d s C Hs(N) f o r ~ 3 0 . ) The s i n g u l a r i t i e s o f c o s P,w and o f s i n p,v a r e r e a d o f f from t h e mown b e h a v i o r o f t h e b a s e N (assuming 2N i s smooth). S i n c e f ( v ) , g ( v ) and c o s v , s i n p v depend smoothly on t h e p a r a m e t e r ~ e ( 0 , ~P ) i, t fo!lows from ( 4 ) t h a t , i n r e g i o n I1 ( i n d e e d , on t h e com-
-‘
plement of t h e c l o s u r e o f r e g i o n 1 1 1 ) , t h e s i n g u l a r s u p p o r t o f l i e s w i t h i n t h e p r o j e c t i o n on C(N) o f t h e ( - b ) - + s i n t(-A)’ 6 >k t h e t i m e - t g e o d e s i c flow on T”(C(N)), r e f l e c image o f T ( r p 2 )
DIFFRACTION OF WAVES BY CONES AND POLYHEDRA
239
t e d from C(N) by t h e laws o f g e o m e t r i c a l o p t i c s ( s e e [ 8 J ) , a t l e a s t i f one makes t h e f o l l o w i n g assumption on t h e geometrv o f N n e a r i t s boundary: I f Y : ( - C , O l 4 N i s a g e o d e s i c i n N t a n g e n t t o 3 N t o i n f i n i t e o r d e r a t Y ( O ) t b N , t h e r e i s an e x t e n s i o n t o T : ( - E , Z ) , a Yk by?,, which w e w i l l g e o d e s i c i n N . Denote t h i s image o f T (rpx2) c a l l the primary The wave f r o n t s e t c o u l d be r e a d o f f from a
wave.
c l o s e r e x a m i n a t i o n o f (18) and ( 2 0 ) - ( 2 4 ) , b u t we h r i e f l v p o s t p o n e d i s c u s s i n g t h e wave f r o n t s e t . Now w e c o n s i d e r t h e b e h a v i o r o f (-a)-’sin t(-i)li 6 in ( 5 ,x.2 r e g i o n 111. From ( 1 9 ) and t h e f a c t t h a t e-” i s a smooth f a m i l y o f smoothing o p e r a t o r s , f o r s 3 6 7 0 , i t f o l l o w s t h a t (-*)-‘sin t(-A)’, i s smooth i n r e g i o n 111. I f r e g i o n 111 i s n o t applied t o b (r2 ,x2) ’ empty, t h e v e r t e x p EC‘“(N) ( t h e m e t r i c c l o s u r e o f C(N)) b e l o n g s t o k; t h e c l o s u r e i n C ( N ) o f r e g i o n 111, and i t i s n e c e s s a r v t o u n d e r s t a n d near the vertex p. t h e b e h a v i o r o f (-A)-’sin t(-A)’ D e f i n i t i o n . L e t b S be t h e domainbLg2&)operator on N f o r s 3 0 , 16’ t h e d u a l o f f o r s C 0 . L e t f a d s . We s a y f b e l o n g s t o ?r 8 Foc(U) f o r some neighborhood U o f t h e v e r t e x p E C (N) i f t h e r e e x i s t s u t d k s u c h t h a t u = f on U.
1
o-’
Theorem 1. I f t > r 2 , t h e n v ( t ) = (-A)-’sin t ( - A ) + 2 ( r x ) be l o n g s k. t o 8 toc(U) f o r some neighborhood U o f t h e v e r t e x p , f8; S k e t c h o f p r o o f . It i s n o t hard t o show t h a t i t s u f f i c e s t o o b t a i n --
ill
(25)
4
v(t)
L2(U)
f o r a l l j.
T h i s f o l l o w s f a i r l v d i r e c t l y from ( 1 9 ) , which g i v e s
provided
Rewrite this as
where F (B2, v) is a bounded family of operators on each Dd (N), for
240
M.E. TAYLOR
I n p a r t i c u l a r , f o r r 2 , t f i x e d , r 1< t - r 2 ’
[ Di
p\+
v(t)l 4 C
so
which g i v e s ( 2 5 ) . T h i s smoothness i n r e g i o n 111 and n e a r t h e v e r t e x f o r t r + r 1 2’ w h i l e e x p e c t e d , i s n o t a t a l l obvious a p r i o r i , and indeed i s t h e major r e s u l t on t h e l o c a t i o n o f t h e s i n g u l a r i t i e s o f (-A)-l’sin t ( - A ) l applied t o R ( r 2 ,x g ) * The i n t e r s e c i o n of t h e c l o s u r e s o f r e g i o n s I1 and I11 i s j r l = t - r 2 ] . The above a n a l v s i s a l l o w s t h e p o s s i b i l i t y t h a t L ( - b ) - * s i n t ( - d ) a b ( r 2 , X ) i s s i n g u l a r on t h i s s e t . I n d e e d , t h e s i m p l e s t n o n - t r i v i a l exam$le, N = [ 0 , 2 n ] , i n which c a s e C ( N ) i s t h e p l a n e , s l i t a l o n g t h e p o s i t i v e r e a l a x i s , w i t h D i r i c h l e t boundary c o n d i t i o n s on t h e s l i t , e x h i b i t s t h i s phenomenon; s e e S t a k g o l d /9], p . 289. A complete d e r i v a t i o n o f t h e e x p l i c i t s o l u t i o n i n t h i s case i s a l s o g i v e n i n [37, from ( 1 7 ) - ( 1 9 ) . We c a l l t h i s s e t the diffracted L a t e r w e w i l l i n d i c a t e a more p r e c i s e a n a l y s i s o f the discontinuitv of (-b)-&sin t (-A)$ a c r o s s 8 t . The f o l l o w (r ,x ) i n g theorem d e s c r i b e s t h e s i n g u l a r suppogt a f t h i s d i s t r i b u t i o n . ,1 1 , Theorem 2. The s i n g u l a r s u p p o r t o f (-A)-’sin t ( - A ) % i s con( r 2 4 t a i n e d i n t h e u n i o n o f ^P and dt. We draw a f i g u r e o f t h e s e s e t s , when N i s an i n t e r v a l o f l e n g t h l e s s t h a n 2 n , which g i v e s a good p i c t u r e o f t h e g e n e r a l c a s e .
at,
wave.
z
We have denoted
5, t h e i n t e r s e c t i o n o f dt w i t h t h e c l o s u r e o f 7)
Another c h a r a c t e r i z a t i o n i s
31
t’
> t = f(t-r2,x1): xlc where 2 i s t h e s e t o f p o i n t s x l b N s u c h t h a t some g e o d e s i c from x 2 ( p o s s i b l y r e f l e c t i n g o f f o r g l i d i n g a l o n g a N , a c c o r d i n g t o t h e laws of geometrical o p t i c s ) h i t s x a t d i s t a n c e e x a L t l v T T . We remark 1’
24 1
DIFFRACTION OF WAVES BY CONES AND POLYHEDRA
that
c
7,
and fit a r e t a n g e n t a t t h e i r i n t e r s e c t i o n . As f o r t h e wave f r o n t s e t of (-C>-'sin t ( - A ) %
= v(t), (r2, x2) f o r t 7 r 2 , we have Theorem 2. WF(v(t)) i s c o n t a i n e d i n t h e normal bundle t o d t U y t .
d$"rt,
Indeed, i f X E ( x , J ) 6 T"(C(N>), 3 n o t normal t o a t u - P t , and i f x l i e s i n t h e i n t e r i o r of C(N), Hormander's p r o p a g a t i o n o f s i n g u l a r i t i e s theorem i m p l i e s t h a t , i f (x,T)&WF(v ( t ) ), t h e n WF(v(t+s)) must c o n t a i n some p o i n t s which do n o t l i e o v e r & t + z ? t + s , f o r s m a l l s , which i s i m p o s s i b l e . I f x l i e s on t h e boundarv of C(N), t h e same r e s u l t f o l l o w s by r83, t o which we r e f e r f o r a d e f i n i t i o n o f t h e wave f r o n t s e t i n t h i s c o n t e x t . The form of t h e s o l u t i o n o p e r a t o r g i v e n bv ( 1 7 ) - ( 1 9 ) , and t h e form of t h e p a r a m e t r i x t o which i t g i v e s r i s e , i s f a i r l v complicated n e a r 3,, as we w i l l s e e l a t e r , which makes i t d i f f i c u l t t o p e r c e i v e from s u c h p a r a m e t r i x t h a t t h e p a r t of W F ( ( - " ) - k i n t ( - A ) % 1
{kai'ba'
over 3, i s c o n t a i n e d i n t h e s e t o f normals t o dt over > t , c o i n c i d e s w i t h t h e s e t of normals t o pt over Y,), though i t i s f a i r l y
-
e a s y t o read o f f WF(v(t)) o n T t from ( 1 8 ) , and a l s o t o read o f f WF(v(t)) on@;>t, g i v e n an a n a l y s i s of v ( t ) n e a r t h e d i f f r a c t e d wave
Bt,
which we w i l l d e s c r i b e s h o r t l v . W e can s i m i l a r l v analvze t h e s i n g u l a r s u p p o r t and wave f r o n t s e t f o r (-A)-*sin t ( - a ) % , cos t ( - A ) k , and e + i t ( - A)". u, g i v e n any u r s S .
I n a s e n s e , t h e major r e s u l t i s t h e f o l l o w i n g . Theorem 4. I f u 6 a S ( C ( N ) ) , w i t h s u p p o r t i n r 2 < R 2 , t h e n f o r t > R 2 , 1, any o f t h e d i s t r i b u t i o n s (-A)-+sin t ( - A ) ' u, e t c . , l i s t e d above
lot( { r l belongs t o # k
t-R2-h))
for a l l k.
Most o f t h e r e s u l t s on p r o p a g a t i o n of s i n g u l a r i t i e s can be deduced from theorem 4 and t h e known r e s u l t s f o r manifolds w i t h smooth bounSee [3] f o r d e t a i l s . s i n t(-A>+ Next we want t o d e s c r i b e t h e behavior of v ( t ) = (-L)k ( n e a r t h e d i f f r a c t e d wave, i . e . a s t ?rl+r2 and a s t J r 1 + r 2 . From (18) and (19) we have dary.
(28) v = where P
(29)
v-4
V-
,-1(rlr2)'-%~~
b(cos a
VV
pl>
Q,-l
x
Fix
i n r e g i o n I1
2
i n r e g i o n I11
(cosh f t )
i s t h e Legendre f u n c t i o n d e f i n e d bv
P,-i(cos
PI)
=
(2/n)
1 (2cos PI
s
-
2cosp,
Note t h a t as t t r l + r 2 ,
p,tv
) - $ cos vs d s ;
and as t $ r 1 + r 2 , f l $ O TO analyze ( 28) i n r e g i o n 11, r e p l a c e s by IT-s i n (295, and,
s e e Lebedev [ 6 ] .
9
4
242
M.E.
with 6
(30)
As
=
PI,
n-
write
PI)
Pv-k(cos
i,$O,
=
TAYLOR
7r
COS TTV
+ sin
{s#(cos6,- cos S)-'COS s w d s TTV
* \&, (cos 6,
,1
cos s ) - % i n sv d s .
t h e second t e r m on t h e r i g h t t e n d s i n t h e l i m i t t o "sin swds, sin lTW sin
i,
;
- cos
S ) " ~ ( C O S
Write t h e f i r s t term a s (31)
-
cos
n
T
j~ 6 , ( c o s 5,
T
+ cos
sv
n v ( ~ ( C O6,S- cos s)-li
-
1) d s
ds.
As h,\LO, t h e f i r s t term here t e n d s i n t h e l i m i t t o cos s v - 1 d s . sin S
cos rrv
The second i n t e g r a l i n (31) i s a s c a l a r , i n d e p e n d e n t o f v , and i t is e a s i l y s e e n t o have a l o g a r i t h m i c s i n g u l a r i t y . More p r e c i s e l y ,
\;(toss, -
cu
cos s)-'
' + CB.J hlj.
ds = (log L ) C A . & j j:o
I
J
j.0
Consequentlv one d e r i v e s t h e f o l l o w i n g . Theorem 5. Fix ( r 2 , x 2 ) and t . Then, a s r1
4t - r 2 ,
As f o r t h e second term on t h e r i g h t i n (32), one h a s :
Proposition
(33)
6.
1;
cos "VC-log
( 2 c o s ;)-'(cos v
+ a:,j
k
v-2j
sv
3+
- c o s ~ y )d s
s i n rr v
=
+ sK(v)
where SK(v) : ~8 * 8 S'2Kfor a l l s . One c a n augment theorem 5 w i t h h i g h e r o r d e r e x p a n s i o n s i n 5, , i n v o l v i n g 6,j and s l J l o g 1 , 0 4 j ,c K - 1 . I n s u c h a c a s e t h e remainder term RK b X s a t i s f i e s t h e l e s t i m a t e 2 1 A c K l o g -. IIRK 6 x 2 1 b - s - K Thus RK 6, converges t o z e r o r a p i d l v as &.LO, i f K i s l a r g e , b u t o n l v i n a weak &ace o f d i s t r i b u t i o n s , It t u r n s o u t t h a t , awav f r o m 3 c N , t h e convergence i s smooth, i . e . , i f UCcNI3, t h e n a s r,& t - r 2 ,
s,
l[RK g X 2 ( r l , * ) l u l ( c k ( u )
5
C
4
6,K l o g 1 f o r a l l 1
k.
243
DIFFRACTION OF WAVES B Y CONES AND POLYHEDRA
Such a "pseudo l o c a l " p r o p e r t y o f convergence r e q u i r e s a more c a r e f u l argument t h a n t h a t s k e t c h e d above t o prove theorem 5 .
In p a r t i c u l a r i t f o l l o w s from t h e uniform a n a l v s i s which we w i l l i n d i c a t e s h o r t l y , which i s a l s o v a l i d i n a neighborhood o f
3
.
I n a s i m i l a r f a s h i o n we c a n u s e (19) i n r e g i o n I I t o g e t : Theorem 7 . For r l f t - r 2'
(34)
where f o r s > (m+1)/2,
L
As i n theorem 5 one c a n g e t h i g h e r o r d e r e x p a n s i o n s and show t h e remainder t e r m goes t o z e r o r a p i d l v , and smoothlv on t h e complement of
3,.
Note t h a t ( 3 4 ) d i f f e r s from ( 3 2 ) bv t h e term n-1(rlr2)'-1i
times
- sin b . (35) x2 T h i s c o n t r i b u t i o n r e p r e s e n t s a iump i n t h e fundamental s o l u t i o n a c r o s s t h e d i f f r a c t e d wave 8 t . Note t h a t ( 3 5 ) i s smooth on N - - 3 , b u t g e n e r i c a l l v i t i s s i n g u l a r on 3 . Of c o u r s e t h e r e i s a l s o t h e l o g a r i t h m i c s i n g u l a r i t v , (r1r2)'-& times 1 2 ( 2 ~ ) - l o g - c o s TTV 6 (36) 6 x2 where 6 = 6, i n ( 3 2 ) and b = f. i n ( 3 4 ) . I n t h e s p e c i a l c a s e where N i s il
a n i n t e r v a l [O,L],
s o dim C(N) = 2, c o s
TTV
6X
i s a sum o f two d e l t a
f u n c t i o n s , and i s s u p p o r t e d on 3 . Thus i n t h s c a s e i f one made an a n a l y s i s o f t h e fundamental s o l u t i o n n e a r t h e d i f f r a c t e d wave "Bt b u t away from 3,, one might miss t h e l o g a r i t h m i c term. We a l s o remark t h a t , i f N i s a subdomain o f t h e u n i t s p h e r e S Z h (of dimension), v a n i s h e s on t h e s e t N \ N o where c o s TTV b x2 N o = { x 1E N : f o r some ycaN, d i s t ( x 2 , y ) i d i s t ( y , x l ) 5 n].
even
Thus t h e l o g blow-up d i s a p p e a r s on N \ N O .
T h i s f o l l o w s from t h e f a c t
t h a t c o s n v 0 = 0 , where V i s t h e o p e r a t o r on S Z k , t o g e t h e r w i t h a f i n i t e p r o p a g a t i o n speed grgument . I n o r d e r t o g i v e a p a r a m e t r i x f o r (-A)-lisin t(-A)li 6 i n r e g i o n 11, good u n i f o r m l v a s r
14
(r2,X ) t - r 2 , e v e n f o r x1 i n a n e i g h b o r -
hood o f 2 c N , we make u s e o f t h e u n i f o r m a s y m p t o t i c e x p a n s i o n s o f t h e We h a v e , f o r c e r t a i n smooth Legendre f u n c t i o n s due t o Szeg6 [lo]. j'
'fi
244
TAYLOR
M.E.
where t h e remainder term s a t i s f i e s t h e e s t i m a t e
The B e s s c l f u n c t i o n s J. ( s ) and Y . ( s ) have t h e f o l l o w i n g b e h z v i o r : J Sj+2~ J bh: s-~' ~ ~ ( ~ ) - r aas, ~s 3 0 and J j ( s ) m ( ~ / * s ) ' c o s ( s - ~ - ~ ) - ( ~ / T T s ) % ~ ( s - ~ - ~ ) s-2h-1 a s s + w ; a l s o Y . ( s ) y d - j sgj+ ...
FcK
+
(sj
as s * O , while f o r s + S , Y has a n e x p a n s i o n ' * ' j 5 s i m i l a r t o t h a t of J . ( s ) , w i t h t h e r o l e s o f s i n and c o s i n t e r c h a n g e d , J S u b s t i t u t i n g t h e o p e r a t o r v f o r 2 i n ( 3 7 ) v i e l d s an o p e r a t o r i d e n t i t y
+d s j l o g s
where ( f o r f i x e d r 2 , x 2 , t ) TK i s C h i n x1 and , / r l - ( t - r 2 ) f o r any h: i f K i s l a r g e enough, The o p e r a t o r s J.(Slv) and Y.(6,v) c a n be s y n J J t h e s i z e d from t h e o p e r a t o r s c o s S V , s i n s v ( lslL 6,) and e-" (s 3 0 ) v i a t h e formulas (40) (41)
Yj ( 2 ) = 7l-l
J
li
sin(Asin t 00
n - l (,e
I-, I
(
, j ~ ~ ( =.2 r )r ( + ) r ( j + + ) l - 1 2')
-
( i - t 2 > j - t cos 2 t d t ,
j t ) dt
'(eJS
+ e-jscos
JT)
ds.
As r l J t - r 2 , t h e o s c i l l a t o r v b e h a v i o r o f t h e R e s s e l f u n c t i o n s and Y i n ( 3 9 ) r e f l e c t s t h e approach o f t h e primarv wave 'p t o
j 5 3t. Note t h a t v-j.l.(&,V) s a t i s f i e s a f i n i t e p r o p a g a t i o n speed con-
;
d i t i o n , bv ( 4 0 ) ; v - j j (6,v)u v a n i s h e s i n U s [ x e. N: d i s t ( x , N \ U ) > &, ( i f u v a n i s h e s i n U. A o n s e q u e n t l v , i f U E & ~ ~ ~ ( U we) s, e e t h a t V-jJ (6,v)u i s bounded i n c 8' (U,) f o r O d S I C s There i s no f i n i t e j p r o p a g a t i o n speed f o r Y (&,vfycbut from (41) we c a n g e t s u c h a "pseudo 1 l o c a l " p r o p e r t v a s above, w i t h l o g terms thrown i n , a s f o l l o w s . Write t h e two i n t e g r a l s i n ( 4 1 ) as Y . ( h , v ) = U (S,V) + V.(&,v). J j 1 Now U (6,~)= n - l ( r s i n ( 6 , v s i n t - j t ) d t c l e a r l v s a t i s f i e s t h e pseudoj l o c a l p r o p e r t v g i v e n above. RewritewVj( 6 , ~ )a s
.
( - 1 ) j \:e-K,vsinh
e-js ds +
1, e-6~\Jsinh.'
eJs d s
+
I,
I
(
1
ds
245
DIFFRACTION OF WAVES BY CONES A N D POLYHEDRA
=
(-dIjG , v > + $j (6,v) + A J. ( 6 , v ) .
For j 1 1 ,
3.J ( A )
(Wj(6lv)
=
i s smooth f o r JC[O,m) and r a p i d l y d e c r e a s i n g as ;\+w; Thus t h e o p e r a t o r s 6 . (6,~) behave l i k e P o i s s o n we sav e $,! (lo,-)). J i n t e g r a l s . Note t h a t g0(1) = To(A). As f o r p ( I ) , i t i s smooth on 5 (0,Oo ) and r a p i d l v d e c r e a s i n g a s a3-, b u t t h e r e i s a blow-up as I + 0 . I t s s t r u c t u r e i s e a s i l v p e r c e i v e d i f we w r i t e
9
,,I
0
e-'IVS
(fi+
(s2+1)-'
s)j
ds.
One o b t a i n s
(&,v)j*. (5 w ) = P j ( 6 , ~ )+ ( A , V ) ~ ~ Q (S,v> ~ log(6,v)-' s o P . ( 6 , ~ )and Q. (&,v) behave where P . ( 2 j :nd Q j (1) b e l o n g t o i ( [ O , @ ) ) , J J J l i k e P o i s s o n i n t e g r a l s . The c o n t r i b u t i o n t h i s g i v e s t o ( 3 9 ) i s o f
t h e form
( 6 , ~ )+ v - 2 j { j (&,v) lOg(6,v)-l
v-'jP.
where
Gj(?)
= $iZjQj(2)E~(CO,-)). I
As f o r A ( A ) = Soexp(-2sinh s ) ( j w i t h an a s y m p t o t i c e x p a n s i o n
) d s , c l e a r l y Aj
6
C"([O,m))
Do
( 2 ) Cw. ~ ~
JIj
6'0
3
j as
-
I+-;
one s a y s A G s - ' ( ! R + ) .
1
These o b s e r v a t i o n s j u s t i f y t h e remarks f o l l o w i n g theorem 5 on smooth c o n v e r g e n c e i f x1 6 N' A . In [ 31 i s proved a c o n v e r g e n c e r e s u l t which, compared t o theorem 5 , e v e n augmented h y t h i s comment, i s s h a r p e r , i n a f a s h i o n we do n o t have t h e s p a c e t o e x p l a i n ( r o u g h l y h a v i n g t o do w i t h s h a r p w a v e f r o n t s ) .
A n e a t r e s t a t e m e n t o f (39) i s g o t t e n bv a d d i n g s i n n v J . ( S , V ) t o J - c o s - w U j ( & , v ) , u s i n g t h e f o r m u l a J.(A) = n-' { : c o s ( l s i n t - jt) dt J (valid for integral j ) t o get
(43)
sin
-
J.(&,w) c o s TTV U . ( S , v ) = J J jasin((n-L,sin t ) v ) cos j t d t
RV
TT-'
+ =
-1
i-~
i0 c o s ( ( r - 6 , s i n
t)v) sin jt d t
w j (6,,v).
Note t h a t , b y svmmetry o f s i n t a b o u t t = n/Z, t h e f i r s t term on t h e r i g h t i n ( 4 3 ) i s n o n z e r o p r e c i s e l v f o r j e v e n and t h e second term i s
rJj(61v)sin -
Now we have
n o n z e r o p r e c i s e l y f o r j odd.
(44)
ajv-j =
v-l
sj
nv
" - j wj(6,,v)
- 6,j v-J ( - 1 ) j p j (6,V)
-
j
(a,w)
cos n w ~
w-jnj(i,w) cos n v c o s llv
246
M.E.
-
q
P
j
(A;.)
+ Gj ( 5 , v )
TAYLOR
l o g ( 5 , v ) - l ] cos n v ,
where W . ( S , , \ , ) i s g i v e n bv ( 4 3 ) , A ( 2 ) E S-'(/T(S), P j , $ j € 2 (IR'), and J j for j # 0 ( I ) 6 2 (fit) while f o r j = 0 t h e t h i r d l i n e of ( 4 4 ) has t h e same form a s t h e l a s t l i n e . I t i s a p p a r e n t t h a t W . ( a , , \ , > 6, J has a s i n g u l a r i t v l o c a t e d a t t h a t o f C O S ( T T - &b,X) \ ~ b u t n o t a t t g a t o f C O S ( T T + &h, ) V ( c o n s i s t e n t w i t h theorem 2 ) . The s $ n g u l a r i t y of l o c a t e d on 3 c a n c e l s t h a t o f $1. (6,v)cos rt\, IT-~W~(&,,V)C x2 , J x2' f o r 4, p o s i t i q e . ,1 TO a n a l v z e (-J)-%.in t(a s r,? t - r 2 , we u s e t h e uniform a s v m p t o t i c e x p a n s i o n good f o r /1' [O,h] ( s e e Szego [lo] ) . It g i v e s , f o r c e r t a i n smooth yj ( f x ) ,
3
sl
p1),
where TK i s as smooth a s d e s i r e d i n x1 and d ( t - r 2 ) - r l m o d i f i e d Bessel f u n c t i o n s K . ( / ? ) a r e g i v e n by
5, e - 2 s
* J c o
KJ. ( > )
= n%(j+ls)-l[q
f o r K l a r g e . The
.
( ~ ~ - 1 ) j d- sl ~
The same s o r t o f a n a l v s i s a s a p p l i e s t o ( 4 2 ) g i v e s ,j
with L
(46)
Kj(i\) j' v-j
MjC
pi
=
Lj(A)
%
(
+
J2jMj(J)
log]-'
G.
(2) = A 2 j M . as well as J J -2j -2j Kj(Pxu) = V L j ( pz"> + J Mj(
(I).
Consequentlv
p z ~ >l o g ( / , v ) - '
u
and Lj ( f k v ) and M. ( f%v) behave l i k e P o i s s o n i n t e g r a l s . J From ( 4 5 ) and ( 4 6 ) , t o g e t h e r w i t h t h e known q u a l i t a t i v e b e h a v i o r o f t h e P o i s s o n - l i k e o p e r a t o r s l i s t e d above, one g e t s smooth c o n v e r gence a s r T t - r
on N - 3 ,a s mentioned i n t h e remark a f t e r theorem 1 2' A l s o , f o r p o i n t s x 63 s u c h t h a t c o s r v g is represented as a 1 s i m p l e p r o g r e s s i n g wave on a neighborhood o f ( l e t ' s c a l l the s e t 1 o f s u c h p o i n t s 30) one c a n see a f a i r l y p r e c i s e q u a l i t a t i v e d e s c r i p t i o n o f t h e wave i n r e g i o n 111 a s r 1' t - r 2 . 1 S The b e h a v i o r o f o p e r a t o r s \ I , l o g v , - v s , and o t h e r o p e r a t o r s o f t h e form p ( v ) , where p ( 2 ) s a t i s f i e s a "svmbol" c o n d i t i o n , a s pseudo
7.
z2
d i f f e r e n t i a l o p e r a t o r s on N , i s s t u d i e d i n d e t a i l i n c h a p t e r 12 o f [13], i n t h e c a s e where a N i s empty; s e e a l s o t h e e x p o s i t o r v a r t i c l e [117. E x t e n s i o n s o f some o f t h e s e r e s u l t s t o m a n i f o l d s w i t h smooth boundarv a r e g i v e n i n s e c t i o n 1 o f [ 31. The c o n s t r u c t i o n o f p a r a m e t r i c e s f o r cos s v when N i s d i f f r a c t i v e ( e . g . , N i s t h e complement o f a g e o d e s i c a l l v convex domain i n a b o u n d a r y l e s s m a n i f o l d ) i s g i v e n i n 173, [ 1 2 3 , and c h a p t e r 10 o f [13J, and t h e s i n g u l a r i t i e s a r e g i v e n
247
DIFFRACTION OF WAVES B Y CONES AND POLYHEDRA
i n [ 8 1 f o r g e n e r a l smooth 2 N , g r a n t e d t h e h v p o t h e s e s made e a r l i e r .
I n 131, a u n i f o r m a n a l y s i s i s made a s ( r l , x ) ? ( t - r 2 , x l ) f o r x1 i n a neighborhood o f p o i n t s i n a c e r t a i n s u b s e t 300 o f J , , d e f i n e d by t h e c o n d i t i o n t h a t l o g terms do n o t a p p e a r i n t h e p r o g r e s s i n g wave e x p a n s i o n o f c o s n v h t h e r e . I n r e g i o n I1 t h e e x p a n s i o n i s X
g i v e n e x p l i c i t l v i n terms o? f u n c t i o n s o f hvpergeornetric t v p e , w i t h a l e s s e x p l i c i t but q u a l i t a t i v e l y f a i r l y precise d e s c r i p t i o n given i n r e g i o n 111. The a n a l v s i s o f 131 d o e s n o t proceed from ( 3 9 ) and ( 4 5 ) , b u t r a t h e r s t a r t s o v e r from (18) and ( 1 9 ) . We remark t h a t , i f N i s t h e complement o f a g e o d e s i c a l l y convex s u b s e t o f t h e u n i t s p h e r e Sm, t h e n
a,,
=
3,.
As mentioned i n t h e b e g i n n i n g , much o f t h e above a n a l v s i s c a n be e x t e n d e d t o t r e a t t h e wave e q u a t i o n i n t h e complement o f a p o l y h e d r a l c o r n e r K i n IRn (and h e n c e , u t i l i s i n g f i n i t e p r o p a g a t i o n s p e e d , one c a n t r e a t t h e d i f f r a c t i o n o f waves bv p o l y h e d r a i n Rn). Such a r e g i o n i s t h e cone o f a subdomain o f t h e s p h e r e which i t s e l f h a s For s i r n p l i c i t v , we w i l l d e s c r i b e t h e c a s e K C l 3 so R3,K '2 i s t h e cone o v e r a g e o d e s i c polygon N i n t h e u n i t s p h e r e S . The main problem i s t o c o n s t r u c t a p a r a m e t r i x f o r t h e s o l u t i o n o p e r a t o r c o s s v t o t h e wave e q u a t i o n on N ( w i t h D i r i c h l e t boundarv c o n d i t i o n s ) . By f i n i t e p r o p a g a t i o n s p e e d , one c a n r e s t r i c t a t t e n t i o n t o a neighborhood o f a c o r n e r po o f N ( t h e o t h e r r e g i o n s b e i n g e a s y ) . One way t o c o n s t r u c t c o s s v 3 here is t o u t i l i z e a l o c a l coordinate x2 t r a n s f o r m a t i o n an$ a change o f d e p e n d e n t v a r i a b l e which t a k e s t h e a 2 wave e q u a t i o n ( -2+ v ) u = 0 on S2 i n t o t h e wave e q u a t i o n on E u c l i d 2 at with e a n s p a c e IR T h i s i s g i v e n i n 151, a s f o l l o w s . Take S 2 C corners.
n3
.
po = ( O , O , l ) . Use s t e r e o g r a p h i c c o o r d i n a t e s 3 on S2 n e a r p o , d e f i n e d 2 2 by x ' = 2 T / ( l + r ), x3 = ( 1 - p 2 ) / ( 1 + P ) , p =l3l2. Thus 3 = 0 a t po. Now d e f i n e a map from ( 1 , t ) s p a c e t o (X,T) s p a c e bv S / P = X / \ X l , 2 2 t = t a n - l ( ( T - ] X I 2 - 1 ) / 2 T ) , where p = ( b + l ) * w i t h b = 2 (T2-IXI2 - 1 ) / 2 X . Note t h a t anv s e c t o r i n S bounded bv g e o d e s i c a r c s m e e t i n g a t po i s mapped i n t o a n a n g u l a r s e c t o r w i t h v e r t e x a t - L t h e o r i g i n , i n b o t h 3 s p a c e and X s p a c e . L e t f ( T , X ) = / X I - ? 2 2 2 f Then, i f u s a t i s f i e s t h e wave e q u a t i o n ( 2 / a t v ) u = 0 on
+
1 '-
f i g s 2 , o r some s u b r e g i o n , i t f o l l o w s t h a t , on t h e image r e g i o n , v(T,X) = f ( T , X ) u ( t , S ) s a t i s f i e s t h e E u c l i d e a n s p a c e wave e q u a t i o n 2 ( T h i s p r o c e d u r e g e n e r a l i z e s t o m - d i m e n s i o n s . ) See (DT - Ax)v = 0 . [5] f o r d e t a i l s . U t i l i z i n g t h i s t r a n s f o r m a t i o n one c a n c o n s t r u c t t h e f u n d a m e n t a l s o l u t i o n s v - l s i n t v and c o s t v on N ( n e a r p o ) from t h e N on a wedge N i n fundamental s o l u t i o n s ( - A ) - % i n = ( - A ) ' and c o s
248
la2.
M.E. TAYLOR N
so t h e p r e v i o u s a n a l y s i s g i v e s But N = C(M) w i t h M = [O,L], N t h e fundamental s o l u t i o n s o f t h e wave e q u a t i o n on N . One c a n a n a l y z e d i f f r a c t i o n o f waves by p o l y h e d r a in//?" by i n d u c t i o n on n . I n [3] t h i s i n d u c t i o n , t h e p a s s a g e from a n ( n - 1 ) - d i m e n s i o n a l p o l y h e d r o n i n Sn-' t o t h e complement o f a p o l v h e d r a l c o r n e r i n lRn-' and t h e n c e t o a n ( n - 2 ) - d i m e n s i o n a l polyhedron i n Snm2, i s accomplished by a c e r t a i n v a r i a t i o n on t h e method o f d e s c e n t , r a t h e r t h a n by t h e t r a n s f o r m a t i o n used above.
Bibliographv 1.
2. 3. 4. 5. 6.
7. 8. 9, 10. 11.
12. 13.
J . Bowman, T. S e n i o r , and P . U s l e n g h i , E l e c t r o m a g n e t i c and
A c o u s t i c S c a t t e r i n g by Simple Shapes (North-Holland, Amsterdam, 1969). J. Cheeger, S p e c t r a l geometrv of s p a c e s w i t h c o n e - l i k e s i n g u l a r i t i e s , t o appear. J . Cheeger and M. T a y l o r , On t h e d i f f r a c t i o n o f waves by c o n i c a l s i n g u l a r i t i e s , t o appear. L. Hbrmander, On t h e e x i s t e n c e and r e g u l a r i t y o f s o l u t i o n s o f l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , L ' E n s e i g n . Math. 17 (1971), 99-163. P . Lax and R . P h i l l i p s , An example o f Huvgens' p r i n c i p l e , Comm. P u r e Appl. Math. 3 1 ( 1 9 7 8 ) , 415-421. N . Lebedev, S p e c i a l F u n c t i o n s and t h e i r A p p l i c a t i o n s (Dover, New Yorh, 1 9 7 2 ) . R . M e l r o s e , M i c r o l o c a l p a r a m e t r i c e s f o r d i f f r a c t i v e boundarv v a l u e problems, Duke Math. J . 42 (1975), 605-637. R . Melrose and J . S j g s t r a n d , S i n g u l a r i t i e s o f boundarv v a l u e p r o b l e m s , I , Comm. P u r e Appl. Math 31 ( 1 9 7 8 ) , 593-627. P a r t 11, t o appear. I . S t a k g o l d , Boundarv Value Problems o f M a t h e m a t i c a l P h v s i c s , v o l . I1 (Mac M i l l a n , N e w York, 1 9 6 7 ) . G . Szega, Uber e i n i g e a s y m p t o t i s c h e entwicKlungen d e r Legendres c h e n f u n k t i o n e n , P r o c . London Math. SOC. 36 ( 1 9 3 4 ) , 427-450. M. T a y l o r , F o u r i e r i n t e g r a l o p e r a t o r s and harmonic a n a l v s i s on compact m a n i f o l d s , P r o c . Symp. P u r e Math., v o l . 35, h e r . Math. SOC. 1979, 115-136. , G r a z i n g r a y s and r e f l e c t i o n o f s i n g u l a r i t i e s o f s o l u t i o n s t o wave e q u a t i o n s , Comm. P u r e Appl. Math.29(1976), 1 - 3 8 . , Pseudo D i f f e r e n t i a l O p e r a t o r s ( P r i n c e t o n Univ. Press, t o appear).
ANALYTICAL AND NIJMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S . A x e l s s o n , L . S . F r a n k , A . van d e r S : L I I S l e d s . ) @ N o r t h - H o l l a n d Pub1 i s h i n g C o m p a n y , 1981
SOME ASYMPTOTIC PROBLEMS I N MECHANICS R . Temam
Mathematiques - B a t . 425 U n i v e r s i t e de P a r i s-Sud 91405 - Orsay, France
INTRODUCTION Our aim i n t h i s l e c t u r e i s t o d i s c u s s some v a r i a t i o n a l problems which a r i s e i n mechanics ( p l a s t i c i t y ) and t o s t u d y t h e r e b e h a v i o u r when some parameter tends t o z e r o . The l i m i t case corresponds t o t h e p e r f e c t l y p l a s t i c case and t h e p e r t u r b e d problems correspond t o v a r i o u s " u n p e r f e c t l y " p l a s t i c b e h a v i o u r ( i n v o l v i n g f o r i n s t a n c e v i s c o s i t y , hardening,
.. . ) .
A f t e r t h i s d e s c r i p t i o n o f t h e p h y s i c a l o r i g i n o f t h e problem considered, l e t us i n d i c a t e t h e mathematical problems encountered : we have a f a m i l y o f v a r i a t i o n a l problems problems
'?:
ya
depending on a parameter COO
,
f o r which we c o n s t r u c t dual
i n t h e sense o f Ekeland-Temam [ 2 ] , R o c k a f e l l a r 191 ( t h e unknowns
a r e r e s p e c t i v e l y t h e displacement and t h e s t r e s s o f t h e mechanical p r o b l e m ) . F o r m a l l y , as
a+O
pa
(or
gence o f
, Ta
and
3)
to
9; go
tends t o (or
2;)
To
and
3: .
The n a t u r e o f t h e conver-
i s n o t always obvious : i n p a r t i c u l a r
t h e convergence o f i n f i m a ( c f . De G i o r g i [ l ] ) ,o r o f t h e s o l u t i o n s . I n S e c t i o n I we i n t r o d u c e t h e v a r i a t i o n a l problems a r i s i n g i n p e r f e c t p l a s t i c i t y and we r e c a l l s e v e r a l r e s u l t s p r o v e d i n 1-121-1131. I n S e c t i o n I 1 we c o n s i d e r t h e p e r t u r b e d v a r i a t i o n a l problems and s t u d y t h e i r b e h a v i o u r as t h e parameter t e n d s t o
I
0
-
.
PLAN V a r i a t i o n a l Problems i n P e r f e c t P l a s t i c i t y
1. General f o r m u l a t i o n o f t h e problem 2. F u n c t i o n a l s e t t i n g - V a r i a t i o n a l problems 3. P e r f e c t p l a s t i c i t y problems 4. The l i m i t a n a l y s i s I1
- The P e r t u r b e d 5. 6. 7.
V a r i a t i o n a l Problems
Hardening problems Examples Another p e r t u r b a t i o n
References.
250
I
R. TEMAM
-
VARIATIONAL PROBLEMS
I N PERFECT PLASTICITY
General F o r m u l a t i o n o f t h e Problem 1. ___
ro
Let R
r
that
be a g i v e n open bounded s e t i n
, r=2
i s a s u r f a c e o f c l a s s Xr
rl
and
lRJ
r
w i t h boundary
; we assume
u n l e s s o t h e r w i s e s p e c i f i e d . We denote by
r , w i t h r o U rl
two d i s j o i n t s measurable p a r t s o f
=
r
.
The s t u d y o f t h e d e f o r m a t i o n ( e q u i l i b r i u m ) o f a s o l i d body occupying t h e domain
R l e a d s under g e n e r a l assumptions t o t h e f o l l o w i n g problems : 3 and a t e n s o r To f i n d a v e c t o r f u n c t i o n u : R --3 IR , u = (u1,u2,u3), f u n c t i o n u : R -+ E , E space o f symmetric t e n s o r s o f o r d e r 2, U ( X ) = (ui j(x))lsi x
,jc3 ( u ( x )
and t h e s t r e s s a t
and u ( x ) x)
r e p r e s e n t i n g r e s p e c t i v e l y t h e displacement o f
which s a t i s f y t h e f o l l o w i n g c o n d i t i o n s :
-Fundamental _-- - l-a -w_o-f _Mechanics-: -(1.1) where
g,i
denote
,
in R
t fi = 0
uij,j
and we have used t h e c o n v e n t i o n o f summation o f r e p e a t e d
i n d i c e s ; f = (f,,f2,fi)
r e p r e s e n t s a g i v e n v e c t o r f u n c t i o n on R ; we assume f o r
instance t h a t (1.2)
f
=u:(n)j .
-Boundary - - - -c g n A i t i g n 2 u = U on
(1.3)
u.v
(1.4)
v = (vl,v2,v3)
where
=
ro,
F on
i s t h e u n i t outward normal on
p r e s c r i b e d displacement and f o r c e s on
U
(1.5)
E:
ro
and
rl
r , and
U
and
F
are the
respectively, w i t h f o r instance
3 H 1 (0)
(1.6) Collstitutivee4u~t~o~s
The b a s i c e q u a t i o n s (1.1), (1.3),
(1.4)
a r e supplemented w i t h a c o n s t i t u t i v e
e q u a t i o n which depends on t h e m a t e r i a l , and c o n n e c t i n g a t e v e r y p o i n t stress tensor (1.7)
U(X) w i t h the s t r a i n tensor Eij(4 =
1
+i,j+Uj,i)
E(u)(x)
l
9
x
, the
w i t h components
.
T h i s c o n s t i t u t i v e e q u a t i o n i s d e s c r i b e d i n p l a t i c i t y by a convex l o w e r semicontinuous f u n c t i o n j f r o m E i n t o IRUIt-1 , and a t e v e r y p o i n t x : (1.8) where (1.9)
dx) aj(6)
a j ( 4 u ) x) 1
i s the subdifferential set o f
{n=E,j(C)+j*(q)
=
5.111 = {
j
n E,j(S) ~
a t paint
-s
, i.e.
j * ( v ) >, 5 . ~ 1 1
,
251
SOME ASYMPTOTIC PROBLEMS I N MECHANICS
S.n
with
c,n
,
nij
= Eij
, and
c.E
j*
j * ( n ) = SUP C S . -~ ~ ( s ) I
(1.10)
j :
denoting t h e conjugate o f
5EE
,
which i s a l s o a convex l o w e r semi-continuous f u n c t i o n f r o m
E
into
R
u
.
Ctm)
It i s well-known t h a t (1.8) i s e q u i v a l e n t t o
(1.11)
.
E(u)(x) E. a j * ( u ( x ) )
For t h e p r o p e r t i e s o f convex f u n c t i o n s and t h e i r a p p l i c a t i o n t o t h e c a l c u l u s o f
J. Moreau 1-81, R.T. Rocka-
v a r i a t i o n , t h e r e a d e r i s r e f e r r e d t o Ekeland-Temam [:2], f e l lar
lie] .
The basic _
2.
fi boundary
equations
thus ( l . I ) , (1.3),
(1.4),
c o n d i t i o n s o f t h e e q u i l i b r i u m problem a r e
(1.8).
Functional S e t t i n g
-
V a r i a t i o n a l Problems
I n o r d e r t o d e s c r i b e t h e f u n c t i o n a l s e t t i n g o f t h e e q u i l i b r i u m problem 2 we c o n s i d e r t h e f o l l o w i n g spaces: L (Q) = space o f f u n c t i o n s f r o m R i n t o IR , 2 which a r e L 2 f o r t h e Lebesgue measure dx , I L 2 ( Q ) = L 2 ( Q ) 3 , L (R;E) = space o f
L2
R into E
functions from
. We
w i l l denote i n d i f f e r e n t l y by
(f,g)
and
If1
t h e s c a l a r p r o d u c t and t h e norm i n one o f t h e s e spaces. We i n t r o d u c e a l s o t h e 1 H1(Q) and M 1 ( Q ) = H ( R ) 3 . We s e t
Sobolev space
(2.2)
2
fa
= { U E L (R;E)
"e,
=
(2.1)
{UE.
H1(R)3
(2.3)
5~
=
I
R
5
ci.
.v
IJ
j = Fi
on
,
a.e.
XE
R
.
R and u s i n g t h e g e n e r a l i z e d Green f o r m u l a [6] , [14] :
-
E(U).(ci-Z)dx =
-
Ui dx +
(uij,j-uij,j)
Iro(o-Z).vU
dr
,
r e a l i z e s t h e maximum f o r t h e v a r i a t i o n a l problem :
sup
0ega
{-I
R
rl)
- ci, u, a r e s u f f i c i e n t l y r e g u l a r s o l u t i o n s t o t h e UC.'~?~) , t h e n f o r e v e r y c i c f d
=
we f i n d t h a t
R,
U on r o l .
j * ( o ( x ) ) 5 j * ( Z ( x ) ) t E(~)(x).(u(x)-~(x))
By i n t e g r a t i o n on
(2.5)
,u
= 0 in
x,
I t i s easy t o see t h a t i f
e q u i l i b r i u m problem ( s a y
, uijYj + fi
j*(o)dx t
o.v U d r ) rO
.
252
R. TEMAM
u E t a , by (1.8),
Similarly, i f
j ( E ( u ) ( x ) ) 5 j(E('li)(x))
(2.6)
O(X).(E(U)(X)-E(U)(X))
t
and u s i n g t h e g e n e r a l i z e d Green f o r m u l a and ( 2 . 1 ) - ( 2 . 2 ) , t h e minimum f o r t h e v a r i a t i o n a l problem :
{j
Inf ucfa
(2.7)
The
tcj
i,fu
dx
-
1
we f i n d t h a t
Fu dr3
u
realizes
.
rl
gr& ( 2 . 7 ) a r e t h e v a r i a t i o n a l problems
v a r i a t i o n a l problems ( 2 . 5 )
n a t u r a l l y associated
-
j(E(u))dx
R
, f o r a.e. X E D ,
t h e e q u i l i b r i u m problem. The r e l a t i o n s between these
problems and a l s o t h e a c t u a l e x i s t e n c e o f s o l u t i o n s i s discussed below, a f t e r making more assumptions on 3.
j
.
P e r f e c t P l a s t i c i t y Problems n
We denote b y vanishing trace,
E
= E
Eu D
t h e subset o f @ IRI
o f symmetric t e n s o r s w i t h a
E
, and we a r e g i v e n a c l o s e d convex s e t
KcE
,
such t h a t
i
KD = OE
K = K ~ @ I R I ,
(3.1)
KD
Then t h e f u n c t i o n s
(3.2)
~
K
,
i s bounded and c o n t a i n s a neighbourhood o f
D E )
.
j* = j o a r e g i v e n as f o l l o w s :
if EE K
AS.S t
(in
x
j = jo and
jt(c) =
0
otherwise
m
,
where we posed (3.3) a,
p o s i t i v e constant, 6 . .
K~
1J
We denote by that
j:
jo t h e c o n j u g a t e o f
i s the conjugate o f
and continuous on
E
jo
,
(3.6)
. Using (3.2) 0
ko(lCDl-l) 6 jo(s
E E
the deviator o f
1ciil2
0 < k,
D
i t was shown t h a t
D + jo(s )
< kl <
5 :
& i j*
x
jo(E;) = ;r
and t h e r e e x i s t s two c o n s t a n t s ,
D
j o which i s a l s o convex L.s.c.
K
(3.5)
- i(6kk)
5 Di j -- E i j
(3.4)
5
t h e Kronecker symbol,
m
D G k1(15 1+1)
such Z h a t
Y
E= E
and i t i s c l e a r jo i s f i n i t e
SOME ASYMPTOTIC PROBLEMS IN MECHANICS
253
Problem ( 2 . 7 ) becomes
(
{T '6
Inf
To)
u s H 1 (Q)3 u=U on r
j
( d i v u)2dx +
jo(cD(u))dx
-
L(u)?
R
R
0
where
and problem ( 2 . 5 )
becomes
u .,.+fi=O iJ J
uij
vj
D
in
= Fi
on
r,
D
u (x)EK a.e. The r e l a t i o n between
(go)and (9;) was
i n v e s t i g a t e d i n 1131 ( c f . a l s o t h e
b i b l i o g r a p h y o f t h i s a r t i c l e ) , and i t was shown t h a t
go =
Inf
(3.8)
(9,)
9; ,
sup
and i f t h i s l a s t number i s f i n i t e ( i.e. >
Problem
t h e dual o f
[ I q . Furthermore
i n t h e sense o f [2],
s o l u t i o n To
(8;) is
-a)
. (go) may
, problem
n o t possess a s o l u t i o n (even i f
(9;)possesses inf
yo >
-a)
a unique
: we r e f e r
t o [ 1 2 ] , [13] and t o f o r t h c o m i n g a r t i c l e s f o r t h e i n v e s t i g a t i o n o f g e n e r a l i z e d solutions o f
(go) which
are less regular
which t h e boundary c o n d i t i o n
u = U
r0
on
(u
E
BD(R) , c f . [12]) , and f o r
i s p a r t l y r e l a x e d ( c f . [13]), as
expected f r o m t h e mechanical p o i n t o f view.
which i s i n t e r e s t i n g f o r t h e sequel i s
Another problem i n v e s t i g a t e d i n [13],
the
Limit A n a l y s i s
problem which a l l o w s t o d e t e r m i n e whether t h e number i n ( 3 . 8 )
i s f i n i t e . We d e s c r i b e t h i s i n n e x t S e c t i o n .
4.
The L i m i t A n a l y s i s The number i n ( 3 . 8 ) i s f i n i t e i f t h e i n f i m u m o f problem
u's
o r e q u i v a l e n t l y i f the s e t o f admissible a set o f data
(4.1)
(f,F)
, of
for
; ' 3
the form N
f = X f ,
Iv
F=XF,
XED-!+,
(go)i s
finite
i s n o t empty. We c o n s i d e r
254
R . TEMAM
?, F‘
s a t i s f y i n g o f course t h e same assumptions ( 1 . 2 ) - ( 1 . 6 )
as
f, F
. We
then
r e c a l l the following : Theorem 4.1 ~-
I c f . 11311 -
The infimwn of
9;
i s f i n i t e , if and only i f ,
O
,
X is t h e infimwn of t h e l i m i t a n a l y s i s problem.
where
L i m i t Analysis problem Inf UEH1(Q)3 %
(QLA)
R
\ E ~ ( u ) Idx3
u=O on To div u = 0 2.
L(u) = 1
where (4.1) The theorem i s proved i n 1131 where we a l s o make e x p l i c i t t h e dual o f
(QLA) :
SUP {A} 2 ucL ( Q ; E ) , 1 X I I r u t hf. = 0 i n Q
ijJ
J
u.v = XF on T1 u D ( x ) K~ a.e.
I1
-THE PERTURBED VARIATIONAL PROBLEMS 5.
Hardening problems We i n t r o d u c e an a u x i l i a r y f u n c t i o n
m e t e r ) where
El
e
: R
-+ El
( t h e hardening para-
i s a f i n i t e dimensional e u c l i d i a n space ; El = E and
El =
IR
i n t h e two f o l l o w i n g examples. We a r e g i v e n a f a m i l y o f c l o s e d convex s e t s
XacE I$ = K (5.1) or
x
El.,
x El
.
ab0, whose i n t e r s e c t i o n w i t h E x lo) i s j u s t K x { O } and such t h a t The c o n d i t i o n s (1.8) and (1.11) a r e r e p l a c e d by ( u ( x ) , e ( x ) ) e aja(E(u)(x),o)
SOME ASYMPTOTIC PROBLEMS IN MECHANICS
[t
255
otherwise.
m
Problem (2.7) i s now r e p l a c e d by Inf
(5.4)
UtH
1( Q ) 3
u=u on
{jR
jcl(e(u),O)
-
dx
j Q f u dx
-
Fu d T l rl
ro
w h i l e problem (2.5) i s r e p l a c e d by
-Remark 5.1.
F o r m a l l y , as
, 16, "converges" t o K
a+O
e
t o t a l l y u n c o n s t r a i n e d , t h e supremum w i t h r e s p e c t t o we r e c o v e r
q:.
x
El , so t h a t
is
0
i s attained f o r
8=0
, and
We a r e g o i n g t o prove t h i s p o i n t .
I
We make t h e f o l l o w i n g assumptions : Ma = { ( u , e ) c y a
(5.6.)
2 L (a;E1)
, ( u ( X ) , e ( x ) ) c K a a.e.
and ua. j J weakly i n L2 (R;E1)
(5.7)
Theorem 5 . 1 .
__I_-
The problems -
Ma
d u a l i t y i n the sense Q [ Z ] , (5.81
Theorem 5 . 2 . --
$ = sup
possesses a unique s o l u t i o n
(Passage t o the limit
furthermore
(5.91
o r equivalently,
that,
as
2
,
, then
(u,e)
u weakly i n L (R;E) c1
j
+. 0
E
Mo
.
q) are i n
[ l o ] , and under t h e asswnption (5.6) : Inf
and problem y3""
t
( h e r e a f t e r denoted
(5.4)-(5.51
x ~ n l
.
i s n o t empty f o r e v e r y COO ) E
we assume --
x
a@)
a 4 Inf
E
w ,
(ca,ga/
-
assumptions as i n Theorem 5 . 1 and
, Pa
3
.
Inf
Po
256
R . TEMAM
p:
sup
(5.10)
Q
-
case,
u
inf
01
converges
2 = sup
9 := 2
converges
&
0
f Theorem 5.1. -Proof _ _o_ _-
q. & unbounded.
sequence Ea
To
,
= H ’ ( Q ) ~ x L2(Q;E1)
Vx
J O
In the gpposite
(?To)
-
ea
, ( c f . p . 60-62)
t h e general s e t t i n g of [2]
We use as i n [13]
v
.
L (Q;E,)
sup
to the solution
L IQ;E)
2
&
- , the --?-
i t s dual,
Y = L ~ ( Q ; E ) x L2 ( Q ; E ~ ) = Y x
A
=
A1
x
=
E(U)
t
(p1,p2)=
=
j
=
u
if
= U
on
To
m
VP
, and,
A ~ ( u ~ S= )0
9
{ - L ( U )o t h e( cr wf .i s e( 3 . 5 ) )
F(u,E) =
r (u,E)E. v
, AI(U,E)
A2
Y :
.
~ ~ ( P ~ ( X ) ~ P dx ~ ( X ) )
R
I t i s easy t o see t h a t
t h e problem
(5.11)
:Ta , and t o determine i t s dual
i s identical t o
(5.12) p
f
x x x
SUP
-
[-F (A p
f E.Y
Gx(-Px)]
we check as i n 121, [13] t h a t ,
dr
G*(P*) and s e t t i n g Now sup
=
-t
1
s2
otherwise
j t ( p x ( x ) ) dx
x x
px = (p1,p2)
9; > -
m
m
x
if pl=LQa
,
= ( - 0 ~ 0 ) , we f i n d t h a t (5.12) i s t h e same as
because o f (5.6) and :
5’:
.
SOME ASYMPTOTIC PROBLEMS I N MECHANICS
257
which i m p l i e s t h a t t h e i n t e g r a l G(P) = i s f i n i t e f o r every theorem,
G
i m p l i e s (5.6),
.
Y = L 2 (R;E) x L 2 (R;E1) By K r a s n o s e l s k i i ’ s and i t i s w e l l known ( c f . [2] p . 7 7 ) that this
p = (pl,p2)e
i s c o n t i n u o u s on
Y
and t h e e x i s t e n c e o f a s o l u t i o n
(ua,<) t o 3: . T h i s s o l u t i o n
i s unique by s t r i c t c o n c a v i t y . P r o o f o f Theorem 5.2. If
inf
go = sup 9;
[
’R and t h e sequence
a;
(5.14)
If sequence
=
qE >
(Sa,ea)
and (5.10) h o l d s , then, as
a
-
AZa.Fa dx
2
i
(Oa.v)U dT
-+
+
cx
0
--f
-a’,
c1
depends on
R and f
t h e n (5.10) i m p l i e s t h a t
i s bounded i n
2
L (Q;E)
compactness, t h e r e e x i s t s a subsequence
sup
2
x L (Q;E1)
(za.,Bi ) ~j
,
,
i s unbounded, s i n c e by a t r a c e theorem i n 1141
( where sup
-
( c f . p. 9 ) :
. gE
i s bounded, and t h e
( u s i n g a g a i n ( 5 . 1 4 ) ) . By weakly convergent t o some
( o , e ) which belongs t o Mo ( c f . ( 5 . 6 ) ) because o f ( 5 . 7 ) . By l o w e r semic o n t i n u i t y , ( a , e ) i s t h e n a s o l u t i o n o f ( 5 . 5 ) f o r a=O , b u t as observed i n : , and i t s u n i q u e s o l u t i o n i s Remark 5.1, t h i s problem i s n o t h i n g e l s e t h a n 9 limit
(0,O). One t h e n proves b y usual methods t h a t t h e whole sequence 2 2 (5,O), f o r t h e norm t o p o l o g y o f L (Q;E) x L (R;E1)
converges t o 6.
Examples We a p p l y t h e r e s u l t s t o two c l a s s i c a l examples ( c f .
Case -
(6.1)
1.
.
KD
i s the b a l l
{(S,e)E E
(gDI 6 1 x
El = R
, and
IR , I S ~ 1 ~t aei
for
.
a20
1141 , [ I l l ) .
,
$
i s the set
258
R. TEMAM
16
These s e t s a r e c l o s e d convex and i t i s c l e a r t h a t = K x El , The assumption (5.6) i s s a t i s f i e d s i n c e 'fa 3$ [E x I011 = K x I01
.
n
i s not
empty and by s e t t i n g e(x) =
(u,e)
we g e t a p a i r
Ma
in
( l u D ( x ) I-1)+ a
. The
assumption ( 5 . 7 ) f o l l o w s e a s i l y f r o m t h e
weak closedness o f convex s e t s .
9 :
is
Au.u dx
-
There remains t o check (5.10). Problem (6.2)
SUP
( u , e ) E 3 a x L*(Q)
IUD ( x ) l . - c 1 t a.e.
I
i,
-
';1
( e ( x ) &*
1
(u.v) W d r l
.
e(x)
~1
XEQ
It i s easy t o performe t h e supremum w i t h r e s p e c t t o
$a
dx t
a.e. x )
a
. We
0
, for u
fixed i n
obtain
and (6.2) ,becomes : (6.4)
To maximize ( 6 . 3 ) f o r
ucc3a.
Problem (6.4) i s a p e n a l i z a t i o n o f problem g: i n t h i s case ( p e n a l i z a t i o n 0 o f t h e c o n s t r a i n t : Iu ( x ) l < 1 a.e.) and we w i l l n o t develop t h e p r o o f o f (5.10) which i s c l a s s i c a l . Case 2. Again
K
D
i s the b a l l
150 I 6 1
, but
now El = ED
a>O ,
and f o r
%
the set
The s e t s
a r e c l o s e d convex and i t i s c l e a r t h a t
I$ n [E x ~-Dl]= K
x
CO)
. The
76 = K x
El
assumption (5.6) i s s a t i s f i e d s i n c e
empty, b y s e t t i n g ( f o r example) e ( x ) = 1 uD( x )
.
The assumption (5.7) i s c l e a r l y s a t i s f i e d . I n t h i s exdmple (5.9) i s s i m p l e r t o check t h a n ( 5 . 1 0 ) .
, kfa
i s not
is
259
SOME ASYMPTOTIC PROBLEMS I N MECHANICS
We have
(6.6)
ja(s,O) =
,,ISD
Sup (ole)=
uD t ,(tr 1
5 ) ( t r u ) - -IuDI2 1
= ( t a k i n g t h e supremum w . r .
tr u
to
= ( t a k i n g f i r s t t h e supremum w . r .
Now i t i s easy t o compute
f u n c t i o n b e i n g L i p s c h i t z i an,
and
%ltr 1
u
, and
to
T
D
setting
T = 0
)
jo(5)
which i m p l i e s t h a t t h e g r a d i e n t o f
Back t o (6.6) we f i n d t h a t
-
2a0
EXE
D
jo(E )
(on
D E )
i s l e s s t h a n one and t h i s
R. TEMAM
260
I n particular (6.11) and we i n f e r f r o m (3.4),
( 5 . 4 ) and (6.11) t h a t
(6.12) where
c2<m depends on 0 and
(5.13)
U :
Inf ( / ( 1 c D ( u ) l + 1 ) 2 XI ueH1 ( Q ) ~' Q
c2
u=U on
.
r0
Whence ( 5 . 9 ) . Because of (6.10), t h e problem least. 7.
?
a
possesses i n t h i s case, one s o l u t i o n a t
Another p e r t u r b a t i o n We c o n s i d e r t h e f o l l o w i n g p e r t u r b a t i o n o f t h e b a s i c v a r i a t i o n a l problems
( c o r r e s p o n d i n g t o t h e g e n e r a l i z e d Norton-Hoff m a t e r i a l , c f . F r i a d [3])
: j x = j:
i s d e f i n e d by
with
Ag
as i n ( 3 . 3 ) and
(7.2)
I t i s clear that
g(c
jt
D
g
t h e gauge f u n c t i o n o f
= i n f {y,cD=
KD
in
ED :
Y K ~ >, 01 ~
i s convex l o w e r semicontinuous and we d e f i n e
ja as i t s
c o n j u g a t e : t h i s i s a convex l o w e r semicontinuous f u n c t i o n t o o and
j,* i s t h e
.
c o n j u g a t e o f ja Problem ( 2 . 5 ) i s now r e p l a c e d by
and we l e a v e (2.7) i n t h e f o r m
SOME ASYMPTOTIC PROBLEMS I N MECHANICS
(7.4) UEH
Inf 1 3
(n)
The problems
&
17.4)
( 7 . 3 ) ( h e r e a f t e r denoted
qcl= sup
inf
L?:
and problem i Pc1 posse sse s As c1 + 0 , -
a unique s o l u t i o n
(7.6)
inf
sup:5
Proof.
>
.
t h e sense of 121, LIO],
17.51
If
L(u)?
r0
u=U on
~Theorem 7 . 1 . i n duality i n -
-
{ \ jcl(E(u))dx R
-
<
= sup
m,
ucl converges
g:
E
4i n f
2
V = H1(Q)3
, V*
Y = L2 (R;E)
= Y
.
= sup :?~
t o the solution
F(u) =
- L(u) t
G(p) =
m
R
i t s dual,
x ,
,
E(U)
if
u = U
uEV
otherwise,
j(p(x))dx
ro
on
,
p~
The problem i n f [ F ( u ) t G(Au)] ucv i s t h e same as (7.4) and i t s dual
(7.7)
x % * SUP t- F ( A P 1 - G * k P * ) I * P GY
i s t h e same as ( 7 . 3 ) ,
setting
p* =
-u
.
,
L'(n;E)
We g e t as i n 1131 t h a t
(7.8)
32)are
.
~r
L IR;E)
~
3, &
B
We use t h e g e n e r a l s e t t i n g o f d u a l i t y as i n [2] p . 60 :
AU =
26 1
.
5;
Q
P: .
262
R . TEMAM
We n o t e t h a t
2 i s everywhere d e f i n e d and c o n t i n u o u s on Y = L ( R ; E )
so t h a t t h e f u n c t i o n
G
I t f o l l o w s ( c f . [2])
that
there exists a solution If sup 3:
u(x)
E
KD
; -m
a.e. a+1
XE:
.
$
= sup , t h i s number i s c l e a r l y f i n i t e , and ._ o f ( 7 . 3 ) ; ucl i s unique by s t r i c t c o n c a v i t y .
i n f :3
",
, t h e r e e x i s t s a u a d m i s s i b l e f o r 9; , i . e . 0c.a R . F o r 5D E KO , g ( 5 0 ) I 1 ; hence g ( a0 ( x ) ) 6 1 ,
[ g ( u D ( x ) ) ] a .I 1 f o r a . e .
x
and
and
(7.9) Writting that
i s m a j o r i z e d by t h e same e x p r e s s i o n w i t h
-
",
r e p l a c e d by u
and t a k i n g (7.9)
.
i n t o account we g e t t h a t sup 9 : remains bounded as 9 : 0 2 0 F o r e v e r y u c L ( R ; E ) , s a t i s f y i n g u ( x ) K~ a.e. we have a.e.
XE
R
, and
gives a t the l i m i t
-$
i f furthermore
c1
A0.u dx
o ~ g ,a t h e r e l a t i o n
-
,
0
---f
-
(a.v)u di- d i-0
and t h i s r e l a t i o n which i s v a l i d f o r e v e r y ( u e g a , u D ( x ) e K a.e.
x
D g(u (x)) s 1
E:R)
lim SUP^':) , c1+0
u which i s a d m i s s i b l e f o r
9:
, implies
Using (5.14) and t h e f a c t t h a t t h e e x p r e s s i o n (7.10) remains bounded, we i s bounded i n L 2 (n;E) and t h a t conclude t h a t t h e sequence Zcl (7.12) There e x i s t s then a subsequence ( s t i l l denoted
ScL) whiEh converges weakly
in
263
SOME ASYMPTOTIC PROBLEMS I N MECHANICS
2 L (R;E)
t o some l i m i t ?i
Z E Y a and u s i n g ( 7 . 1 2 ) we see t h a t f o r $ on R :
; clearly
every p o s i t i v e r e a l continuous f u n c t i o n
,< ( b y H o l d e r i n e q u a l i t y )
1
<
( b y Lebesgue's theorem and ( 7 . 1 2 ) )
(i,
< Therefore a.e.
s(;~(x))
,
x E
i.e.
a 4
i1 h
G
a -
$ ( x ) dx)
f o r a.e.
x ER
i s admissible f o r
(sup $)
s
( c ---) at1
lim
I
-
: . :A R
, and
u = u0 Tv
. As
i
(Z.v)U d r
sup
9;=
for
.
i s solution o f
T,,
.
t h e sequence
$(x)dx
TO
usual one proves t h a t t h e whole sequence 2 t h e s t r o n g t o p o l o g y o f L (R;E)
If
R
semi-continuity,
T h i s i n e q u a l i t y combined w i t h (7.11) shows t h a t .<
i
t h i s means t h a t z D ( x ) = K
9;. By dx +
=
a
a+0
sup $:
-+
9; and
converges t o
0
thus for
. Indeed t h e
as
a
t
p r e c e d i n g p r o o f would show o t h e r w i s e t h e e x i s t e n c e o f some
7
admissible f o r
--m
by c o n t r a d i c t i o n w i t h
sup
$=
--m
--m
.
0
0
9Ey
The p r o o f i s complete.
Footnote
( 1)
O f course
EE K + ,+.
5 D ~l KO
REFERENCES
[l] E. De G i o r g i r-convergences, Congrgs I R I A , V e r s a i 1l e s Decembre 1979. 121
I. Ekeland, R. Temam Convex Anal s i s and Variational Problems
N%li="*lrn.
[3]
A. F r i a a
La l o i de Norton-Ho BnBralise'e en p l a s t i c i t & g visco-plusticite' v e r j i t e p P a r i s 7 T
TGX K i
R. TEMAM
264
141
B. Halphen, Nguyen Quoc Son
15.1
C. Johnson On
-Sur l e s matdriaux
standards g i n 4 r a l i s 4 s Journ. de M @ c a n i q u m 7 5 ) , p.39-63.
fit-nd
1161
lasticit
with hardenin
A p p d 9 7 8 ) , p.325-336
J.L. L i o n s , E. Magenes Non homo eneous boundar
Fi&,+
+problems & applications
171 J.J. Moreau La notion de sur-potentiel e t l e s l i a i s o n s u n i l a t g r a l e s en 4 l a s t o --s t a t i ue C.R. !cad. Sc. P a r i s , s e r i e A, t . 2 6 7 (1968), p.954-957. [8]
J.J. Moreau Fonctionnelles Convexes Seminaire s u r l e s e q u a t i o n s aux d e r i v e e s p a r t i e l l e s , C o l l e g e de France, 1966.
191
R.T. R o c k a f e l l a r
i n v o l v i n g convex functia?s
[loj
R.T. R o c k a f e l l a r Convex Anal s i s P r i n c e t h r s i t y Press, 1970
[ll] G. S t r a n g
08
model pyblerns i n p l a s t i c i t y I n Procee i n m i r I n t e r n a t i o n a Symposium on Computing Methods i n A p p l i e d Sciences and Engineering, L e c t . Notes i n Comput. Sciences, S p r i n g e r - V e r l a g ,
A family
1121 R. Temam, G. S t r a n g Functions 0f- bounded de ormation Arch. Rat. M e m l y s f s , t o appear. [13J R . Temam, G. S t r a n a Duality and r e l m a t i o n in the v a r i a t i o n a l problems Journ. d e e c a n i q u e , t o appear 1141 R. Temam
Navier-Stokes e uations (Znd e d i t i o n ) North-Hol land,-
& plasticity
ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N ANALYSIS S . A x e l s s o n , L . S . F r a n k , A . v a n her S l u i s ( e d s . ) @ N o r t h - H o l l a n d P u b l i s h i n g C o m p a n y , 1981
ASYMPTOTIC AMPLITUDE AND PHASE FOR ISOCHRONIC FAMILIES OF PERIODIC SOLUTIONS B. Aulbach Mathematisches Institut der Universitat Wiir zburg West Germany
Consider an autonomous differential system admitting an isochronic family of periodic solutions with a minimal number of characteristic exponents on the imaginary axis. A global characterization is given for the solutions approaching the invariant manifold of periodic orbits with asymptotic amplitude and phase.
INTRODUCTION In this paper we are concerned with the approach of solutions of autonomous differential systems to invariant manifolds formed by the orbits of families of periodic solutions. It is known that in general each orbit corresponding to a member of the family of periodic solutions has a stable and an unstable manifold. This immediately shows that solutions approach the invariant manifold with asymptotic amplitude and phase if they eventually enter a member of the family of stable manifolds corresponding to the family of periodic orbits. The problem we are concerned with is to state conditions for the approach with asymptotic amplitude and phase in terms of the invariant manifold itself. PROBLEM AND RESULTS Consider an autonomous differential system
x
(1) where f: lRd follows.
+
IRd
=
f (x)
is of class
C2. The ploblem data are as
i) System (1) admits a m-parameter family of periodic solutions x(t,a,p), a E w I R ~ - ’( W open ) , p € IR. a, p are regarded as amplitude and phase parameters respectively, i.e. x(t,a,p) = q(t+p,a), where q(t+wo,a) = = q(t,a) for all a E W, t E IR. w > 0 is constant and q is continuous and of class C2, C ’ as function of a , p respectively. ii) For all a E W
and
p
E
IR 265
B . AULBACH
266
rank
xa(O,arp), xp(O,a,p)l
=
m.
i) and ii) simply mean, that the family x(t,a,p) smooth m-dimensional invariant manifold G := i
X E R ~:
x
=
constitutes a
x(O,a,p), acw, ~ E I Ri
in the phase space lRd. We say a solution x(t) of ( 1 ) approaches G with asymptotic amplitude and phase, if there exists . a (S W and po E IR such that lim [ x(t) - x(t,ao,po) I = 0. t, + -,
The assumption upon the flow on G is stated in terms of the linear variational equation (2) Due to
y i) and
=
fx(x(t,a,p)) Y.
ii) there are always
II:
characteristic exponents of
(2) with zero real parLs. The main assumption made in this paper and all the quotes mentioned below is, that iii) the remaining d-m characteristic exponents stay away from the imaginary axis while aCW and ptlR. The known results concerning our problem read as foliows. The Theorem of Andronov and Witt (see e.g. 1 2 1 , 13,Th.2.2) (m=l) and the papers [ 3 ] and [4 1 deal with the stable case, where no characteristic exponent has positive real part. In this case the manifold G is asymptotically stable and every solution coming sufficiently close to G approaches G with asymptotic amplitude and phase. However this case is of no practical interest since families of periodic solutions appear in general in Hamiltonian or reversible systems which implies that the characteristic exponents are in the complex plane symmetrically located with respect to the origin. This general (unstable) situation has been treated in [ I ] and the sufficient condition for a solution x(t) to approach G with asymptotic amplitude and phase is: the o-limit set of x(t) is contained in a compact part of G determined by (a,p) t W'xIR, where IJ' is a compact subset of the amplitude parameter domain W. While the above mentioned results concern families of periodic solutions with continuously varying period we have devoted this paper to isochronic families, i.e. families with constant period, independent of the amplitude parameter a. As increase in information we obtain a sufficient condition that is also necessary, In addition there is no compactness condition involved. The difficulties arising
267
ASYMPTOTIC AMPLITUDE AND PHASE
from the lacking compactness can be overcome by means of the uniform stability property of isochronic families: (3)
For any I, , O there exists a positive following property:
1 1 x(to,ao,po) - x(to,al,p,); I i=0,1, implies 1 1 x(t,aolpo)
Li='.'
( v ) with the
f o r tot:B, ail!W, pifIB x ( t , a l r ~ lI 1 ). n for all t C n
; I
-
The obvious proof depends upon the fact, that one may restrict t to a compact interval of length of the period . independent of a8-W. The result of this paper is as follows. THEOREM: Consider system (1) with i), ii) and iii). Then a solution xo(t) approaches G with asymptotic amplitude and phase if and only if the couple of conditions a) and b) is satisfied: a)
lim dist(x
b)
xo(t) has a ,.-limit point on G.
0
t-
t),G)
=
0,
PROOE'
The necessity part follo s immediately from the relations
11
dist (xo(t), G )
xo(t) - x(t,aotpo)I~ for all t
and x (io ) 0
0
-
~ ( i ~ ~ ~ , =a x~ , ( pi ~~ ) -~ x(O,ao,po) ~ ) 0
for i=O,1 ,...
The sufficiency part will be proved in four steps.
-
1st step: Let
x(O,ao,po) be an 'L-liniitpoint of xo(t) on G , i.e. as i ' ' Since x (t,ao,Fo) has lim x0 (t1. ) = x (0,ao,co) where t . 1-1 period 0 there exists a bounded -sequence t:El 0, L'O ) , t:1 ~ 1t mod . 0 such that x(ti,ao,po) = x(ti,ao,po) for all i. Let T be the limit of a converging subsequence ti of ti and set po:=po- T. Then k x(ti ,ao,po) converges to x(T,ao,po) = X ( O , ~ ~ , and ~ ~consequently ) k Xo(ti ) - x(ti ,ao,po) converges to 0. Hence as result of the first k k step we yet a sequence ti-' as i-.,and aotW, potB such that - 7
.
lim lxo(ti) - x(ti,ao,po)l = 0. i* In the sequel ao,po will be kept fixed and they will be proved to be the asymptotic amplitude and phase respectively of xo(t). (4)
I
2nd step: By means of three consecutive transformations we shall put the equation ( 1 ) near x(t,ao,po) in a form that meets two requirements. First it is amenable to local center manifold theory
268
B . AULBACH
and secondly it can eventually be delt with as a linear one. The details can be found in [ I ] . There exists a real 2a0
matrix function M(t), periodic with period
CL
such that the transformation
ifi,O,Q)
=
M(t) [ x
- x(t,ao,po)I
carries ( 1 ) into
G G
+ P I (t,G,O,G)
= NCi =
+
PO
Q =
P2(t,d,O,G) P3 (t,G,O,Q),
where N, P are constant matrices whose eigenvalues have negative and positive real parts respectively. This system has near (Q,f,Q)= = (O,O,O) a smooth center stable manifold MZ with local representation 0 = ^s(t,fi,G) and a smooth center manifold Mo with local parameter representation (u,v,w)(t,a,p) := M(t) [ x(t,ao+a,Po+p)
- x(t,aorp0) 1 .
The effect of the next transformation ( C i , O , O ) = (;+u(t,w) ,G+v(t,G),w(t,i)) is, that the transfornied center manifold corresponding to M
0
has a local representation );,;( = (0,O). The function v = s(t,u,w) de0 scribing the transformed center stable manifold corresponding to Mis used in the third transformation
-
_ _ -
-
- -
- - -
(u,v,w) = (u,v-s(t,u,w),w). Then in the resulting system u v
(5)
= =
Nu Pv
+ rl (t,U,V,W)
+
w =
r2(t,u,v,w) r3 (t,u,v,w)
beside the center manifold (u,v) = (0,O) also the center stable manifold'is locally linearized with equation v = 0. All of this implies r 1 (t,O,O,w) = 0, r3(t,0,0,w) = 0, r2 t,u 0,w) = 0 for 1 1 u / ( small. Thus ( 5 ) may be written as and / I
wII
(6)
u = Nu v = Pv w =
+ +
A1 (t,u,v,w) u B2(t,u,v,w) v
+
B1 (t,u
V
A3(t,u,v,w) u + B 3 t,U,V,W) v where the matrices Ak, B 1 are given by 1
Ak(ttu,v,w) :=
(rk)u(t,us vs,w)ds 0
k=l ,3
269
ASYMPTOTIC AMPLITUDE AND PHASE
and
1
B1 (t,u,v,w) :=
I
(rl)v(t,us,vs,w)ds 1=1,2,3. 0 Obviously these matrices are continuous, periodic in t and vanish for (u,v,w) = ( O , O , O ) . Beside the particular form of (6) we shall use two features of the above over all transformation which are stated explicitly. By (u(t),v(t),w(t)) = 2 (x(t)) we denote the transform of the solution x(t) of (1). (7)
a) There exist positive constants r, c, d with the
following properties: If 1 1 xi(t) - x(t,ao,po)lI~r , i=1,2 for all t in a subset S of IR, then the estimate
c 1 1 x 1 (t)-x2(t)lI s 1lXxl(t)) holds on S. Note x(x(t,ao,po)) : 0. (8)
b) There exists a positive
s
p
p
for all telk implies
Z(x2 (t)) / I s dll x1 (t)-x, (t)ll
-
such that 1 1 x(t)-x(t,ao,po)ll u(t)=O, v(t)=O
for all tern.
3rd step: By (4) there exists a sequence ti with lim [ xo(ti) i+m
-
x(ti,ao,po) I
=
o
implying lim (u,v,w)0 ( t1. )= 0 1-t-
where ( ~ , v , w ) ~ ( t:=)
a(xo(t)) is the transform of xo(t). The aim of ) limit this and the next step will be to prove that ( u , v , ~ ) ~ ( thas 0 as t-.
We suppose the contrary, i.e. that f o r any sufficiently smally> 0 there exists a sequence T.+m as i-f- with / I ( u , ~ , w ) ~ (1 1T 2~ )Y . Without loss of generality we may assume that 1 y c c min i r, p , 2 a ( p ) (9) (for the meaning of the constants see ( 7 ) , (8) and (3)) and that there exists a sequence of intervals Ii := (ti,T.) such that = ~0) lim ( u , v , ~ ) ~ ( t iim
11
(12) for all
...
i=0,1,2
( U , V ~ W ) ~ (( 1T=~ )Y
.
>
0
Because of ( 7 ) , (12) and (9) we get
(13) for all i. x(Ti,ao,po) is bounded and thus there exists a subsequence
270
B . AULBACH
’
of Ti such that x(Ti,ao,po) and ec ips0 xo(T;)
T:
is convergent,
The aim of the current step of the proof is to show that uo(T:) and vo (T: ) tend to zero as i >‘x’ , where uo(t) , v 0 (t) denote the respective For this purpose let F be any positive numcomponents of (u,v,~)~(t). 1 ,j)d. By assumption a) the -limit point :x cf ber smaller than ?
t
~
/II
(
xo(t) cannot be an exterior point of G , hence there exists a sequence x(O,aj,pj) of points on G such that x Choose
jo=jo ( b
)
m
-
lim x(O,aj,pj). O j-m such that =
Inferring as in the first step there exists a p i.IR
lim x(T;,a
such that
j,
-
,p ) = x(O,a ,P ) . 7 0 10 10 7 0 Together with (14) and ( 1 5 ) we get 1 fm
(16) for
i
11
-
xo(T;)
x(T;,a l o t p l o
$ )
l
l
T1 ” ( 0
3
s
sufficiently large. Moreover by (13) and (16) we obtain the
estimate
(1
-
x(T;,aolp0)
x(T~,alof~ ) l II o
1 (i
)
for sufficiently large i. Using (3) we get
/I
x(t,aorpo) - x(t,aj , p j ) 0
0
(1
<
P
for all E I R
and (8) shows, that the u- and v-components of rPj 1) 10 0 vanish identically. With (16) this leads to the estimate 5 (x(t,a
(urvrw)(t,jo) : =
11
(urV,W)o(Ti)
-
(O,Otw(Tl,jo)
11
c
for sufficiently large i. Writing again Ti for TI we arrive at lim uo(T.) = 0 , lim v ~ ( T ~= )0. i fa, i+m 4th step: By means of utilizing the special form of (6) we shall show, that lirn wo(Ti) = 0 which together with (17) contradicts (12). i-im ( u , v , ~ ) ~ ( tsatisfies ) on each Ii the system of equations (6) and also the linear system (17)
u = Nu ;= Pv
w =
+ Ti 1 (t)u + B.,
+
(t)v
B2(t)v
A3(t)u
+
B3(t)v
,
ASYMPTOTIC AMPLITUDE AND PHASE
27 1
where Ak(t):= A ( t , ( u , ~ , w ) ~ ( t ) )k=1,3, , Rl(t):= B ( t , ( ~ , v , w ) ~ ( t ) ) , k 1 1=1,2,3. Because cf (11) and the properties of the matrices Ak, B1 we manage by proper choice of y that in particular the estircates
1 1 A1 (t) 1 1 hold true for
0
<
u.
, I1 B2(t) II
<
'A
as small as desired. This implies, that the linear
systems (18)
li
=
[
(19)
G
=
[ P
Pj
+
Al(t) 1 u
and
+ B2(t)
I
v
are uniformly asymptotically stable for positive, negative time respectively. Denoting by Fl(t,s), F2(t,s) the fundamental matrices (F1(t,t), F2(t,t) the respective identity matrices) of (la), (19) componentes of ( u , v , ~ ) ~ ( ton )
t A3(s)uo(s)ds + B3(s)F2(s,t)vo(t)ds. ti ti Doing some estimations and integrations we get from this to an estimate wo(t) = wo(ti) +
11
Wo(Ti) 1 1
-rl
I/
t
f
1
Wo(ti)
11
+ y2
11
Uo(ti) 1 1
+ y3
11
Vo(Ti)
11
with positive constants y 1 r y2, r3. Using (17) and (10) the right hand side and eo ips0 the left hand side tends to zero as i - + c - . As mentioned above this leads to a contradiction and the Theorem is proved.
REFERENCES 1 1 1 Aulbach,B., Behavior of Solutions near Manifolds of Periodic Solutions, J.Differentia1 Equations (to appear) [ 2 ] Coddington,E.A. and Levinson,N.,Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955). [jl
[4
Hale,J.K. and Stokes,A.D., Behavior of Solutions near Integral Manifolds, Arch.Rat.Mech.Ana1. 6 (i960) 133-170.
1 Yoshizawa,T. and Kato,J., Asymptotic Behavior of Solutions near Integral Manifolds, in: Hale,J.K. and LaSalle,S. (eds.), Differential Equations and Dynarnical Systems (Academic Press, New York, London, 1967).
This Page Intentionally Left Blank
QUASIOPTIMAL FINITE ELEMENT APPROXIMATIONS OF FIRST ORDER HYPERBOLIC AND OF CONVECTION-DOMINATED CONVECTION-DIFFUSION EQUATIONS 0.Axelsson a u d I.Gustafsson Mathematical Institute Catholic University Nijmegen, The Netherlands
We show quasioptimal error estimates for bi-lagrangian finite elements of odd degree over quasiuniform meshes, applied on first order hyperbolic and on singularly pertubed second order problems with small enough parameters affecting the highest derivatives. INTRODUCTION Consider the first order hyperbolic problem (1)
k.Vu + cu
where
u
r-
=
E
=
f, x E R,
is given at the inflow part of the boundary, that is, at
r; k * i < 01,
where
4
is the outward pointing normal of
r
=
20.
It is well known that in general we get only an o(hp), t i 4 0 rate of convergence in the L -norm, when we approximate ( 1 ) by Galerkin finite elements based 2 on piecewise polynomial approximations of degree p. For the model equation u + u = 0, with solution extended by periodicity, it was shown by Dupont ( 1 9 7 3 ) t x 2 that for uniform meshes one gets an optimal rate of convergence ( o ( h ) , 11 + 0 ) when one uses piecewise linear approximations. We extend this result to second order problems of the type
(2)
div(-EVu)
+ b-Vu + cu
=
f, E
=
diag(E.1, 0 <
E. < t 1 -
b
xE
R
and to quasiregular meshes (see below) and show that we get an optimal rate of convergence if E is small enough for all piecewise polynomial tensorproduct Lagrangian approximations of odd degree. The influence of downstream boundary conditions is eliminated by using exponential weighting functions in the layer elements as described by Axelsson (1980) or by using Neumann boundary conditions Apart from this trick, we use classical Galerkin approximations. The above result is obviously also valid for the reduced equation where c = 0. The ability to solve the reduced problem effeciently is crucial in the solution of the singularly pertubed problem. For instance, in the calculation of a n asymptotic series approximation of the solution, the most common method used needs repeated solutions of the reduced problem. When solving the reduced problem numerically one has sometimes observed a quasioptimal rate of convergence. In this contribution we explain this phenomenon when using Lagrangian type basis functions. NOTATIONS AND BASIC RESULTS Let the operator
LE be defined by 273
214
0. AXELSSON and I. GUSTAFSSON
where
0
5
5
E~
E
are parameters, bi, i = I , . . .,n are continu-
<< I , i = I,...,n
ously differentiable and
c
is uniformly bounded on
E. r
with sides parallel to the coordinate axes, see Figure 1.
r-
and on
and on
Yo =
r+ =
{x E r;
b.5
{x E r; k.i > O }
=
0)
r+
On the inflow boundary
we have essential boundary conditions, u = y,
we may have essential or natural boundary conditions.
We also assume that we have no layer at ditions on
is defined by a polygon
Yo.
The effect of essential boundary con-
do not influence the solution in the interior of R and can be
It follows that apart from an error ~ ( J E )we , may assume that we have Neumann boundary conditions on r+.
weighed away as described by Axelsson (1980). of
Figure 1.
The subdivision of the boundary for two different directions of the flow.
This can often be obtained by a variable transformation of u tial factor, see Axelsson (1980). Let
v = {v E H 1 (n); v = y on r-ur 0 v0 = {V E H1 (n); v = 0 on r-ur 0 1 , (u,v)
=
IIvII
=
luv do, = I uv ds, R f r+ t (v,v) , IVI =
.
b of
by e.g. an exponen-
275
QUASIOPTIMAL F I N I T E ELEMENT APPROXIMATIONS
The variational formulation of ( 3 ) is aE(uE,v) = (f,v)
V v
E V 0,
av
+
uE
E V,
where aE(u,v)
=
J[ZE.-aU
b.Vu v + cuv]dR
E V, v E V0
V u
lax.ax. 1 L
E V c V E,h h (f,vh) v vh E vh0 c v 0
and the Galerkin approximation u ac(UE,h,vh)
=
.
is defined by
From the assumptions ( 4 ) it easily follows that the bilinear form 0
coercive over V XV
0
,
(5)
aE(v,v)
2
where
0 < p
min(c o,
We let
=
aE(",,)
is
that is,
+ l
p[(v,v)
t/
v E
v 0 , tJ
E
0,
do).
Rh be a mesh consisting of hyperrectangles being constructed by a smooth
mapping of a uniform hyperrectangular mesh defined by the stepsize parameter h. For an example of such a mesh see Figure 2. For less elementary examples we refer to Gordon et al. (1973) and Axelsson et al. (1977). F o r notational simplicity, in the following we assume that already Rh is a uniform hyperrectangular mesh.
Figure 2. An example of the mesh On each hyperrectangle of
,$i
Oh'
we define the usual tensor product Lagrangian
interpolation polynomials of odd degree p, over corner and on regularly placed edge and interior nodes. Since
E~
may be zero, ( 3 ) represents singularly pertubed parabolic and hyperis a first x € R, u = y V € rLou = f V -
x
bolic problems. The reduced problem
Our aim is to order hyperbolic problem. We assume that its solution uO E HPt'(Q). prove an error estimate in L2-norm for the Galerkin method which is quasioptimal. DISCRETIZATION ERROR ESTIMATES Let
Gh be the interpolant in Vh
of
u
over
Rh and let
276
0 . AXELSSON and I. GUSTAFSSON
uE
-
ch,
@
Apparently, uE
-
u
=
nh
(6)
=
~
h
= u
-
E,h
,rlh ~ - Oh
h' and
@
0
h E V h . Further from
aE(uE - u ~ , ~ , v= ~0) V vh we have with
v
h = h '
aE(eh,eh)
" {$ili=l
Let
=
aE(nh,eh)
be a basis for Vh, the space consisting of continuous functions
in the form of piecewise polynomials of degree p, and let 0
sponding basis for Vl,, N
N
{$i)i=l be the corre-
We have
and = 1y.J h*Vn $.dR, /~-V~,el1dQ
h J
J
R
j' of area O(hn) is the support of
where R
j associated with node j.
4
j'
the basis- (and test-) function
Let be the leading polynomial in the Taylor expansion of q h 'h,j b at j. We have and let b . be the value of -
around
j
-J
l/&*VnhBhdQI
(8)
=
R
IEy./[b:Vfi, Ja,-J J
.$. +
,J J
Here, the two last terms are bounded by max/(b
-
.
b.>W $.I h J
-J
=
O(hp + I ) ,
b;(Vnh-Vfih $ 3. )
+
O(hPtltn)Z1yjI
maxIb.*(Vqh - Vfih,j)$jI
(b - hj>V\$j]dQl. since
=
o(hptl) and
x€n J j area(Q.) = o(hn). Now,from Bh = Cyj$j, (Oh). = y we have C ( Y j l = cl(eh)j( J ~j over each element Qe (area(Re) = I ) ( h n ) ) and we easily find by considering ( O h l X G
that
leh\dR
2
e'
(9)
I lehl 1 2 c
-J
area(Re) C Iy. I jEQe J /(ehldf2
R
C
and hence also
area(Re)CIYj
1.
Here and in the following C stands for a constant independent of necessarily the same at different instances.
h, not
277
QUASIOPTIMAL F I N I T E ELEMENT APPROXIMATIONS
From (8) and (9) we have
and by Chauchy-Bunyakowski’s inequality we get from ( 7 )
In the following we let
E =
2
O(h ) . Then the first three terms of the righthand
side of (10, are all o(h2(p+1)) shall prove at first that
and we pay our attention to the last term. We
if p is odd. To illustrate the idea we at first consider linear basis functions in one dimension. We observe that and Vfih,j are and odd functions ‘h,j even and hence (11) follows, respectively with respect to node j . Further, @j is see Figure 3 .
-
Figure 3 . -$I~, th,
and
even
Vfih,j at an interior node
j
For higher order f.e.m. of odd degree p we get a similar cancellation if the basis functions Gj are chosen in a proper way. In Figure 4 we show these functions in the case when V consists of continuous piecewise cubics, i.e. for h p = 3 . We note that the three quadratic functions (solid lines) make up a complete basis for piecewise quadratic f.e.m. Hence, adding one new cubic function at each element gives a basis for Vh. Now all basis functions even (with respect to
are
Vfi . is odd again leads to h. 1 For larger odd values of p we can in a similar way consider a basis consisting of a complete basis for piecewise polynomials of degree p-l plus two functions of degree p (over each element), all of them being even. Observe, however, that since uE,h is independent of the basis actually chosen for Vh, in practice we may use standard Lagrangian f.e.m. basis functions as well. the midpoint of their support) and the fact that (11).
0 . AXELSSON and I. GUSTAFSSON
Figure 4 . Basis functions for Vh, p
=
3.
This result is readily carried over to two (or more) space dimensions by using b i Lagrangian f.e.m. where the basis functions are tensor products of the one-dimensional ones. That we have a similar cniicellation as in the one-dimensional case follows from the fact that the basis functions are even with respect to both space variables x and y and that the interpolation error has the following form; Let
Q
P re
(x,y) be a two-dimensional interpolation polynomial of degree
f(x,y) i.e. Qp,p(xi,~i)= f(xi,yi), i,j = 0.1, R(x,y),
where R(x,y)
=
I aP+' -f(S,y) ( p i - 1 ) : aXpt1
...,P. Then
f(x,y)
=
(p,p)
Q, SP (x,Y)
to +
P P 7 aP+l ll(x-x.) + -.f(x,n) n(Y-Yi) i=o ( p + i ) ! aye+' i=O
Here we do not bother about the last term since it is of order O(hZpf2). For the first two terms we notice that the corresponding first and second terms of the gradient are odd with respect to x and y respectively. Hence for all interior nodes we get cancellation for these terms, that is, (11) is satisfied. Now, consider
in the case n = 2, similarly for n > 2. (For n = 1 vious.) From (11) follows that S
Then in a S
< -
the result below is ob-
279
QUASIOPTIMAL F I N I T E ELEMENT APPROXIMATIONS
1
< 2c , C1 being the constant in (10). Combining this
where we may choose 0 < C
1
with (10) we finally get
1 l'hl I 2
' l'hl
<
2
o(h2(P+1))
-
and hence from (6) I l u c - U ~ , h l /5
llnhll
5 u(hPfl),
/I'hlI
+
i.e. an quasioptimal error estimate.
In the above analysis we have assumed that
is smooth up to the boundary when
uc
we estimated the interpolation error. In general uc solution of the reduced problem and coefficients in - =
an
r+
0 at
has to be
Lo
110
E
=
u o + GE, where
is a layer term.
and given data are such that uo
we readily find that
o(hptl)
that there i s
u
1 lVZE1 I
< _
uo
is the
(We have assumed that
is smooth enough.)
~ ( J Eand )
Since
it follows that
E
for the analysis to be valid. Note that since we have assumed
layer at
r0 , uE
-- uo
at
yo.
NUMERICAL EXPERIMENTS The following testproblems were run for p
=
1 and p
=
3.
_Problem _ _ _ _1 _(one-dimensional): ______ + u; +
-Ell';
with
f
uE=
f
U,(O)
=
1
Ub(l)
=
0
,
0< x < 1
chosen such that the reduced solution is
uo = (1 - 2x)
4
_Problem _ _ _ _2 _(two-dimensional): _______ -EAuE+
b.VuE+
cuE=
in R
f
a u ~ =0 an with
5
that uo
= =
on
[I, It, c = 1 and f (1 - 2x) 4 (1 - 2y) 4
.
=
(O,l)x(O,l)
r+ and the boundary conditions on
r-
chosen s o
-Problem - _ _ 2: The same as P oblem 2 but with u
The errors
u =
E
-
2
t
- xy/2,1 + xy/2], and
u ~ , ~measured , in the discrete I2-norm (I
...,uNIt),
[uI,u2,
1 = [I
2
24
1 ui/N)'
for
uo = sin IT(X +y ) (u) = (
2 -
are given in Table 1 for different values of
N
i= 1 h
and
E.
One observes an o(hP+') behaviour of the error for sufficienly small values of E, which is in agreement with the theory presented in the previous section.
.
280
0 . AXEISSON
Table 1 .
The errors in h, p
and
1
and I. GUSTAFSSON
2-norm for the test problems for different values of
E.
CONCLUSION We have shown a quasioptimal error estimate for first order hyperbolic and for singularly perturbed second order problems with small parameters affecting the highest derivatives. We have used tensor product (in two dimensions bi-Lagrangian) f.e.m. basis functions of odd degree over quasiuniform meshes. In this way we have contributed towards an explanation of when one may get a quasioptimal rate of convergence. REFERENCES [I]
Axelsson, 0. and Ngvert, U., On a graphical package for non-linear partial differential equation problems, i n : Gilchrist, B. (ed.), Information Processing, IFIP (North-Holland, Amsterdam 1977).
[Z]
Axelsson, O., Stability and quasioptimality of Galerkin finite element approximations for convection-diffusion equations, Report 8 0 0 5 , Mathematical Institute, Catholic University, Nijmegen, The Netherlands (1980).
[31
Dupont, T., Galerkin methods for first order hyperbolics: An example, SIAM J. Numer. Anal. 10 (1973) 890-899.
141
Gordon, W.J. and Hall, C.H., Construction of curvilinear co-ordinate systems and applications to mesh generation, Int. J. Numer. Meth. Engng. 7 (1973) 461-477.
ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S. A x e l s s o n , L.S. F r a n k , A . v a n d e r S l u i s ( e d s . ) @ N o r t h - H o l l a n d P u b l i s h i n g C o m p a n y , 1981
HOMOGENEIZATION OF THE EQUATION OF STATIONNARY DIFFUSION I N CYLINDRICAL DOMAINS CAILLERIE Denis LABORATOIRE DE MECANIQUE THEORIQUE ( L . A. 229) U n i v e r s i t 6 P. e t PI. Curie Tour 66 4 p l a c e J u s s i e u 75270 PARIS CEDEX 05 FRANCE
le s t u d y t h e e q u a t i o n s of s t a t i o n n a r y d i f f u s i o n i n a t h i n c y l i n d r i c a l domain o f t h i c k n e s s 2 e , which p r e s e n t s a p e r i o d i c s t r u c t u r e i n i t s p l a n e , t h e o r d e r o f t h e p e r i o d b e i n g E. We s e e t h a t t h e two proc e s s e s : e -. 0 and E -0 do n o t commute. We s t u d y t h e p r o c e s s e -- 0 i r i t h c = A e ( A f i x e d r e a l number) ; t h e n we prove t h a t by making i, go t o i n f i n i t y ( r e s p . z e r o ) , we o b t a i n t h e same r e s u l t as i n t h e 0 then E 0 (resp. E 0 then e 0). process e
-
-
-
-.
1 . INTRODUCTION The s t u d i e d problem i s t h e homogeneization of t h e s t a t i o n n a r y h e a t - e q u a t i o n i n a f l a t c y l i n d r i c a l domain o f R3 which p r e s e n t s a p e r i o d i c s t r u c t u r e i n i t s p l a n e . There a r e t w o small p a r a m e t e r s , one of them e i s t h e t h i c k n e s s of t h e domain, t h e o t h e r E i s t h e s i z e o f t h e p e r i o d of t h e c o e f f i c i e n t s of t h e e q u a t i o n . The purpose o f t h i s work i s t o s t u d y , i n t h e l i m i t , t h e b e h a v i o u r o f t h e s o l u t i o n o f t h i s problem o f d i f f u s i o n when e and c tend t o z e r o , t h i s b e h a v i o u r depends on t h e r a t i o E /e.
-
Different processes a r e considered : (e 0 then E 0 ) o r (E 0 thae 0), l a s t l y e and E t e n d t o z e r o t o g e t h e r , b e i n g p r o p o r t i o n a l ( E = h e , h f i x e d r e a l number). I n t h e f i r s t c a s e , t h e t h i c k n e s s e i s much s m a l l e r t h a n t h e s i z e o f t h e p e r i o d , i n t h e second one i t i s t h e c o n t r a r y , i n t h e t h i r d c a s e , t h e t h i c k n e s s and t h e s i z e o f p e r i o d a r e comparable. Following t h e s e t h r e e p r o c e s s e s , t h e l i m i t problem a r e t h r e e b i d i m e n s i o n a l problems of s t a t i o n n a r y c o n d u c t i o n whose 2 x 2 m a t r i c e s are d i f f e r e n t i n e a c h c a s e . Hence t h e two p r o c e s s e s e -. 0 and t 0 do n o t commute. When e and E t e n d t o z e r o s i m u l t a n e o u s l y , t h e r e s u l t i s i n t e r m e d i a t e , tends t o i n f i n i t y between t h e r e s u l t g o t by t h e t w o o t h e r p r o c e s s e s , i n d e e d , i f % 0 t h e n E -. 0 ) and t h e t h e new l i m i t problem i s t h e same as i n t h e p r o c e s s ( e 0 ) i f A tends t o zero. same as i n ( E 4 0 t h e n e -+
+
-
+
The p r e c i s e s t a t e m e n t o f t h e problem i s g i v e n i n s e c t i o n 2, i n the t h i r d s e c t i o n w e l a y down a p r i o r i e s t i m a t e s and some r e s u l t s a b o u t t h e l i m i t s o f t h e f l u x e s which e n a b l e u s t o s t u d y t h e d i f f e r e n t p r o c e s s e s i n t h e s e c t i o n 4, 5 and 6 . The methods used a r e t h e homogeneization methods which a r e exposed i n [ l ] and [ 5 ] and o t h e r methods used f o r o t h e r problems [ 2 ] [4]. The main r e s u l t s a r e g i v e n by t h e theorems 4.1, 4.2, 5.2, 6.1 and 6.2. These res u l t s w i l l be p u b l i s h e d i n a d e t a i l l e d v e r s i o n [?], t h e n some p r o o f s of t h i s pres e n t p a p e r are j u s t o u t l i n e d , t h e d e t a i l s b e i n g a v a i l a b l e i n t h e p r e v i o u s r e f e r e n c e . 3R 1
D. CAILLERIE
282
2. STATEMENT OF THE PROBLEM Let (r) be a bounded domain o f lR3 w i t h a r e g u l a r boundary t h e c y l i n d r i c a l domain of R3 d e f i n e d as :
Q~
E
=
R'
I
(XI,
x2)
c
w
, 1.~1
<
el
e
".
y, and l e t
be
> o
,eE
We s t u d y a problem of s t a t i o n n a r y d i f f u s i o n i n The 3 x 3 m a t r i x ( a i j ) of cond u c t i v i t i e s i s p e r i o d i c i n t h e v a r i a b l e s ( x l , x 2 ) , t h e p e r i o d b e i n g of o r d e r c for s i m p l i c i t y r e a s o n s , i t does n o t depend on x3, l a s t l y i t i s s u c h t h a t ( e which may c o n s i d e r e d as t h e g l o b a l c o n d u c t i v i t y of Qe, does n o t depend on e . The unknown f u n c t i o n i s supposed t o be e q u a l t o z e r o on t h e s i d e r," ; t h e f l u x e s and r e a r e g i v e n f u n c t i o n s g+ and -g-. through r: The t i c k n e s s of Qe is n o t c o n s t a n t and by t h e change of v a r i a b l e s : of IF?. z
n
L
X
a
a
=
z)
1, 2
z3
r+
=
is
pe
:$),
-
transformed i n a f i x e d domain
x3/e
Then: ' l ( r e s p . r,, r becomes ( r e s p . I'-, r o ) . The p r e c i s e s t a t e m e n t of t h e problem i s g i v e n i n '2. F i r s t we d e f i n e t h e c o n d u c t i v i t y m a t r i x : 2 Let a i . i , j = 1 , 2 , 3 be nine p e r i o d i c bounded f u n c t i o n s d e f i n e d i n !R , t h e Y{) x 0 , Y2 [). They a r e supposed t o s a t i s f y t h e c o e r c i period i e i n g Y = ( veness r e l a t i o n :
(1
10,
3
m
>
0 such t h a t
mwiw.
V w
=
E
( w l , w 2 , w3)
lR3
<
a . . w . w . a . e . in y (2.1) 1J 1 J Here and f u r t h e r we u s e t h e convention o f r e p e a t e d i n d i c e s . The L a t i n i n d i c e s den o t e t h e i n d i c e s 1 , 2 , 3 : t h e Greek ones d e n o t e 1 and 2 .
Then we d e f i n e L
a 1. J ( z l , z 2 )
a I j ( z l , z 2 ) and =
ec a 1J . . ( z 1 ' 22) :
aij ( z l / € , z ~ / E ) i, j
=
I , 2, 3
The v a r i a t i o n n a l f o r m u l a t i o n o f t h e problem i s :
283
STATIONARY DIFFUSION I N C Y L I N D R I C A L DOMAINS
If
i s supposed t o be l e s s t h a n 1 , by t a k i n g
e
( 2 . 1 ) , i t i s e a s y t o prove t h a t : Y E H~ n U, < ae.c.
v
wi
12
i
vi
;d;
%) (Id
w = ( l r , , IT2,
a. o , in
in
IR')
z.
Then a c c o r d i n g t o t h e theorem o f Lax Milgram, t h e r e i s one unique s o l u t i o n o f t h i s problem, i t i s denoted by uec. The purpose o f t h i s work i s t h e s t u d y o f t h e l i m i t of t o z e r o , one a f t e r t h e o t h e r or t o g e t h e r .
ueE
when
e
and
c
tend
3 . A PRIOR1 ESTIMATES AND PRELIMINARY RESULTS 3.1 ---___-----------A priori estimates
I n (2.3) we t a k e v = i s a norm o n t h e space
[ c
u e c , we u s e ( 2 . 1 ) and we n o t e t h a t V, t h e e s t i m a t e s (7.1) follows :
JQ
(e)? dz I-? i'*
< c on
E
C
.
Then, as
V
(3.1)
d e n o t e s d i f f e r e n t c o n s t a n t s which do n o t depend on
e
neither
(Q) i s a H i l b e r t s p a c e , we may s t a t e t h e f o l l o w i n g lemma :
[LEMMA 3.1
I
When e o r which t e n d s t o
tqnd t o z e r o we may e x t r a c t a subsequence ( s t i l l denoted by u*. weakly i n V ( a ) .
uec)
I f , f o r i n s t a n c e , we f i x c and we make e tend t o z e r o , t h e weak l i m i t of u e c depends on b u t i t i s bounded by t h e same C as i n ( 3 . 1 ) , t h e n we may e x t r a c t from i t one o t h e r subsequence which converges weakly.
Three p r o c e s s e s w i l l be c o n s i d e r e d , t h e l i m i t s w i l l be denoted by : ueE
+
uef
u*' *e
e
when
c
+ u f A when
e
u
uec
when
-,0
t f i x e d and
-.
e f i x e d and
0
and
E
+
0
u*'
-+
u*'
when
&
-. 0
u
4
u**
when e
-. 0
0 =
*e
A e
A f i x e d real number.
Remark ------
The u n i q u e n e s s o f t h e s e l i m i t s which w i l l be proved l a t e r , a l l o w s t o s a y t h a t t h e whole sequences converge. I Moreover we may e a s i l y have one o t h e r e s t i m a t e s :
< c And we have t h e lemma 3.2. rLEMMA 3.2 The l i m i t s u*' 6- U*L = - = 6 u*x bZ
3
az
3
and
u * ~s a t i s f y :
0
(7.2)
D. C A I L L E R I E
284
LThen
u*'
and
may be i d e n t i f i e d w i t h f u n c t i o n s of
u*'
HA
ee
(@).
(0)
a r e bounded i n L2 which converges t o ai
tends t o zero, the ( s t i l l denoted by
and we may x t r a c t weakly i n L ( Q )
5
The proof of t h i s lemma i s s t r a i g h t forward w i t h t h e u s e of t h e e s t i m a t e s ( 3 . 2 ) and o f t h e p r o p e r t i e s of t h e a . . ' s . 1J Notation --------
The n o t a t i o n s of t h e l i m i t s o f
aah : t:'
=
ueE
J:,
a
m
a r e used f o r t h e l i m i t s o f
dZ7
a
.
1,
2
satisfy : d w
Proof -----
V v € Hl ( w )
This lemma i s proved by t a k i n g v n o t depending on z and i d e n t i f i e d w i t h a f u n c t i o n of Ho ( w ) i n (2.3), t h e n by p a s s i n g t o t h e l i3m i t e and E-. 0 w i t h ' = h e .
*E
~~~~i~~ a3
i s e q u a l to z e r o .
Proof -----
a*f 'p dz
We prove t h a t
t h i s i s c a r r i e d o u t by t a k i n g the l i m i t
e
i s e q u a l t o z e r o f o r any f u n c t i o n
v
= 13q(zl, z
-. 0 a f t e r h a v i n g m u l t i p l i e d by
4. HOMOGENEIZATION
2'
s ) ds
of
a
('2) ;
i n ( 2 . 3 ) and by going t o
e.
rn
OF AN INFINITELY FLAT BODY
*
4 * 1 !!!?:~2!?_E:~eZ!b_Y--!I n t h i s s e c t i o n , f i r s t e t e n d s t o z e r o and we g e t a t w o dimensionnal d i f f u s i o n
problem, t h e n we homogeneize which i s done by making From (7.3) and ( 2 . 2 ) we have : ec Q B uet ( y3 -d )U -e E 1 = 3a d z a e a e g 3 From (2.1), i t i s e a s y t o s e e that73a;3 t o d i v i d e by a' 33'
ae=
-
Then i n (2.3) we t a k e
v
et L 3 --
a~ 3a
E
tend t o zero.
d z
(4.1)
'a a' is3~ounde~3below and t h a t i t i s j u s t i f i e d
n o t depending on
z3
and we u s e (4.1) :
285
STATIONARY DIFFUSION I N CYLINDRICAL DOMAINS
We t h e n p a s s t o t h e l i m i t and
-.
0, as
e
'
' Li z,
tends t o
t e n d s t o z e r o (lemma 7 . 5 ) w e a k l y %
(I"
3
au* p d Z
L2 (61) we h&e
=
I, 2
(lendla
3.1)
:
/,
Ye d e f i n e
As
b
=
a9
2 (aaB
-
d o e s n o t depend on
u*'
i.
U*L
Lj
(lemma 3 . 2 ) we have :!
z7
v
(4.7)
'
s a t i s f y a two-dlmensional s t a t i o n n a r y d i f f u s i o n problem, a g e n e r a l s t a t e Then u* ment of which may be g i v e n by : For a given 2 x 2 matrix 011 u v d w =
S,
t g-)
(gi
b'
1
u E Ho ( w )
E
Y v
v a w
;::
such t h a t :
P (d).
he denoted by Study o f
) find
d = (d aP
(w)
= (big)
2
2
Let (u,, u 2 ) be a n element of R , we t a k e i n ( 2 . 1 ) w = L, , u2 ' - aJ3 u u ) and we have : I 2 2 a. e . i n ( z l , 2 m ( u +u2) < u b z2) 1 a aP "P Then t h e f i r s t s i d e of ( 4 . 3 ) i s a n c o e r c i v e b i l i n e a r form and a c c o r d i n g t o t h e theorem of Lax Milgram, we have t h e theorem 4.1. t o zero,
t e n d s t o u*' weakly i n V ( a ) , u*' does n o t dew i t h a f u n c t i o n of Ho ( w ) i s t h e unique s o l u t i o n of
ue'
______________-
4.2 Homogeneization
The c o e f f i c i e n t s b' a r e periodic, the period being aP z e r o , u s i n g t h e methods exposed i n [ l ] or i n [ 5 ]
.
Xe d e f i n e some f u n c t i o n s
lowing problems.
x
Find and
r Yv
I
(V
=
I , 2) E
d Y =
We now d e f i n e 1
GI
x
Y
c Y . We make
C
tends t o
(y = 1 , 2 ) which a r e t h e unique s o l u t i o n s of t h e f o l -
W (Y) =
Iv
H1
(Y)
1
v
I
=
'a =Y
'a=,
a = 1, 2
such t h a t :
c
c = ap mes Y The m a t r i x c =
a, P
=
1,
2.
and we have t h e theorem 4.2.
2 86
D. CAILLERIE
F.
tends t o zero, t h e problem
u*' tends t o P (c)
u*'
weakly i n
HA ( w ) ,
i s t h e unique
u*l
5. TWO-DIMENSIONAL APPROXIMATION OF A HOMOGENEIZED DOMAIN 5.1 Homogeneization of t h e three-dimensional domain ............................................... I n t h i s s e c t i o n , we f i r s t homogeneize t h e three-dimensional domain, we u s e t h e metends t o zero. thod exposed i n [ I ] o r [ 5 1 t o sturly t h e l i m i t when We d e f i n e t h r e e f u n c t i o n s T , ~ (k = 1 , 2 , 7 ) s o l u t i o n s o f t h e problems
Then t h e c o e f f i c i e n t s
And t h e
p.
3 m1 >
s
. I
1J
0
p;;
a r e d e f i n e d by :
satisfy : such t h a t
w
V
=
( w , , w2, w3)
E
m'
R'
-----Remark
w
i
w. i
<
p . . w. w. 1J 1 J
We have t o d e f i n e t h e c o e f f i c i e n t s p e , which a r e t h e homogeneized c o e f f i c i e n t s o f eE t h e a . .,. b u t i t i s e a s y t o prove t h a t J t h e s e c o e f f i c e n t s may be deduced from t h e
.
There i s between
p. . 1J
t o zero,
and
p e , t h e same r e l a t i o n as between 1J
ueE t e n d s
u*e
to
weakly i n
a!, 1J
*e V (P), u
and
a". 1J
i s t h e unique
_--____________-_-___________
5.2 Two-dimensional l i m i t problem
A s f a r as t h e dependance o n e i s concerned, t h e s i t u a t i o n i s t h e same as i n t h e s e c t i o n 4, t h e p e * l s r e p l a c i n g t h e ae .Is. Then we have t h e theorem :
15
1J
THEOREM 5.2 t e n d s t o z e r o , u*e t e n d s t o uX2 lweakly i n V (n), u * ~ does n o t depend and, i d e n t i f i e d w i t h a f u n c t i o n o f H (w), i s t h e unique s o l u t i o n of t h e problem P (9) where q i s t h e 2 x 2 m a t k c d e f i n e d by :
When
on
L
e
e
STATIONARY DIFFUSION I N CYLINDRICAL DOMAINS
Remark _--_-and uX2 a r e t h e s o l u t i o n s o f t w o similar problems of two-dimenThe f u n c t i o n s u*' s i o n a l s t a t i o n n a r y d i f f u s i o n , but t h e c o n d u c t i v i t y m a t r i c e s of which have t w o d i f f e rent expressions.
It may be proved t h a t t h e s e m a t r i c e s a r e a c t u a l l y d i f f e r e n t , i n t h i s a i m we may s t u d y ( t h i s i s done i n [7]) t h e c a s e when t h e a . .Is depend o n l y o n mne -.-ariable (zl f o r instance). =J a 6 . CASE WHERE
c
AND
e
ARE OF THE SAME ORDER
*1 6.1 Two-dimensional problem f o r u -______-__----------------------
Let L
1.
E
be e q u a l t o
he, where
A
i s a p o s i t i v e r e a l number. (we keep t h e n o t a t i o n
From t h e lemma 3.1 we know t h a t ueC cog e r g e s t o a c e r t a i n a d we have t o i d e n t i f y t h i s f u n c t i o n u
K.
-THEOIlEM
u*'
weakly i n
V (Sj
6.1
e and c t e n d t o z e r o (E = h e , h non z e r o r e a l number, u e E c o n v e r g e s t o ( a ) ; u*' does n o t depend o n z 1 ' t h e n i t may be i d e n t i f i e d w i t h weakly i n a f u n c t i o n of Ho ( w ) and i t i s t h e unique s o l u t i o n of t h e problem F ( r A )where r i s d e f i n e d by :
y
-
L
2aP
=
1
a, $ 0
,
=
The p r o o f o f t h i s theorem i s very n e a r o f the p r o o f described i n [ i s not given h e r e , i t may be found i n [3]. I t r e l i e s on t h e lemma and on t h e f o l l o w i n g one, t h e p r o o f o f which i s elementary.
1, 2
11
or [7.1],
[ 51 [3.4]
and
288
D. CAILLERIE
LENMA 6.1 l e t
$ (c.'
be a p e r i o d i c f u n c t i o n of
(z,, z2, z7)
mesY c
=
6 Y7) d
( Y l , Y2'
(z,/t,
", 1
2
loc
z2/E,
d y2
(R2 x e3)
z
weakly i n
31, E
1 [ ) o f p e r i o d y , then
Q converges t o :
L2 (62).
Remark This "homogeneization" i s d i f f e r e n t from t h e ones o f s e c t i o n 4 and 5 , indeed t h e o n e s , f o r C$' is f u n c t i o n C$ depend on zz ; i t i s a l s o d i f f e r e n t o f t h e e . on y_ 2 + 1 n o t r e q u e s t e d t o be e q u a l dn t h e upper and lower f a c e s of 5 and y j = - 1 ) b u t o n l y on t h e l a t e r a l s i d e s . W >
6.2 ................................................... L i m i t s of u*" f o r A t e n d i n g t o i n f i n i t e o r z e r o .
It i s c l e a r t h a t t h e f u n c t i o n s Cph depend on A and, t h e r e f o r e t h a t t h e r UB I t may be proved t h a t ( s e e u*' depend on i t t o o :
and
.
[?I)
t o i n f i n i t y ( r s p . z e r o ) , u (rapA) t e n d s t o u*l ( r e s p . u* ) weakly i n Ho ( w ) .
5
cR?
'
's
( r e s p . qa3 ) and
REFERENCES
[I ]
BENSOUSSON A., LIONS J . L . , PAPANICOLAOU G . , p e r i o d i c s t r u c t u r e s ( T o r t h H o l l a n d , 1978).
[2]
CAILLERIE D . , E q u a t i o n aux d 6 r i v Q e s p a r t i e l l e s & c o e f f i c i e n t s p 6 r i o d i q u e s dans d e s domaines c y l i n d r i q u e s a p l a t i s , CRAS P a r i s , t . 290, s k r i e A 143-146
Asymptotic a n a l y s i s f o r
[3]
C A I L L E X E D . , HomogQneisation d e s Q q u a t i o n s d e l a d i f f u s i o n s t a t i o n n a i r e dans d e s domaines c y l i n d r i q u e s a p l a t i s , J o u r n a l de Mathkmatiques p u r e s e t appliquees. (TO appear).
[4]
PHAM H . , SANCHEZ-PALENCIA E . , Phknombnes de t r a n s m i s s i o n h t r a v e r s d e s
[5]
couches minces d e c o n d u c t i v i t k klev6e. J o u r n a l of Math. Anal. and Appl. vol.' no 2 (1974) 284-709. SANCKEZ-PALENCIA E . , Non homogeneous Media and v i b r a t i o n t h e o r y , L e c t u r e s N o t e s i n p h y s i c s , ( S p r i n g e r - V e r l a g 1980).
ANALYTICAL AND NIJMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S . A x e l s s o n , L . S . F r a n k , A . van d e r S l u i s ( e d s . ) @ N o r t h - H o l l a n d P u b l i s h i n g Company, 1981
SINGULAR PERTURBATIONS OF AN ELLIPTIC OPERATOR WITH DISCONTINUOUS NONLINEARITY L.S.
Frank and E.W.
van Groesen
Mathematical Institute University of Nijmegen The Netherlands.
A family of nonlinear elliptic boundary value problems, affected by the presence of two positive parameters X and
E,
is considered. These problems arise in the
theoretical treatment of membranes with enzymic activity, realistic situations a small parameter compared to A . mulation of the reduced problem which corresponds to of X such that for X
there exists a critical value X are regular and for X > A estimates for X
E
being in several
We give an appropriate for0. On the other hand
E = 2
X
the problems with
E
> 0
singular perturbations of the reduced problem. Some
as a functional of the boundary function
explicit formula for the first variation of X
0 are derived and an
with respect to I$ is established.
Newton-Kantorovich iterative procedure is applied in order to approximate the solutions and to derive some asymptotic formulae. This procedure gives rise to an interesting class of linear coercive singular perturbations with discontinuous coefficients.
1.
Introduction. - In the theoretical treatment of membranes with enzymic activity
one uses, as a mathematical model, a quasilinear second order elliptic operator, in order to describe the steady state distribution of the substrates' concentration. The nonlinearity of this operator is affected by the presence of two positive parameters:
E,
the so-called reduced Michaelis constant, and h , the latter being
connected with the ratio of initial concentrations of the enzyme and the substrate. For several realistic membranes For a given
E
>
E
is very small compared to
(see C 9 l ) .
0, the mathematical model mentioned above fits the classical
framework of Frechet-differentiable nonlinear elliptic operators (see, for instance,
1 2 1 , 151 and the literature there). One can also view this model as a family of perturbations of some reduced nonlinear elliptic operator corresponding to
E
=
0. It turns out that the reduced
operator has as its nonlinearity a discontinuous function of the concentration. Moreover, there exists a critical value X
of the parameter X such that for I? 5 X
the original problem is a regular perturbation of the reduced one, while for h > hc it becomes a singular perturbation, characterized by the presence of an
internal boundary layer (located in some neighbourhood of a free boundary, tightly 289
L.S. FRANK and E.W. VAN GROESEN
290
connected w i t h t h e reduced problem ( s e e [ 8 1 ) ) . To be more s p e c i f i c , w e c o n s i d e r t h e f o l l o w i n g boundary v a l u e problem
01'
:
(1.1)
nO
A u (x') = E
$(XI)
au
E
I
where U c l R n i s a bounded domain w i t h Cm-boundary aU, A i s t h e L a p l a c e o p e r a t o r ,
II
0
i s the r e s t r i c t i o n operator t o
au,
$ : aU
+
l R + i s a smooth s t r i c t l y p o s i t i v e
a:
f u n c t i o n on t h e boundary. The problem
h a s been p r e v i o u s l y s t u d i e d by s e v e r a l a u t h o r s ( s e e 131 and
a:,
the references there). Along w i t h
(1.2)
where
Qch,
x+(u) (x) i s
aA
w e c o n s i d e r t h e reduced problem
A A u (x) rl
= A
X+(u
A
)
(x)
A
0
u ( x ' ) = $(XI)
:
X € U
I
,
E
au,
I
t h e c h a r a c t e r i s t i c f u n c t i o n of t h e s e t E ( u ) = {x c
The e x i s t e n c e and u n i q u e n e s s o f s o l u t i o n s uA and u A t:
t h e problems
a%
u(x) > 0 ) . and
r e s p e c t i v e l y , are proved and t h e r e g u l a r i t : of u A and uA i s i n v e s t i g a t e d . A t o uA ( a s E++O) i n Holder .;aces C l r a ( G ) , Va E C 0 , l )
The convergence o f u
and t h e Sobolev s p a c e H (U) i s e s t a b l i s h e d , a s w e l l as t h e estimate of t h e 1 A d i f f e r e n c e u - uA i n H ( U ) i s g i v e n as a f u n c t i o n of t h e p a r a m e t e r E ( s e e a l s o 1 C31 where s i m i l a r r e s u l t s a r e o b t a i n e d w i t h a d i f f e r e n t i n t e r p r e t a t i o n of u a s a
x
s o l u t i o n t o some v a r i a t i o n a l i n e q u a l i t y ) . Furthermore,
a:problems:) aA, .
some monotone i t e r a t i o n s and t h e Newton-Kantorovich
are used i n o r d e r t o approach t h e s o l u t i o n s t o t h e problems
procedure
and
( s e e a l s o [ 3 1 where t h e monotone i t e r a t i o n s are a p p l i e d t o t h e
These i t e r a t i v e p r o c e d u r e s g i v e r i s e t o an i n t e r e s t i n g c l a s s of l i n e a r c o e r c i v e singular p e r t u r b a t i o n s with discontinuous c o e f f i c i e n t s (see
C71
f o r t h e c a s e of
m
coercive s i n g u l a r p e r t u r b a t i o n s with C - c o e f f i c i e n t s ) . Some e s t i m a t e s a r e g i v e n f o r t h e c r i t i c a l v a l u e A
of t h e p a r a m e t e r A a s a
f u n c t i o n a l o f t h e boundary f u n c t i o n $ > 0 and an e x p l i c i t formula i s d e r i v e d f o r t h e f i r s t v a r i a t i o n 6 $ h c ( @ ) of A c ( $ ) .
I n t h e s p h e r i c a l l y symmetric c a s e a n
e x p l i c i t formula i s g i v e n f o r t h e s o l u t i o n ux o f
are d e r i v e d f o r t h e s o l u t i o n ux E
of@,
V
1 >
ax
and some a s y m p t o t i c formulae
0.
The r e s u l t s i n t h e p a r a g r a p h s 4 , 6 and Theorem 1 . 2 are due t o t h e second author.
in
[el.
The d e t a i l e d p r o o f s of t h e r e s u l t s p r e s e n t e d i n t h i s p a p e r c a n b e found
29 1
E L L I P T I C OPERATION WITH DISCONTINUOUS N O N L I N E A R I T Y
The reduced problem.
2. E
E
IR
-
Throughout t h i s p a p e r
( w e u s e t h e s t a n d a r d n o t a t i o n IR
denotes a p o s i t i v e parameter :
E
f o r t h e p o s i t i v e h a l f l i n e ) . We i n t r o -
duce a l s o t h e f o l l o w i n g n o t a t i o n . For a g i v e n f u n c t i o n u
U
+
0 -
IR, u t C ( U ) ,
-
r e s p e c t i v e l y , d e n o t e t h e f o l l o w i n g Sets i n U:
E + ( u ) , E O ( u ) and E - ( u ) ,
I
E+(u) = { x E
u ( x ) > 01, E o ( u )
1
E - ( u ) = tx i
x+(u), x 0 (u) and x-(u)
:
=
u(x) < 0 )
{x
E
U
.
I
u ( x ) = 01,
a r e t h e c h a r a c t e r i s t i c f u n c t i o n s of t h e s e t s E (u), E O ( u )
and E- (u), r e s p e c t i v e l y . F o r m a l l y , w e s e t up t h e boundary v a l u e problem
h
where h
E
=
X x+(u) (x)
u (x')
=
$(x')
and $ : aU
+
IR+ i s a g i v e n smooth s t r i c t l y p o s i t i v e f u n c t i o n on
A u (x)
(2.1)
h
n0 iT7
i n the following fashion:
X € U xi
au
t h e boundary :
(2.2)
$
cm(au),
E
ah.
4
>
0.
We need t h e f o l l o w i n g d e f i n i t i o n i n o r d e r ' t o g i v e a r i g o r o u s i n t e r p r e t a t i o n of t h e problem
ah,
D e f i n i t i o n 2 . 1 . - A f u n c t i o n uX s o l u t i o n o f t h e problem
i f uh
E
+ IR
:
Co
i s s a i d t o be a d i s t r i b u t i o n a l
(GI, Il0
u(x') = @ (XI), V x'
E
aU,
and t h e
d i f f e r e n t i a l equation i n (2.1) i s s a t i s f i e d i n t h e d i s t r i b u t i o n a l sense.
0
Using s t a n d a r d Sobolev embedding theorems and r e g u l a r i t y r e s u l t s f o r l i n e a r e l l i p t i c boundary v a l u e problems (see t i o n t o t h e problem
Qc"
in its definition:
[ 1 1 ) , one shows t h a t
a d i s t r i b u t i o n a l solu-
h a s , i n f a c t , some more r e g u l a r i t y t h a n t h e o n e , r e q u i r e d
X - L e t u b e a d i s t r i b u t i o n a l s o l u t i o n of t h e problem
Proposition 2.2.
O'Lh
w i t h $ s a t i s f y i n g ( 2 . 2 ) . Then u
X
E
c
l,a -
( U ) , Va E
C0,l).
0
The n e x t p r o p o s i t i o n i s proved by t h e c l a s s i c a l minimum-principle argument. h P r o p o s i t i o n 2.3. - L e t u (x) b e a d i s t r i b u t i o n a l s o l u t i o n of w i t h i$
mh
satisfying (2.2). h
Then
u (x)
2
0
,
V X E U .
0
One e a s i l y c h e c k s t h a t f o r h > 0 s u f f i c i e n t l y s m a l l t h e s o l u t i o n uA to&' h loses
i s , i n f a c t , s t r i c t l y p o s i t i v e . I t t u r n s o u t t h a t f o r X l a r g e enough u
.!
t h i s p r o p e r t y and becomes o n l y n o n n e g a t i v e on h P r o p o s i t i o n 2.4. - L e t u be a d i s t r i b u t i o n a l s o l u t i o n of f y i n g ( 2 . 2 ) . Then t h e r e e x i s t s a c r i t i c a l v a l u e A
ah
with $ satis-
> 0 of h such t h a t
L.S. FRANK and E.W. VAN GROESEN
292
One proves this last proposition using the following argument. Let @(x) be the harmonic function on U such that Il
0
@(XI) = $(XI), and denote by V ( x ) the solution
of the Poisson's equation: -A V(x) = 1, V x tion: I[ V(x') = 0, V x '
A
0
au. Then for
E
E
U, with homogeneous boundary condi-
h > 0 sufficiently small, the function
MA.
u (x) = @(x) - A V(x) is a distributional (in fact, classical) solution of
For the critical value A
one has the formula:
It is obvious that A = A ($,U) is a functional of the boundary function $ and the c c shape of the domain U. For A
A
the function u (x) = @(x) - A V(x) is nonnegative and represents
A
5
the unique solution of
a'.
m , whereas for A A
>
A
O(x)
C'
- A
V(x) becomes negative
on some subset of U with a positive measure, so that it can no longer be a solu-
.
A
Hence, one must have: meas E ( u ) > 0 for V A > A 0 We end this section by exhibiting an explicit formula for :he
tion of
the case of a spherical symmetry of the problem U = B1 = {x
1
IRn
E
ac". Let
1x1 < 1) and $(XI) = 1, V x'
solution uA in
aB1 (The case when U is a ball
E
of a given radius R and $ is a positive constant $ on a B can be reduced to R 0 R the normalized problem with R = 1, $o = 1 by an appropriate dilatation of the
B
independent variables x and the corresponding scaling of the function u A 1 . The uniqueness of the solution ux of
A
symmetry of u (x) with respect to x
Ac
(2.4) (2.5)
=
A c (1,Bl )
A
u (x) = 1 -
A -
=
2n
@
(see
3) guarantees the spherical
§
B1, so that one immediately finds:
E
,
(1 - 1x1
2
AC
)
for
A
2
A
C
and (2.6)
A
u (x) = c1
-
A
-{l-Ixl
2
+
2 n-2 6n (l-lxl2-n)}I
H(lx1-6)
for
A
>
A C'
AC
the parameter 5
E
C0,ll in (2.6) being the solution of the algebraic equation:
(For n = 2, the formulae ( 2 . 6 1 , expressions as n 3.
+
2)
(2.7) are to be replaced by their limiting
.
Existence and uniqueness of the solutions to the perturbed and reduced problems
via a minimization formulation.
-
E L L I P T I C OPERATION WITH DISCONTINUOUS N O N L I N E A R I T Y
293
= max{u,O} and d e n o t e
Let u
U
( u ) = -2-
f
(3.1)
E
t
f E : IR
'
U
-f
.
IRt
With t h e n o t a t i o n ( 3 . 1 ) w e rewrite t h e boundary v a l u e problem
fi;
mentioned
i n t h e i n t r o d u c t i o n as f o l l o w s :
nO
Just as
u"(x') = $ ( x ' )
&:
E
mx,
w e c o n s i d e r t h e problem
boundary f u n c t i o n s
0
satisfying (2.2).
au. o n l y f o r p o s i t i v e v a l u e s o f h and
We u s e t h e r e l a t e d m i n i m i z a t i o n problems i n o r d e r t o p r o v e t h e e x i s t e n c e and u n i q u e n e s s of t h e s o l u t i o n s ux and uA t o t h e problems
at ax and
respectively.
I t i s n a t u r a l t o s t a t e t h e c o r r e s p o n d i n g m i n i m i z a t i o n problems i n a H i l b e r t
s p a c e f o r m u l a t i o n . W e u s e t h e s t a n d a r d n o t a t i o n H (U) f o r t h e Sobolev s p a c e o f t h e f u n c t i o n s u such t h a t u space of functions u
-
Let @ : U
2
I Vul
E
H (U) whose t r a c e on
E
1
L2 ( U )
au
m=
{O}
I t i s obvious t h a t F E
E
11 ( U ) b e i n g t h e sub-
vanishes i n H
1
4
(au). E
aU.
t h e following hyperplane i n H ( U ) :
1
.
+
L e t F ( u ) be t h e p r i m i t i v e f u n c t i o n o f f
v
l o , H1 (U) c
IR b e t h e harmonic f u n c t i o n s u c h t h a t llo @ ( x ' ) = $ ( X I ) , V x '
-f
W e d e n o t e by
(3.3)
L (U),
E
:
IR
+
IR
(u) t h a t v a n i s h e s f o r u
= 0:
i s a n o n n e g a t i v e convex f u n c t i o n o f u
E
IR,
R+. Along w i t h F E ( u ) , w e c o n s i d e r t h e f o l l o w i n g f a m i l y o f f u n c t i o n a l s
DA :
+
(3.4)
IR+:
x
D E ( u ) = U/
1% lVuI2 +
h FE(u)ldx.
F u r t h e r m o r e , l e t q be t h e f u n c t i o n a l
(3.5)
q(u) =
u/
where, as p r e v i o u s l y , u
u+(x)dx
= max{u,O}.
We define a functional D
(3.6)
x
D (u) =
,
,I
x
:
-f
I R + as f o l l o w s :
{ % lVu(x)12 t xu+(x)}dx.
h
and Dx are t h e sum of t h e s t r i c t l y convex, lower s e m i c o n t i n u o u s , c o e r c i v e 0 2 f u n c t i t n a l D (u) = i 4 IVu(x) dx on and convex, non-negative lower s e m i -
Both D
U
I
a7i!
294
L.S.
FRANK and E.W.
VAN GROESEN
J A F E ( u ( x ) ) d xand u.r X u ( x ) d x , r e s p e c t i v e l y . U Hence, u s i n g s t a n d a r d Convex A n a l y s i s methods one g e t s t h e f o l l o w i n g r e s u l t :
continuous functionals Lemma 3 . 1 .
-
::
A
The f u n c t i o n a l DX ( r e s p e c t i v e l y , D ) h a s f o r
X
( r e s p e c t i v e l y , uX
6
p r e c i s e l y one c r i t i c a l p o i n t
t
IR+,
E E
m). E
IR+ 0
Using a s t a n d a r d argument, one checks t h a t t h e c r i t i c a l p o i n t uX i s a s o l u -
as w e l l , V
t i o n of t h e p r o b l e m m ; ,
B+,V X
E t
E
IR+. Indeed, f o r E € E I R + t h e
f u n c t i o n a l DA i s F r e c h e t - d i f f e r e n t i a b l e and i t s f i r s t v a r i a t i o n y i e l d s
a:.
Consequently, one h a s t h e f o l l o w i n g r e s u l t :
-
Theorem 3.2.
The c r i t i c a l (minimal) p o i n t uX
of t h e p r o b l e r n f l i ,
V
Remark 3.3. - For V p o s i t i v e on
fi
IR+, V X
E E
IR+, V A
E
E
E
of Da i s t h e o n l y s o l u t i o n
E
0
B+. t
X
IR+, t h e s o l u t i o n u ( x ) o f
is s t r i c t l y
A
(see P r o p o s i t i o n 4 . 1 ) . Because u ( x ) > 0 , c l a s s i c a l r e g u l a r i t y
r e s u l t s f o r l i n e a r e l l i p t i c c o e r c i v e boundary v a l u e problems (see c11) y i e l d :
Since t h e functional q
:
%7
-f
IR+ i s not Gateaux-differentiable,
there is
a need f o r some non-standard t e c h n i q u e i n o r d e r t o prove t h e e q u i v a l e n c e between A Indeed, one h a s t o use and t h e minimization problem f o r t h e f u n c t i o n a l D
.
some r e s u l t s i n Convex A n a l y s i s o f s u b d i f f e r e n t i a b l e f u n c t i o n a l s ( s e e , f o r i n s t a n c e , [ 4 , 6 1 ) and t o a d j u s t i n a s u i t a b l e way t h e t e c h n i q u e t h e r e i n o r d e r t o prove t h a t t h e minimal p o i n t uX of DX i s a s o l u t i o n of f a c t t h a t t h e c o n v e x i t y and c o n t i n u i t y o f q
Qc^.
E s p e c i a l l y , one h a s t o use t h e
: L2(U)
-f
-IR+,
guarantee i t s subdiffe-
rentiability. Denote by a q ( u ) t h e s u b g r a d i e n t of q a t t h e p o i n t u. Then t h e mapping:
u
-f
a q ( u ) i s a maximal monotone m u l t i v a l u e d o p e r a t o r and a q ( u ) c L (U) i s a s u b s e t 2
d e f i n e d by t h e formula: aq(u) =
{x
(u)
+
u
j
:
U-.
Hence, t h e unique minimal p o i n t uA o f DX
IR, o
:
5 u
-f
s
x,(~)}
I R + h a s t o b e a s o l u t i o n of t h e
multivalued operator equation: A X aD ( u ) 3 0 ,
X
i . e . u ( x ) must s a t i s f y t h e i n t e g r a l i d e n t i t y : (3.8)
+ -
w i t h some f u n c t i o n a : U (3.9)
A<(X+(U A )
+
0 5 u(x) 5
+
u) ,v>
=
0
,v
v
E
+)
1R which s a t i s f i e s t h e i n e q u a l i t i e s :
x0 ( uA (x)) ,
Using ( 3 . 9 ) , (3.9) one shows t h a t
v
x
E
i.
.
295
E L L I P T I C OPERATION WITH DISCONTINUOUS NONLINEARITY
(3.10)
a ( x ) :0 ,
V x
E
U
.
A Hence, u ( x ) i s a d i s t r i b u t i o n a l s o l u t i o n of
ax.
On t h e o t h e r hand, i t i s o b v i o u s t h a t a d i s t r i b u t i o n a l s o l u t i o n of A c r i t i c a l p o i n t of D
.
ax
is a
These r e s u l t s are summarized i n t h e f o l l o w i n g Theorem 3 . 3 .
- The c r i t i c a l p o i n t u h
ah,
s o l u t i o n o f t h e problem 4.
V A
Convergence of ux t o u
x.
E
of Dx i s t h e o n l y d i s t r i b u t i o n a l
E
El+.
0
- The r e s u l t s i n t h i s s e c t i o n have a g r e a t d e a l i n
[31. The i n t e r p r e t a t i o n t h e r e , of ux , a s a
common w i t h t h e convergence r e s u l t s i n
s o l u t i o n t o some v a r i a t i o n a l i n e q u a l i t y , i s however, d i f f e r e n t from o u r s . Let h
E
IR+, M =
x'
x
and
JR
E E
max E
au
$(XI)
.
Denote by $ ( x ) t h e s o l u t i o n of t h e f o l l o w i n g l i n e a r c o e r c i v e s i n g u l a r p e r turbation (see
C71) :
+
(-EA
(4.1)
h h ) Q E ( x )= 0
x no $ E ( X ' ) =
,
x c u ,
,
$(XI)
x'
3u
E
.
One uses t h e maximum p r i n c i p l e f o r t h e L a p l a c e o p e r a t o r i n o r d e r t o p r o v e the following r e s u l t .
-
P r o p o s i t i o n 4..1.
With ux and u
x
s o l u t i o n s t o t h e problem
r e s p e c t i v e l y , t h e following holds: (4.2)
x
$,(X)
The mapping IR
J E
Moreover,
h
A
u (x)
5
+
u (x) 2
u
0 -
h
6 C
h
U<(X)
,
5 M
V x
ma.
The s e t
and
E
,
(U) i s monotone:
, v
x c
u,
v
E
For a g i v e n h > 0 , c o n s i d e r t h e s e t S h -
to
a: mx,
x { f E ( u E ) j E c I Rb e i n g +
-
> 0.
0
A
%Lm+
where u h i s t h e s o l u t i o n
bounded i n L (U), V p P
E
[l,+m],
r i t y r e s u l t s f o r l i n e a r e l l i p t i c c o e r c i v e boundary v a l u e problems
the regula-
(see [11) and
compact embedding theorems g u a r a n t e e t h e s e t S x t o be precompact i n Holder s p a c e s
- for V a t [0,1). C l , a (u) u s i n g t h e m o n o t o n i c i t y of t h e mapping I R + 3 p a c t n e s s of S h i n C1
(u), V a
E
E +
uh
E
C o ( u ) and t h e precom-
[ O , 1 ) , one c o n c l u d e s t h a t any s e q u e n c e
L.S. FRANK and E.W. V.W GROESEN
296
{ u ; , ) ~c ~Sx ~ converges t o some u 1 t h e s o l u t i o n of
Qt^ by
x
i n C1"(U),
when
t a k i n g t h e l i m i t ( a s E.++O) 1
E . +
1
i n the integral identity
We summarize t h i s r e s u l t i n t h e f o l l o w i n g
-
Theorem 4.2.
V
c1
For V
A
E [0,1) t o th e s o l u t io n
a,"
lR+ t h e s o l u t i o n uA of
E
ux o f
0-
ax
as EWO.
0 . One shows t h a t uA i s
converges i n C
1,a
-
(U),
t h e convergence i s
:oreover,
monotone from above i n C ( U ) :
The f o l l o w i n g p r o p o s i t i o n g i v e s t h e r a t e of convergence of u
A
.
t o uA i n H (U) 1 P r o p o s i t i o n 4.3. - L e t uA and uA b e t h e s o l u t i o n s of t h e p r o i l e m and
mA,
a;
e x i s t s a c o n s t a n t c > 0 which depends o n l y on
r e s p e c t i v e l y . Then the:e
X
and U, such t h a t
P r o o f . - The d i f f e r e n c e uA - uA i s a s o l u t i o n o f t h e f o l l o w i n g boundary v a l u e problem:
This y i e l d s the i n t e g r a l i d e n t i t y :
IV(U
(4.5)
I
A X 2 -U
dx = A
A
u J I x + ( ~) -
One r e w r i t e s t h e r i g h t hand s i d e of
A
A
X
f E ( u E ) l ( u E - u) d x
.
(4.5) a s f o l l o w s :
and u s e s t h e l a s t formula i n o r d e r t o estimate t h e f i s t member i n (4. ) i n t h e following fashion: A
( U -U
S
A
f A E+(u )
Since u
A
- u
h
E
o H1(U),
Remark 4.4.
-
E
dx 5
A
E meas
(U)
h
)dx 2
.
the Poincare's inequality yields ( 4 . 2 ) .
The same e s t i m a t e (4.3) was e s t a b l i s h e d p r e v i o u s l y by a d i f f e -
r e n t argument i n C31, where uA w a s i n t e r p r e t e d as a s o l u t i o n t o some v a r i a t i o n a l inequality.
U
297
E L L I P T I C OPERATION WITH DISCONTINUOUS N O N L I N E A R I T Y
- One c a n show t h a t f o r 0
Remark 4 . 5 .
A
luc - u
(4.6)
1*I
where
11
lk
5 ak E
,
V k
X , t h e following estimates hold:
5
E
N+
,
k-
i s t h e norm i n C (U) a n d t h e c o n s t a n t s ak d o n o t d e p e n d upon
Remark 4 . 6 . - F o r
where
X
X
5
X
> h
C
i s t h e norm i n t h e S o b o l e v s p a c e W
*
E.
and n = 1 one e s t a b l i s h e s t h e f o l l o w i n g e s t i m a t e s : 1
1 #P
1rP
(U)
,
and t h e c o n s t a n t s c
d e p e n d o n l y on p . P Our c o n j e c t u r e i s t h a t ( 4 . 7 ) h o l d s f o r n > 1 , as w e l l .
5.
.
The c r i t i c a l v a l u e h
-
I n t h i s section t h e c r i t i c a l value
d e f i n e d by t h e f o r m u l a ( 2 . 3 ) i s i n v e s -
X
t i g a t e d a s a f u n c t i o n a l o f t h e b o u n d a r y f u n c t i o n j, f o r a f i x e d domain U. We u s e the notation
X ( 4 ) f o r X ($,U). 6
I$
Let Cy(au) =
E
: C
m
m
(au)
C
t
(u) t h e
c (3U)
I
$ ( X I ) > 0 , V x' F a u } . We d e n o t e b y
P o i s s o n o p e r a t o r c o n n e c t e d w i t h t h e D i r i c h l e t problem f o r
t h e L a p l a c e o p e r a t o r i n U, i . e . E $ ( x ) = " ( x ) , x
U, i s t h e h a r m o n i c f u n c t i o n on
t
U which s a t i s f i e s t h e b o u n d a r y c o n d i t i o n : liO Q ( x ' ) =
V x'
$ ( X I ) ,
E
aU.
U s i n g t h i s n o t a t i o n o n e c a n rewrite ( 2 . 3 ) i n t h e f o l l o w i n g f a s h i o n :
where V(x) i s t h e same a s i n ( 2 . 3 ) . Proposition 5.1. (i)
hc
:
-
-f
x c ( t ~ )= (ii) Ac
The f u n c t i o n a l
m
C+(aU)
A
( $ ) has t h e following properties:
IR+ i s p o s i t i v e l y homogeneous of d e g r e e o n e , i . e .
t xc(j,)
m
: C + ( a U ) + 1R+ i s
Xc($,) 5 hc($,)
, v
t
0 , v $ c cy(aui
;
monotone i n c r e a s i n g , c o n c a v e f u n c t i o n a l : for 0 < Q1 5 $2
I
and Xc(y$l + ( l - y ) $ 2 ) 2 Y X C ( Q l )
+ (1-Y)Xc($2) V $j
m
E
I
v Y
c+(3U), j
E
=
C0,lI 1,2.
We a r e g o i n g t o i n v e s t i g a t e r e g u l a r i t y p r o p e r t i e s of t h e f u n c t i o n a l XC($) : C p u )
-f
m+.
W e d e n o t e by E ( $ ) t h e f o l l o w i n g s e t i n
u:
r
:
L.S. FRANK and E.W. VAN GROESEN
298
(5.2)
Ec($) = {x
E
I
7
.
( E $ ) (x) = Ac($) V(x)}
m
One easily checks that E ( $ 1 is strictly contained in U, V $
E
C+(3Ul. More-
over, one has:
V(E$) ( n )
(5.3)
=
,
ic($) V V ( r l )
V rl
t
Ec($).
m
Remark 5.2. - It is easily seen that if $
C (aU) vanishes somewhere on the
E
boundary aU, then the corresponding set E ( $ 1 might contain more than one point. For instance, if $(XI) = 1+2x;x; on E
($) =
{x
I
B1
E
x 1+X2
=
a
B1
{x'
E
I
-IR2
/x'I = 1 1 , then
0).
Conjecture 5.3. -For a large class of
m
$ 6
C
(au)
the set E ( $ 1 consists only of
a finite number isolated points. The following result yields the Gateaux-differentiability of A,($)
with
respect to $ .
-
Theorem 5.4. point 5 = 5 ( $ )
E
Assume that E ( 4 ) for a given $
m
E
C
(au)
consists only of one
U. Then Xc($) is Gsteaux-differentiable at $ , and its first
variation along a given direction
IJJ
has the form:
Proof. - We restrict ourselves to a non riqorous but transparant argument which
.
m
leads to (5.4) (see [ 8 ] for a rigorous proof of this formula) Let $ c C+(aU), +€Cm(aU)
and
S E
[-6,61
C
W with 6 sufficiently small. Let S ( s ) E Ec($+s$). Then
A straightforward computation in (5.5) and the formulae (5.21, (5.3) yield (5.4).
0
From now on we restrict ourselves only to the case when U =
B1 = {x (5.6)
E
lKn
1
1x1 < 1). 1
'In this case, V(x) can be found explicitly:
V(x) = Ac(l) ( 1
-
2 1x1 ) ,
~ ~ ( =1 2n )
,
and €4 has the following integral representation: (5.7)
1 (E$) (x) = n'
where
n
is the area of aB
-M$(w)dw ,
J' --1
/ o \ = l\ x - w ~
V x
E
B
1
1'
We use the notation S ( $ ) for the arithmetic mean value of $ aB
1'
1 n
m
E
C+(aB
1
over
299
E L L I P T I C OPERATION WITH DISCONTINUOUS NONLINEARITY
and G ( $
for its geometric mean value:
(5 9 )
G($) =
exp(S(ln$))
Theorem 5.5. - For
v
.
m
$
C+(aB1), Xc($) satisfies the following inequalities:
E
One proves (5.10) using the mean value theorem for harmonic functions and Harnack's inequalities. The inequality (5.11) is proved using the fact that a geometric mean value does not exceed the arithmetic one. 6. Monotone iterations.
-
In this section the monotone iterative procedure, considered previously in
C31, is applied in order to approach the solution uA of
a2
from below and above.
This procedure can also be used to approximate the iolution uA of from above. 0and a function u E C (U), u t 0, we define a mapping
is uniquely resolvable, its solution ux is the only
Since the problem fixed point of the mapping
Let M =
max
$(x'). Using the maximum principle argument one establishes the
x'Eau
A9
following properties of (i)
maps the set
A:
into itself. (ii)
then &t(vl)
& :.(v)(x)
is monotone on K
(x) S A L ( V 2 ) (x), 'd
(iii) The mapping IR
1
:
Eq
at.
L
+
3
(v)(XI,V x
if v .
3 -
E
+A;"
E E
X E
u, V v
K, j
= 1,2,
and v (x) 1
S
v2(x), V x
E
U
U.
is monotone, i.e. if E
o
< E
< E~
then
K.
0
Denote by u (x) and u (x), respectively, a lower and an upper solution of the 0
problem
Then (i), (ii) and the uniqueness of the fixed point u:
O f M :
guarantee the following monotone convergence properties of the mapping&:
:
300
(A:)"i s
where
t h e n - t h i t e r a t i o n of
In p a r t i c u l a r t a k i n g uo
E.W. VAN GROESEN
FRANK and
L.S.
#:
0 and u
=
.
(see, for instance, r21) max
= M =
one g e t s t h e f o l l o w i n g
$ ( X I ) ,
a:
X'Eau
result: Proposition 6 . 1 .
The solution u x of
-
t h e sequences: ( X ) A U CA ( X ) ,
tJ;,"(O)
0 i s a monotone ( i n C ( U ) ) l i m i t
v
( A p M ) (X)bU;(X),
a s n++m
ah
x
E
.
Of
u, 0
Using theorem 4 . 2 and t h e m o n o t o n i c i t y of t h e m a p p i n g d t w i t h r e s p e c t t o E
E
IR
+'
one can approximate t h e s o l u t i o n uA of
b e i n g a sequence s . t . t: \O
as n
converges t o uh monoton2cally
( i n Co
from above. Namely,
t h e d i a g o n a l sequence
+ a,
(t))
from above.
A (AE n
)"(M)
'En'ncN
Inem
+
+
A d i f f e r e n t monotone p r o c e d u r e i s p o s s i b l e , a s w e l l :
P r o p o s i t i o n 6.2. - L e t
{E
1
n ntm
b e a sequence s . t .
E
U
O a s n++m. Then t h e
sequence i v ( x )1 d e f i n e d by t h e formulae: n ncm+
=J:
v(x) ' M , 0 vntl(x)
V
n+ 1
converges t o t h e s o l u t i o n uA of
7.
The Newton-Kantorovich
X
C
U
(vn)( x )
,
v
x e
u,
n c
m+ ,
aA,
m o n o t o n i c a l l y (in Co(,))
method f o r
a'. '
from above:
- I n t h i s s e c t i o n , we a p p l y t h e Newton-
Kantorovich i t e r a t i v e p r o c e d u r e t o t h e problem I t t u r n s o u t t h a t t h e Newton-Kantorovich
ah.
with
p r o c e d u r e n o t b e i n g a p p l i c a b l e t o t h e reduced problem
Co(U))
v
X
E
IR+,
E
E
IR+, t h i s
method g i v e s r i s e t o a monotone ( i n
A
convergence from below of a p p r o x i m a t i n g f u n c t i o n s t o t h e s o l u t i o n u ( x ) o f
aA,
V E c IR +,V A c I R , when t h e i t e r a t i v e p r o c e s s i s star:ed w i t h AE t h e s o l u t i o n u (x) o f t h e problem T h i s r e s u l t e n a b l e s one t o d e r i v e an A a s y m p t o t i c formula f o r u ( x ) , i n t h e c a s e of t h e s p h e r i c a l symmetry of t h e problem
t h e problem
a'.
The Newton-Kantorovich method i s d e f i n e d a s t h e i t e r a t i v e p r o c e d u r e g e n e r a t e d A h by t h e n o n - l i n e a r mapping u + N (u), where t h e f u n c t i o n N ( u ) ( x ) i s t h e s o l u t i o n t o t h e f o l l o w i n g l i n e a r boundary v a l u e problem:
(7.1)
A
(-A
+
h f ' ( u ) ) N (u) ( x ) = A ( u f ' ( u ) - f
no
X
,
N ~ ( u ( )X I ) = + ( X I )
XI
(u))( x )
,
X € U ,
au ,
d f E( u ) with f ' ( u ) = du * h T i e uniqueness of t h e s o l u t i o n u . ( x ) t o t h e problem
a:,
V A
E
IR+, V
E E
B+,
E L L I P T I C OPERATION WITH DISCONTINUOUS N O N L I N E A R I T Y
x
guarantees the existence of the unique fixed point uh for N_
X
A
E
E
A
N ( u ) = uE
(7.2)
v E , m
,
+ '
v
h
301
:
lR,
E
Using, once again, the maximum principle argument, one can prove the f o l l o wing properties of the mapping N
-
Lemma 7.1.
If 0 5 v(x)
(i)
X
'
5 u
A
x:
(XI, V x
-
A
U, then N-(v)(XI
E
x
u (x), V x
E
' A (ii) If v (x) satisfies the inequalities: 0 Iv (x) < uE(x),
N-(vo)( x )
2
0 v (x), v x 0
-
X n+l (vO)(XI U, then (NE)
E
(x) : Ji, (x), where
(iii) N:(O)
h
$;(x)
Aon (v ) (x), v x E U, b' n E IN E . 0 is the solution of the linear coercive
@,
the following inequality holds:
singular perturbation (4.1). (iv) uX being the solution of N
A
h
(U
)
(X)
2 U
A
(X)
U.
v
r
2
(N )
v
h i IR+r
E
€
m+ ,
t/ X E
6.
+
0
As a consequence of lemma 7.1 one gets the following monotone convergence result: Theorem 7.2. - The sequences
A n
(NE)
A n
and
(0)jnFN
0 -
(N )
X
(u 11
nc lN
A
:
u_(x) of
monotonically (in C (U)) from below to the so1ut:on
x A n A A (NE) ( 0 ) ( x ) a u E ( x ) , (NE) (u ) (x)*uE(x), A n
V x
converge
+
-
E
U, as n
+ m
.
0
From now on, we consider the first approximation in the Newton-Kantorovich
x
ah.
method, assuming that the iterative procedure is started with the solution u ( x ) of the reduced problem
We introduce the notation:
A
h
X
vE(x) =,NE(u1 (x)
x
,
x
-
E U.
The function v ( x ) is the solution of the following linear boundary value problem: A
x
h
h
h
(-A + Xf' ( U ))v~(x)= X ( U (x)fA(u ) (XI - fc(U 1 (X)) t
nO vEx
(7.3)
where f' ( u )
=
d
du
f
E
$(xi)
=
(u).
,
X E
U
au
A
As a consequence of Theorem 7.2, v (x) satisfies the inequalities:
(7.4) For A S A
X
h
A
u(x) < v ( x ) < u ( x ) , C
V x ~ f i , V X c l R + , V t d R +'
the boundary value problem (7.3) is a regular perturbation of the
linear elliptic problem: h
Av(x)=h (7.5)
,
X E U
n0 vX (XI) =$(XI).
We restrict ourselves to the most interesting situation, that where A > A
.
.
302
FRANK and E.W.
L.S.
VAN GROESEN
i s a s i n g u l a r p e r t u r b a t i o n of t h e reduced problem
In t h i s case,
ax
with
x
i n t e r n a l boundary l a y e r s l o c a t e d i n a neighbourhood of t h e f r e e boundary a E (u )
x u (x)
with
x
the solution of
and t h e s e t E
x
0
x (u )
0
d e f i n e d i n s e c t i o n 2 . A s i t was
already pointed o u t i n t h i s case, u (x) has t h e following r e g u l a r i t y p r o p e r t i e s :
ux
Clra(U)
E
,
V a
[O,l) b u t u
E
x
2 -
C (U)I t h e s i n g u l a r i t i e s of ux b e i n g l o c a t e d
h on t h e f r e e boundary a E ( u ) . On t h e o t h e r hand, a s i t was a l r e a d y mentioned p r e v i o u s l y , ux c
0
cm(ii), v c
E
m+,v x
c TR+.
Using t h e formulae
x
o n e can show t h a t t h e s o l u t i o n w - ( x ) t o t h e boundary v a l u e problem
x
-
(EA (7.7)
x
x
Axo(u ) ) w E ( x ) = E A X + ! U
II w O
A
,
=
E
i s an a s y m p t o t i c approximation of v h as
XI
E
(x)
x
E
U
a u
E++o.
The problem ( 7 . 7 ) i s a l i n e a r c o e r c i v e s i n g u l a r p e r t u r b a t i o n ( s e e
171)
with a
d i s c o n t i n u o u s c o e f f i c i e n t and a d i s c o n t i n u o u s second member. The s i n g u l a r i t i e s o f
xo (ux ) ( x )
and
x+ (ux ) ( x ) ,
A
,
l o c a t e d a t aEo (u, )
l e a d t o t h e s i t u a t i o n when wx i s n o t
a s r e g u l a r as vx and i t h a s , g e n e r a l l y speaking, t h e same r e g u l a r i t y p r o p e r t i e s a s
u
A
.
I t i s n a t u r a l t o e x p e c t an i t e r a t i v e p r o c e s s t o improve t h e r e g u l a r i t y of
approximating s o l u t i o n s a f t e r each c o n s e c u t i v e s t e p and t h e a s y m p t o t i c p r o c e d u r e not t o destroy t h i s regularity.
mi,
We p r o v i d e such a c o n s t r u c t i o n i n t h e case of a s p h e r i c a l l y symmetric problem i.e. f o r
U = B
C
1
iRn
,
, v x u (x)
$(XI) E 1
I n t h i s c a s e , an e x p l i c i t formula f o r
x'
E
aB
1'
was g i v e n i n t h e s e c t i o n 2 . I n
p a r t i c u l a r , i n t h i s case, the set E (u) is the b a l l B with the r a d i u s 5 defined 0 5 by ( 2 . 7 ) . Using t h e H e a v i s i d e ' s f u n c t i o n H , t h e problem ( 7 . 7 ) can b e r e w r i t t e n as follows :
(EA
-
A
h H ( 5 - l x ( ) ) w E ( x )= ~ X H ( ( x l - 5 ),
B
X E
1
h S i n c e t h e s o l u t i o n w ( x ) of t h e problem ( 7 . 8 ) g e n e r a l l y s p e a k i n g does n o t Z E y i e l d t h e r e q u i r e d C - r e g u l a r i t y , w e c o n s i d e r , i n s t e a d of ( 7 . 8 ) ' t h e f o l l o w i n g f r e e boundary s i n g u l a r l y p e r t u r b e d problem: CEA
(7.9)
-
A
A H ( Q - ~ X ~ ) zI E ( x ) =
IIO
x z (xi) E
=
I , xi
E
~ ~ ~ ( 1 x l -, v )-X
a
B~
E
B
1
E L L I P T I C OPERATION WITH DISCONTINUOUS NONLINEARITY
whose solution
(Z
h
(x),TI)satisfies the condition:
A
(ZE(X), r l )
(7.10)
303
The problem (7.9)
E
-
2C (B1) X (0,1)
.
(7.10), being spherically symmetric, one immediately
finds its (spherically symmetric) solution, which is expressed in terms of the
- 1. Besides, one finds n as the 2 (O,l), with G an explicitly given
modified Bessel function I (x) of order u = solution to an equation G function of
E
and
=
0, q
E
TI.Moreover, with 5 the solution of the equation (2.71, one
finds the following asymptotic formula for the radius
I- = q ( ~ )of
the free
boundary: n(E) =
5
+(:)’
+
o(E)
,
as
E + 0.
References:
1.
S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I; Comm. Pure Appl. Math. 12 (1959), p.p. 623-727.
2.
H.
3.
C.M. Brauner, B. Nicolaenko, On nonlinear eigenvalue problems which extend into free boundaries problems, report no 79-1, Juin 1979; Univ. de Lyon.
4.
H. Brezis, Operateurs maximaux monotones, North-Holland, 1973.
5.
M.G. Crandall, P.H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal. 58 (1975), p.p. 207-218.
Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), p.p. 125-146.
6. I. Ekeland, R. Temam, Convex analysis and variational problems, North-Holland, 1976. 7.
Frank, Coercive singular perturbations. I. - A priori estimates, Ann. Math. Pura ed Appl. (iv), vol 119 (19791, p.p. 41-113.
L.S.
8. L.S. Frank, E.W. van Groesen, On nonlinear elliptic singular perturbations in the theory of membranes with enzymic activity, report, Univ. of Nijmegen (to appear). 9. R. Goldman, 0. Kedem, E. Katchalski, Papain-Collodion Membranes. 11. Analysis of the kinetic behavior of enzymes immobilized in artificial membranes, Biochemistry 10 (1971), p.p. 165-172.
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A N A L Y T I C A L AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS I N A N A L Y S I S s. A x e l s s o n , L.S. F r a n k , A . van der s i u j s ( r d s . ) @ N o r t h - H o l l a n d P u b l i s h i n g Company, 1981
COERCIVE SINGULAR PERTURBATIONS Asymptotics and reduction to regularly perturbed boundary value problems L.S. Frank and W.D. Wendt
Mathematical Institute, Catholic University, Ni jmegen , The Netherlands
An elliptic singular perturbation arising in the theory of thin
elastic plates is reduced to a regular one. High order asymptotic formulae for the solution are given. The method presented can be extended to general linear coercive singular perturbations. §
1. Introduction.
The algebraic coerciveness condition (see 151) enables one to reduce a coercive singular perturbation to a regular one using a natural extension of the WienerHopf type algebra introduced in 111. For simplicity, the results are presented for the linearized model of thin elastic plates ( s e e 191, 1101 for the corresponding nonlinear problem). A solvability theorem in Sobolev type spaces (see 121, [51) is stated and a recursive process is indicated in order to construct high order asymptotic formulae for the solution.
5 2.
Notation and statement of the problem.
where < C >
2
:=
1 + / C j 2 and O ( E 5 )
Then H(s)(IRn)
:= { u
I
I
EE
=
F
X+C
k u denotes the Fourier transform of u .
/ I u E I / (s) <
sup
(O,E~I
-1
is a Banach space (see [ 2 1 , C 5 l ) .
In a standard way, the spaces H ( s ) (V) are defined for any s
E
I R 3 where V is a
smooth manifold. Consider the singularly perturbed boundary value problem (2.1)
(E
2 2 A -A)u_(x)
(2.2)
(2.3)
where
U c
TI
f(x)
,
x
E
U
710uE =
al(x~) ,
au
=
92 (xi) ,
au
a
--u
0 aN
=
E
lRn is a bounded domain with smooth boundary a U , n o is the restriction
operator to
au
and N(x') denotes the inward normal at a point x'
f , @ l and @ 2 may depend upon
E
c (O,E
1. The solution 0 305
u
E
aU.
The data
is sought in the space
306
L.S.
H ( U ) = H ( s ) (U), s
IR
€
3
. The
FRANK and W.D.
WENDT
problem ( 2 . 1 ) - ( 2 . 3 ) o c c u r s i n t h e l i n e a r i z e d t h e o r y of
and w i t h t h e reduced problem
u0 = f ( x ) , x
(2.5)
-A
(2.6)
r 0 uo = $ , ( x ' ) , x ' c aU
(2.7)
K0 =
t
U
the operator
(iA . 0
We assume t h a t t h e c o n d i t i o n s
a r e s a t i s f i e d . Then t h e t r a c e theorem f o r s p a c e s H (2.9)
$" E
(see
(S)
151) y i e l d s :
Hom(H(U),K(U))
where (2.10) K(U) : = H
(s-(0,2,2))
d e n o t e s t h e space of t h e d a t a .
(u)
H(s-(o,+,o)) (au)
I n s e c t i o n 4 w e s h a l l show t h a t t h e a l g e b r a i c
c o n d i t i o n s o f e l l i p t i c i t y and c o e r c i v e n e s s ( s e e [ 3 1 , t h e problem ( 2 . 1 ) - ( 2 . 3 ) , e n a b l e one t o r e d u c e
a"
(see
C41 f o r t h e case n
3 3) ( a u ) H ( s+ S - - , o , s +S -1 2 2 2 3 2
a'
r 5 l ) which a r e s a t i s f i e d by
t o a r e g u l a r p e r t u r b a t i o n of
= 1 ) . S i n c e t h e reduced problem
(2.5), (2.6j is
u n i q u e l y r e s o l v a b l e , one g e t s i n t h i s way t h e s o l v a b i l i t y r e s u l t f o r t h e p e r t u r b e d problem: (2.11)
8"
6
I ~ ~ ( H ( u ) , K ( u ) )f o r eo s u f f i c i e n t l y s m a l l
Moreover, a s y m p t o t i c formulae f o r t h e s o l u t i o n of
(2.1)-(2.3)
w i l l be g i v e n (see
section 5 ) . The method d i s c u s s e d i n t h i s p a p e r can be extended i n o r d e r t o r e d u c e a g e n e r a l c o e r c i v e s i n g u l a r p e r t u r b a t i o n t o a r e g u l a r one ( s e e 1 7 1 ) . I n t h i s way, one g e t s isomorphism theorems l i k e ( 2 . 1 1 ) under t h e assumption t h a t t h e reduced problem i s u n i q u e l y r e s o l v a b l e . I n t h e g e n e r a l c a s e l o n e o b t a i n s t h e r e s u l t t h a t t h e index of t h e problem
8" d o e s
n o t depend upon
E
c [O,col.
I n p l a t e t h e o r y , some c o e r c i v e
boundary v a l u e problems w i t h boundary c o n d i t i o n s d i f f e r e n t from t h e D i r i c h l e t o n e s
are of g r e a t i n t e r e s t . Some examples a r e d i s c u s s e d below i n s e c t i o n 6 . !i 3 .
The a p r i o r i estimate f o r
Since (2.1)-(2.3)
aE.
s a t i s f i e s t h e c o e r c i v e n e s s c o n d i t i o n , one h a s t h e f o l l o w i n g
307
COERCIVE SINGULAR PERTURBATIONS
two-sided a p r i o r i e s t i m a t e ( s e e C 5 l ) :
(3.1) where
/I
11
(s,)
1 1 UEll
(s)
(11
%
( f 1 @ 1 * @ 2 ) 1K 1(u)
11
i s any norm weaker than
11
The c o n s t a n t s i n t h e equivalency r e l a t i o n uniquely r e s o l v a b l e f o r
5 4.
E
E
(O,E
0
1
with
11 u E I I
( s t ) )
depend only upon s , s '
/I u E l l ( S ' ) i n
We s h a l l s e e l a t e r t h a t t h e term
(s)'
+
E~
(3.1)
and E 0' can be d e l e t e d s i n c e
3'
is
s u f f i c i e n t l y small.
Construction of the reducing o p e r a t o r .
W e s h a l l give an a l g e b r a i c c o n s t r u c t i o n of a s i n g u l a r l y perturbed Wiener-Hopf type operator I?"
( s e e [ l ] , C71) which s a t i s f i e s :
(4.1)
8" =
Ro +
RE
eE
E'
with some p o s i t i v e number y and an o p e r a t o r
QE E
Hom(H(U),K(U)). (Here K ( u ) i s
t h e space of t h e d a t a introduced i n ( 2 . 1 0 ) ) . The o p e r a t o r
RE,
c a l l e d reducing o p e r a t o r , w i l l have t h e form
[
E
(p
(4.2)
RE
=
Q:
T:
)
E
Hom(W(U),K(U)).
Q2 Here P i s a p s e u d o d i f f e r e n t i a l operator with r a t i o n a l symbol of v e c t o r i a l order
(0,0,2), E
n H
1
(s-(0,2,2))
(au) into (s-(0,4,0)) a r e t r a c e o p e r a t o r s which map
i s a Poisson operator which maps H (U) (see 1611, T
1
and T
2
(au)
and H ( s .+S -3,'2,O,s2+s3-3,'2) 1 2 r e s p e c t i v e l y , Q, and Q2 a r e p s e u d o d i f f e r e n t i a l o p e r a t o r s on a U and
(s-(0,2,0))
(u) into
(s-(0,?1,0))
(au),
= H (aU). I n t h e case considered here, it w i l l H(s-(o,4,0)) (s-(0,2,0)) t u r n o u t t h a t T1 and El can be chosen a s zero s i n c e t h e t r a c e operator which 2 2 appears i n ( 2 . 2 ) does not depend upon E and s i n c e E A - A can be f a c t o r i z e d i n t o
W(U)
t h e reduced d i f f e r e n t i a l operator and an e l l i p t i c d i f f e r e n t i a l operator
a
P = P ( E -).
ax
For t h e s i m p l i c i t y of p r e s e n t a t i o n , we r e s t r i c t o u r s e l v e s here t o
t h i s case; t h e general case i s discussed i n From ( 4 . 1 ) follows t h a t
6'
C71.
can be w r i t t e n a s t h e product of
0
R E and & modulo a
small o p e r a t o r . As we s h a l l s e e l a t e r , t h e coerciveness of t h e o r i g i n a l problem
(2.1)-(2.3) E~
implies t h a t
RE
i s an isomorphism of t h e above described spaces f o r
s u f f i c i e n t l y small. W e s h a l l prove t h i s r e s u l t by c o n s t r u c t i n g a quasi-inverse
Wiener-Hopf type o p e r a t o r S',
(4.3)
so t h a t one has
zESE = Id
K (U)
with some y > 0 and an o p e r a t o r 9; S'
w i l l have t h e form
E
+
E~Q'
1
Hom(K(U)).
308
L.S. FRANK and W.D. WENDT
where P' is a pseudodifferential operator with rational symbol, E;,
E' are Poisson 2 Q' are pseudodifferential operators on 2 T': with trace operators TI.' and Poisson operators
operators, T' is a trace operator and d u . G' has the form
G'
=
E'I.
C E',' 0
j
Ql,
7
l
7
3
From the invertibility of the operators 2' and
@,"
for
E
c
(O,tOI ,
E~
sufficiently small
follows the claim (2.11).
In order to construct RE, we consider first an auxiliary problem in the half-space 11.- 1 7x+n : = {(X',Xn) x' t IR ,x 01.
1
Here
11"
denotes the restriction operator to IR
n- 1
x{0},
1 1 ajk(x)/ I
, x
c IRn,
is a
positive definite nxn matrix the elements of which depend smoothly upon x, and r(x,t-,c) is a polynomial in 5 and a symbol of lower order, i.e. the inequality
holds with some constant C which does not depend upon x,
E,
and 5 . In (4.5)
O p b(x,c) denotes the pseudodifferential operator associated with the symbol b:
Op b(x,S)u = F-'
C-fX
b(x,5)
zF5,
where F and F-l denote the direct and inverse
Fourier transforms, respectively. The reduced operator of (4.5) has the form
We shall construct an operator (4.7) with some
QE
ff f
c Hom(H(lRy),
=
R:
1
R;
Kim:)).
which satisfies 3.12-s
+
E
2QE
Here f / ( i R y ) , K(IR:)
are the spaces of the
solution and of the data, respectively, which are defined similarly to H(U) and K(U).
Further, one defines w ( I R y ) . For x = (x',x
)
E
l R n and E; = ( C ' , E ; ) E IRn
define the function h(x,S') by: (4.8)Ca. (x)E.S + 1 = ann(x)(iEn 1k 1 k
+
A(x,C'))(-iC
Further let A(x',€,') = X[x',O,<') and a(x) A parametrix for
8'
is given by
=
+ T(x,c')) , Re A
a (x),a(x') := a (x',O). nn nn
>
0
COERCIVE SINGULAR PERTURBATIONS
3 09
where n + d e n o t e s t h e r e s t r i c t i o n o p e r a t o r t o IR;,
1 u i s t h e extension by zero to 0 - A ( x ' , C ? ) xn I R n o f a f u n c t i o n d e f i n e d o n I R n a n d t h e P o i s s o n o p e r a t o r Op e ( s e e [61) i s d e f i n e d by: (Op e
(4.10)
-A (x'
&'
Multiplying
,C')Xn
(4.11)
z'1
;s
=
( 2 n ) l-n
by t h e p a r a m e t r i x ( 4 . 9 ) f o r
one g e t s
We h a v e
I$) ( x ' , x n )
R
Op(1 +
E
2
Ca.
IR n-
@o
7
i S ' x ' - X ( x ' ,5') xn
a n d o m i t t i n g t e r m s o f lower o r d e r ,
5.C )
0
1 a ( x ) (-icn+x(x,c'))
'0
(F$) ( 5 ' ) d C ' .
Ik 1 k
0
=
f
'0
1
AId ( x ',C')
-OP
c Hom(/I(IRy), K ( I R y ) ) .
R;,
( 4 . 7 ) h o l d s w i t h t h i s c h o i c e of
using the f a c t t h a t for 3.'2-S2 ) in a n y symbol b ( x , c ) o f o r d e r 0 o n e h a s m o O p b ( x , c ) l , = O ( E 3 (IR;), H ( o ) ( I R n - l ) ) w i t h U := (s + s - 3,'2, 0, s 2 + S 3 - 5 ) . Hom(H One c h e c k s t h a t
E2
(S)
W e are g o i n g t o c o n s t r u c t a m a t r i x o p e r a t o r S which i s q u a s i - i n v e r s e t o R: 1 3 t h e s e n s e o f ( 4 . 3 ) w i t h y = - - s a n d K(U) r e p l a c e d by K ( I R y ) . The p r o b l e m R;(v,JI)
T
=
2 2 (f,$ ,$ I T w i t h g i v e n d a t a ( f , @ ,$ 1
2
1
f o l l o w i n g way:
(4.12)
i
w h e r e b ( x , F , ) : = ( a ( x ) (-it;
2
)
K , can be r e w r i t t e n i n t h e
E
2 Op(l+r I d , (x)F 5 ) v ( x ) = f ( x ) 'j k
jk
n o Op b ( x , c )
+ x(x,('))
-1
in
l0v
=
i
=
,
x c I . :
I$2 + O p A ( X ' , ~ ' ) $ ,
.
The P o i s s o n operator F d e f i n e d by
+
x(xt,5'))
i (S')dF,'
h a s t h e f o l l o w i n g p r o p e r t i e s ( s e e 161): (4.14) (4.15)
F t H o m ( H ( 0 )( I R n - ' ) ,
1 1 O p ( l + t :2 Za.lkCI. 5k )F$lI
(4. l b )
IT
0
(s-(o,2,2))
e q u a t i o n s of
+
5 C
i
c
(IRy)) (0)
Op b ( x , c ) l F = I d 0
where t h e c o n s t a n t C d o e s n o t d e p e n d upon follows t h a t F($2
H(s-(0,2,0))
+
O(E)
V$ c H i
(u)
Hom(H
(0)
(IRRI1-l) (En-'))
( 0 , 1~ a n d $ . From ( 4 . 1 5 ) a n d ( 4 . 1 6 ) 0
Op A ( X ' , ~ ' ) $ ~ s a) t i s f i e s a s y m p t o t i c a l l y t h e f i r s t t w o
(4.12) w i t h f
2 0. S i n c e O p ( ( l + € Ca. ( x ) t j ( , )
Ik
-1
1 l o f i s an asymptotic
310
L.S. FRANK and W.D. WENDT
s o l u t i o n of t h e f i r s t e q u a t i o n of operator
( 4 . 1 2 ) (see C S l ) , one can choose f o r Sf t h e
2 ( n -Fno Op b ) O p ( ( l + E Za
, t
(4.17)
S;
=
\
jk
(x)S.S ) - l ) l 0 , F I k
0
Op h ( x ' , S ' )
,
F
Id
,
o
\,
~
0
I
W e a r e going t o c o n s t r u c t a r e d u c i n g o p e r a t o r RE f o r t h e problem ( 2 . 1 ) - ( 2 . 3 ) and a q u a s i - i n v e r s e SE, u s i n g t h e c o n s t r u c t i o n s g i v e n above f o r t h e problem ( 4 . 5 ) i n a h a l f space. For x i n a s u f f i c i e n t l y s m a l l neighbourhood V o f aU, w e d e f i n e x ' Ix-x'
I
min Ix-y'
=
I
d ( x , au)
=
YlEau
Then one can choose
/
RE=
a
/
1-E
2
. Denote
aU by
a Id
0
m)-')
+
E
by A ' t h e L a p l a c e - B e l t r a m i o p e r a t o r on a U .
The o p e r a t o r n ( ( - may b e d e f i n e d as f o l l o w s . L e t x ( x ) be a smooth 0 aN f u n c t i o n which i s i d e n t i c a l l y one on x E [O,r] and t h e s u p p o r t of which i s
. Further,
The p o s i t i v e c o n s t a n t
T
let <S'>n:=
m.
Then d e f i n e
i s supposed t o b e s u f f i c i e n t l y s m a l l s u c h t h a t t h e
s u p p o r t o f x ( d ( 9 , a U ) ) i s c o n t a i n e d i n V. I n s t e a d of
-K one
may t a k e t h e p s e u d o d i f f e r e n t i a l o p e r a t o r
- 0 ~ 1 6I ~(where 5 :
d e n o t e s t h e c o t a n g e n t i a l v a r i a b l e ) which i s d e f i n e d w i t h a p a r t i t i o n of u n i t y . Hence, t h e c o n s t r u c t i o n o f
RE
is purely algebraic.
L e t n and lo d e n o t e t h e r e s t r i c t i o n o p e r a t o r t o z e r o o u t s i d e U, r e s p e c t i v e l y . Then one can choose
U and t h e e x t e n s i o n o p e r a t o r by
311
COERCIVE S I N G U L A R PERTURBATIONS
4 5.
Asymptotic formulae f o r t h e s o l u t i o n .
L e t u ( x ) be t h e s o l u t i o n o f
(Z.llb(2.3) for
E
E
( 0 , 1, ~ 0
E~
s u f f i c i e n t l y small.
Using t h e f a c t o r i s a t i o n ( 4 . 1 ) modulo a s m a l l o p e r a t o r and a q u a s i - i n v e r s e t o R E , w e shall c o n s t r u c t a sequence o f f u n c t i o n s u ( k ) E H ( s ) (U), k 2 0 , s u c h t h a t f o r
all r 1 0 one h a s
where
i s a c o n s t a n t which depends o n l y upon r and y =
Cr
3
.-
2
-
s2 > 0
a s above. The f u n c t i o n s u ( ~ a) r e d e t e r m i n e d r e c u r s i v e l y by an i t e r a t i o n p r o c e d u r e where i n e v e r y s t e p one h a s t o f i n d t h e s o l u t i o n of t h e r e d u c e d problem ( 2 . 5 ) , ( 2 . 6 ) , t h e (0) , ,u ( k - 1) , f , $ 1 ,$ 2 and E . L e t u~"), k 2 0 , be d e t e r m i n e d by
d a t a b e i n g known f u n c t i o n s of u.
. ..
where t h e sum i s by d e f i n i t i o n z e r o f o r
k
= 0.
One p r o v e s ( 5 . 1 ) u s i n g ( 4 . 1 ) , ( 4 . 3 ) and t h e f a c t t h a t one c a n d e l e t e t h e t e r m
1 1 uEII (s,)i n
the a p r i o r i estimate (3.1) since
REi s an
isomorphism.
I n o r d e r t o be more s p e c i f i c , w e w r i t e down t h e problem t h e s o l u t i o n of which i s (0) t h e z e r o t h o r d e r approximation u _
.
One h a s t o s o l v e t h e problem
where g
C
lids t h e f o l l o w i n g i n t e g r a l r e p r e s e n t a t i o n i n terms of t h e g i v e n d a t a :
Here G ( 1x1 ) i s t h e fundamental s o l u t i o n o f 1-A I t can b e e x p r e s s e d by a Hankel f u n c t i o n :
2-n -
2 w i t h some known c o n s t a n t C L e t the function L
.
be d e f i n e d h y
i n I R n which v a n i s h e s a t i n f i n i t y .
312
L.S. FRANK and W.D. WENDT
where J
v
denotes the Bessel function of order u .
Then the kernels S , P and P can be expressed in terms of the functions G n nl I1 2 L and their derivatives:
,
are sufficiently smooth and independent upon Assume now that the data f , $ ,@ 1 2 Then the asymptotic formulae can be simplified.
E.
-
Let S' denote the operator
We define the functions v ( ~ ' , k t 0, recursively by
where the sum is by definition zero for k = 0. Hence, in order to find the zeroth order approximation v(O), one has to solve
=
$l
The first order approximation v . following problem for v (1).
=
(5.6)
T~
E
v(O)
. v(O) t
& v(')
can be determined by solving the
313
COERCIVE SINGULAR PERTURBATIONS
The d i f f e r e n c e between t h e p r e c i s e s o l u t i o n u approximation v
i
E
o f t h e p l a t e problem and i t s f i r s t
c a n be e s t i m a t e d a s f o l l o w s :
T h i s e s t i m a t e g i v e s a b e t t e r c o n v e r g e n c e , namely i n H (U), when one s u b t r a c t s t h e 2 main s i n g u l a r term v from t h e s o l u t i o n . The same k i n d o f improved e s t i m a t e w d s c e s t a b l i s h e d i n C81 f o r t h e one d i m e n s i o n a l l i n e a r i z e d model o f t h i n e l a s t i c p l a t e s . One g e t s a l s o t h e f o l l o w i n g e s t i m a t e i n Schauder norms f o r t h e d i f f e r e n c e u.-v L
( 5 . 1 0 ) [ u -v i
A special case of
I
E1,lX
v
5 c(f,@l,@2,'A)E1-(X
'X
iI
f-
:
C0,l).
( 5 . 1 0 ) w a s p r e v i o u s l y g i v e n by F i f e f o r t h e two d i m e n s i o n a l
D i r i c h l e t problem f o r t h e e q u a t i o n ( 2 . 1 ) w i t h homogeneous boundary c o n d i t i o n s .
W e end up t h i s s e c t i o n by s k e t c h i n g t h e p r o o f of t h e error estimate ( 5 . 1 0 ) .
To
t h a t end, w e s h a l l make u s e o f t h e c l a s s i c a l S o b o l e v imbedding theorem and t h e g l o b a l Schauder e s t i m a t e s f o r s o l u t i o n s o f t h e r e d u c e d problem. B e s i d e s , a n k r analogue of t h e e s t i m a t e ( 5 . 1 ) with u 1 v ( ~ i)n s t e a d of Ik=O t k r 2u - - kCO E u ( ~ ) w i l l be u s e d , as w e l l .
'
i
S i n c e t h e d a t a a r e supposed t o b e s u f f i c i e n t l y smooth, one can choose s2 = 1 and
s3 s u c h t h a t s t S >"2. 2 3 2 Then t h e S o b o l e v imbedding theorem y i e l d s :
w i t h some c o n s t a n t C which d o e s n o t depend upon E and w. L e t k be a l a r g e n a t u r a l number which w i l l be c h o s e n l a t e r on. The l e f t hand s i d e
of
( 5 . 1 0 ) can be e s t i m a t e d i n t h e f o l l o w i n g way: k' k k
rut
-
V t ~ l , as
cu -
Hence,
z
k'=O
k E
E
1,u
k' -
k' -
k'=O
s i n c e , as i t h a s been a l r e a d y p r o v e d ,
&
+
k
k' 2 (k') v
-
7 : I E
E
4,cc
i s an isomorphism.
W e s h a l l show f i r s t t h a t t h e r e e x i s t s a c o n s t a n t C = C ( f , $ l , $ 2 , a ) which d o e s n o t
depend upon E s u c h t h a t
314
L.S.
FRANK and W.D.
WENDT
We s t a r t by p r o v i n g t h e f o l l o w i n g a u x i l i a r y r e s u l t . L e t g respect t o
E
I.
I
C n i ( u ) uniformly w i t h
where m i s s u f f i c i . e n t L y l a r i j e . Assume t h a t supp CJ
some small neighbourhood o f 2U. T h c n t h e s o l u t i o n z -
-Az
t.
L
is contained i n
of tlie problem
d(x, J U ) __-
= e
ij,
(x)
can be e s t i m a t e d i n t h e f o l l o w i n g way:
Here we have used t h e c l a s s i c a l g l o b a l Scliauder e s t i m a t e f o r t h e s o l u t i o n o f t h e D i r i c h l e t problem f o r t h e L a p l a c i a n . U s i n g orif more term of boundary l a y e r t y p e i n t h e asymptotic expansion f o r z
and a p p l y i n g a g a i n t h e c l a s s i c a l Schauder t y p e
k
e s t i m a t e , one can p r o v e t h e l a s t e s t i m a t e f o r
IX
= 0 , as w e l l ,
so t h a t
We s h a l l a l s o need one more a u x i l i a r y r e s u l t . i n o r d e r t o prove ( 5 . 1 2 ) . The s o l u t i o n w
of t h e problem -
(5.14) 5
with g
E
(-Aw
'.
= -l
'-
e
d (x,JU) __t
cIL (x)
m -
C (U) u n i f o r m l y w i t h r e s p e c t t o i , h a s t h e f o l l o w i n g p r o p e r t y :
aw
where C d o e s n o t depend upon
One e a s i l y c h e c k s t h a t
i.
I n d e e d , one h a s
315
COERCIVE SINGULAR PERTURBATIONS
w i t h some smooth u n i f o r m l y bounded w i t h r e s p e c t t o
+
(5.13) t o w
Y
E
f u n c t i o n g (x). Applylng E
--d ( x , au) E
E
e
q E , one g e t s ( 5 . 1 5 ) w i t h k = 0. Using h i g h e r o r d c r
asymptotic formulae f o r t h e s o l u t i o n w
o f ( 5 . 1 4 ) and e s t i m a t i n g t h e boundary
l a y e r t y p e t e r m s i n t h e same way as p r e v i o u s l y , one e s t a b l i s h e s ( 5 . 1 5 ) f o r
VP,<m. We a r e now i n a p o s i t i o n t o p r o v e ( 5 . 1 2 ) . The f u n c t i o n c v ! ~ ) i s t h e s o l u t i o n o f t h e boundary v a l u e problem
30 ( E
(2))
= S E ( ( f , @,@
1
-E = S
2
)T
-
+ &
OE(V(O)
v
p
T
((hErOrljiE)) d ( x , aU)
where
-
:= f
(-E
2
+
A
+
1)
&-VL1’)
From now o n , we u s e c o n s i s t e n t l y t h e n o t a t i o n g . (x), j t 1 , f o r d i f f e r e n t smooth uniformly with r e s p e c t t o Since (-E
2
A
+
JE
functions.
E
_ _d_ (_x , au)
_ _d_ (_x , a u )
E
1)t-l e
= e .
one can r e w r i t e h E , a p p l y i n g ( 5 . 1 5 ) t o v o
hE = f
-
(-E
2
Af
+
f) -
2 -(-E
A
+
(-E
1)
2
2
-1 1 ) ( ~ e
+
A -1
E
“Lo),
a
E
E
$2
Applying one more t i m e ( 5 1 5 ) , one shows t h a t
s T h e r e f o r e , one h a s :
CE
.
qlE ( x)
as f o l l o w s :
- -d (x,au)
-~d (x,au) e
6
x W 2 -
To
a 5 V E( O ) ) +
E
2
Af)
316
L.S.
To
Applying ( 5 . 1 3 ) t o
E
F R A N K and W . D .
WENDT
(E v ( 2 ) ) = 0 ,
v ( ~ ) one , g e t s (5.12).
By i n d u c t i o n , one shows t h a t
h o l d s f o r k s u f f i c i e n t l y l a r g e and t h a t
where C does n o t depend upon c. The estimates ( 5 . 1 1 ) ,
( 5 . 1 6 ) and ( 5 . 1 7 ) y i e l d ( 5 . 1 0 ) .
Using a v e r y s i m i l a r argument, one e s t a b l i s h e s a more g e n e r a l e r r o r e s t i m a t e i n Schauder t y p e norms:
CUE
-
2r-1
c
k'=O
E
k' 2
v (k') E
],,a
l+r-L-a
5
V L < r
where C d o e s n o t depend upon t . The r e d u c t i o n method p r e s e n t e d i n t h i s p a p e r can b e e x t e n d e d t o g e n e r a l l i n e a r coercive singular perturbations.
§
6.
Other c o e r c i v e s i n g u l a r p e r t u r b a t i o n s a r a i s i n g i n p l a t e t h e o r y .
For s i m p l i c i t y , w e r e s t r i c t o u r s e l v e s h e r e t o p l a t e s c o n s i s t i n g o f a homogeneous and i s o t r o p i c m a t e r i a l . The problem ( 2 . 1 ) - ( 2 . 3 ) w i t h U c I R 2 , i s a model of a t h i n e l a s t i c p l a t e b e i n g clamped a l o n g i t s boundary. With t h e t h e o r y developped above, one c a n a l s o t r e a t 3 d i f f e r e n t s i t u a t i o n s . Denote by - t h e t a n g e n t i a l d e r i v a t i v e a t au and by u t h e aT
P o i s s o n ' s r a t i o o f t h e p l a t e . The problem (6.1)
2 2
(c A
-
A ) ut
onau
n u = O
(6.2) (6.3)
in U
= f
O
II
(--
a* -
O aN2
t
a2
a -)u aT2
=
0
on
au
i s a model o f a t h i n e l a s t i c p l a t e b e i n g s i m p l y s u p p o f t e d a l o n g aU. O n e can check t h a t t h e problem ( 6 . 1 ) - ( 6 . 3 )
s a t i s f i e s t h e c o e r c i v e n e s s c o n d i t i o n s t a t e d i n C5l.
Hence, t h i s s i n g u l a r p e r t u r b a t i o n can b e r e d u c e d t o a r e g u l a r one. Iloreover, s i n c e
317
COERCIVE SINGULAR PERTURBATIONS
the problem
- n u0 = f
in U
no u0 = $5
on 3U
is uniquely resolvable,one can derive asymptotic formulae and establish error it was done above for the problem ( 2 . 1 ) - ( 2 . 3 ) .
estimates for ( 6 . 1 ) - ( 6 . 3 ) as
One can also consider the case that aU has two connected components 2 , U and
so that one has au =
a 1U (E
a 2 U.
u
2 2
The problem
- A) u c
h
r0
a
uE
=
f
in U clR
=
0
on
2
a,u
which describes a thin plate being clamped along
a 2 U,
a 2 U,
a 1U
and simply supported along
can be treated in an analogous way (see 141 for related two point boundary
value problems). Finally, consider the singular perturbation (E
(-
ir
O
d
m2
2 2 A - A ) u~~ = f
- u 1-
d
3T2
u
=
0
in U cIR on
2
au
which is a model of a thin elastic plate being free at its boundary. One can check that this problem is coercive as well. Hence, it can be reduced to a regular perturbation.
318
L.S. FRANK and W.D. WENDT
References.
111
L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta 14ath. 126 (1971), p. 11.
12 1
A. Demidov
,
Elliptic pseudo-differential boundary value problems with
small parameter in the coefficients of the leading operator, USSR Sbornik 20 (19731, p. 439. c3 I
P. Fife, Singularly perturbed elliptic boundary value problems I, Annali di Mat. Pura ed Appl. Ser. 4, 90 (1971), p. 99.
141
L. Frank, General boundary value problems for ordinary differential equations with small parameter, Annali di Mat. Pura Appl. Ser. 4, 114 (1977), p. 27.
rsi
L. Frank, Coercive singular perturbations I, Annali di Mat. Pura Appl. Ser.
4, 119 (1979), p. 41. I61
L. Frank and W. Wendt, Isomorphic elliptic singular perturbations, Report
8003 (1980), K.U. Nijmegen. L71
L. Frank and W. Wendt, A Wiener-Hopf type algebra of singular perturbations. Preprint K.U. Nijmegen, to appear.
81
L. Frank, Coercive singular perturbations: stability and convergence, in: P. Hemker and J. Miller (eds.), Numerical analysis of singular perturbation
problems, Academic Press 1979. c9
I
K. Friedrichs, J. Stoker, The non-linear boundary value problem of the
buckled plate, Amer. J. Math., V o l . 63 (1941), p. 839.
I lOj
T.
von KSrmdn, Festigkeitsprobleme im Ilaschinenbau, Encyklopddie der
Plathematischen Wissenschaften, Vol. IV-4, C, Leipzig, 1907-1914
EFFICIENT TWO-STEP NUMERICAL METHODS FOR PARABOLIC DIFFERENTIAL EQUATIONS Arieh Iserles Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge CB3 9EW, England
We develop two-step A-stable methods of maximal order for the numerical solution of ordinary differential systems. When these methods are applied to the stiff, large systems which originate from parabolic differential equations they yield a set of algebraic equations of special form. This set is easier to solve than the algebraic equations which are obtained when using one-step methods of the same order. When the spatial variables of a parabolic differential equation are discretized either by finite differences or by finite elements, one obtains a stiff ordinary differential system of the form
u'
(1)
=
+ f(t,u), u(to)
Mu
=
uo
,
where M is a large, sparse, constant, stable matrix, p ( M ) > > O , and the function f has a Lipschitz constant much smaller than the spectral radius of M.
To illustrate the approach of our paper, let us compare two onestep methods for the system (1): (i) diagonal Obrechkoff method [7]: U1
where
u1
=
'
- 'jhul
u(to
+
+
1 2 " 12 h u 1 = u 0
-
+
%hue'
+
1 2 " 12 h u0
h);
(ii) Ndrsett-Makinson restricted Obrechkoff method [ 4 ] ,
.
[6]:
Both methods employ the first two derivatives of u They are A-stable. The first is of order 4 , while the second is of order 3. While solving (2), it is u s u a l to evaluate an algebraic linear system with a matrix 1 2 2 I - S h M + - h M . 12 The term M 2 causes large fill-in, which makes the application of either iteratives or direct methods for the solution of linear algebraic systems expensive. However, method (3) was proposed because the matrix of its linear system has the form 319
320
(I
A . ISERLES
-
'.
(Ji+fi/G)hM)
T h e r e f o r e t h e s y s t e m c a n b e s o l v e d i n two s t a g e s ,
where t h e m a t r i x o f e a c h s t a g e i s I advantage of avoiding f i l l - i n .
- (f +
J 3 / 6 ) h l l , which h a s t h e
Hence, t h e method ( 2 ) ( a n d t h e d i a g o n a l O b r e c h k o f f m e t h o d s i n general [81) is superior a s f a r a s t h e order i s concerned, while t h e method ( 3 ) ( a n d t h e r e s t r i c t e d O b r e c h k o f f m e t h o d s i n g e n e r a l ) y i e l d s a more c o n v e n i e n t a l g e b r a i c s y s t e m . W e p r o p o s e m e t h o d s which s h a r e t h e m e r i t s o f b o t h d i a g o n a l and r e s t r i c t e d O b r e c h k o f f m e t h o d s . The p r i c e w e pay i s moving from o n e s t e p t o t w o - s t e p m e t h o d s . L e t u - ~= u ( t o - h ) . An e x a m p l e o f a method w h i c h c o r r e s p o n d s t o ( 2 ) and ( 3 ) i s
u1
-
(I+J?/3)hu;
+
( 1 / 3 + f i / 6 ) h 7 u i = -2fi/3huA
+
(l+6/3hull
+
+
(1/3+fi/6)h2u11
u - ~+
.
I t i s A - s t a b l e and o f o r d e r 4 and y i e l d s a n a l g e b r a i c s y s t e m w i t h
t h e matrix (I
- (f +
0/6)hM)'.
I n g e n e r a l , w e c o n s i d e r m e t h o d s o f o r d e r 2n o f t h e form
where a > 0 , i n o r d e r t h a t t h e m a t r i x o f t h e l i n e a r s y s t e m h a s t h e form ( I
-
I t i s c o n v e n i e n t t o a n a l y s e m e t h o d s ( 4 ) i n terms o f a p p r o x i m a t i o n s t o t h e exponential function. I n d e e d , f o r ut;C" ( 4 ) i s e q u i v a l e n t , i n operational notation, t o
n hD {(l-ahD) e where D =
(5)
d
.
ez(l-az)"
-
n
1 b k ( h D )k - n1 c k ( h D ) k e - h D } u ( t o ) = 0 k=o k=o
,
C o n s e q u e n t l y ( 4 ) i s o f o r d e r 2n i f and o n l y i f n
- 1 bkzk k=o
e-'
n
1 ckzk
k=o
= O(z
2n+l
)
.
Equation (5) g i v e s a u s e f u l t o o l f o r t h e d e r i v a t i o n of t h e e x p l i c i t form o f t h e c o e f f i c i e n t s bk and c k . Theorem 1:
-
L e t co = 0 ,
-
bo = 1
and
321
E F F I C I E N T TWO-STEP N U M E R I C A L METHODS
k = l ,
. .,
k = l , . Then the method (4) with bk = bk, ck order 2n.
=
.,
n;
n
.
ck, k = 0,
...,
n,
is of
In the proof of the theorem we show that the given coefficients satisfy (5). This involves long and tedious algebra and numerous applications of the theory of hypergeometric functions. For any fixed N (5) reduces to 2n + 1 linear equations in 2n + 2 unknowns. It is an easy exercise in linear algebra to show that the matrix of this system is of rank exactly 2n + 1. Therefore we have an extra degree of freedom, in addition to a . Theorem 2: The general form of the coefficients of the method (4) of order 2n is
(2n-k)! C k = C k+ - (2n)!
(:)a
, k
=
0,
...,
n;
5
where bk and ck
are given by (6) and
is an arbitrary constant.
The proof of this theorem rests upon the fact that we can perturb the second and third terms in (5) by -BPn(z) and BQn(z) respectively, where Pn/Qn is the n-th diagonal Pad6 approximation to exp(z) and is an arbitrary constant, without changing the order. Using the theory of weakly nonlinear methods [5] we are able to show that method (4) is zero-stable [3] if and only if -1 < B 51. It is evident that the classical definition of A-stability [ 3 ] is equivalent to /R+(z)I, IR-(z)I < 1 for every z, Rez < 0, where R+ and R- are the solutions of the quadratic (7)
A R ~- BR A(z) = (
method (4) is
- c = o ,
322
A. ISERLES
Theorem 3: The method (4) is A-stable if and only if the three following conditions are satisfied: (i)
(ii)
> 0,
-1 < p c
[A(it)1 ’
2 IC(it)
ci
1 ; 7
.
{/A(it)I2-1C(it)I for every real t ;
It is L-stable if and only if in addition
deg8, degC
5 n-1 .
As usual with A-stability analysis, the proof is based upon the application of the maximal modulus principle. Hence, for ci > 0 it is sufficient to investigate the solutions of (7) along the imaginary axis. It is easy to see that the A-stability is equivalent to the satisfaction of the root condition by the quadratic (7) for z = it, --m < t < m Using a suitable modification of Schur-Cohn criterion [ 2 ] , we are able to establish (ii) and (iii). The condition for L-stability follows from an investigation of the asymptotic behaviour of R+ and R- when I z l + m .
.
Figure 1 gives the A-stability regions in the ( a , B ) plane for 151153. Note that the regions are closed, with the exception of the portion of the boundary where
a=
-1.
There are no L-stable methods for 1 & n 3, with the single exception of the second order formula of Gear 181, namely n = 1, a = -23 r P = - 17 . Furthermore, computer search showed that there are no A-stable methods (4) for n = 4. By noticing this and the steady contraction of the stability regions in Figure 1 for increasing n, we conjecture that there are no A-stable methods (4) for n 2 4
.
Finally we examine the effect of solving the algebraic system which is obtained by various methods from the diffusion equation
where u = u(t;xl, ..., xm) and A is the Laplace operator. We assume that the equation is solved in an equidistant square spatial grid and the second derivatives in respect to xi, i = 1, ..., m, are discretized by the usual three-point formulae. The comparison is between the usual methods (for example the diagonal Obrechkoff), which produce a system with a matrix which is a polynomial in M,
323
E F F I C I E N T TWO-STEP N U M E R I C A L METHODS
n=2
n=1 1
Figure 1
324
A . ISERLES
and the restricted methods (like the restricted Obrechkoff and method ( 4 ) ) , which produce a system with a factorizable matrix (I-ahM)n, where h is the time step and a > 0. If a direct method, like Hockney or CORF scheme [l], is used to solve a system with the matrix I-ahM, the computation consists mainly of a large number of solutions of (2m-l)-diagonal q x q systems, where q is the number of internal points in the spatial grid in any direction. However, if such a method is used to solve a system with the matrix P(hM), where P is a polynomial of degree n, one must solve instead (2n(m-l)+l)-diagonal q x q systems. Hence, if NR and No are the numbers of multiplications involved in the solution of systems with the matrices (I-ahM)" and P(hM) respectively, then
-
NR
=
C
+
No
=
C
+ (n(m-1)(2n(m-l)+1)+1)D
n(2m'
3m
+
2)D
;
where C and D are constants of equal order of magnitude. Table 1 displays the number of multiplications which are required The advantage of to solve the problem for m = 2 and m = 3 method ( 4 ) is obvious.
.
Now we examine the efficiency of iterative schemes [9] for sparse algebraic systems, in conjunction with the restricted and the usual methods. We restrict ourselves to two space variables only and to n = 2 . Let g be the size of the space grid and u = h/g2 . The spectral radius of the Jacobi method for the matrix I - clhM is
while the spectral radius of the SOR method for this matrix is 4aucos 2q+l
However, it is possible to show that the spectral radius of the Jacobi method for the matrix I - fhM + x1 h 2 M 2(which originates from the second diagonal Obrechkoff) satisfies
Computer calculations estimated the spectral radius of the SOR method for this matrix as l-2/max{qIu} + O(l/(max{q,a~)' ) . Hence,
325
E F F I C I E N T TWO-STEP NUMERICAL METHODS
Two space variables: diagonal Obrechkoff
+
restricted Obrechkoff
method (4)
order 4
C
11D
C
+ 12D
C
+
8D
order 6
C t 37D
C
+ 20D
C
+
12D
Three space variables: diagonal Obrechkoff
+
order 4
C
order 6
C t 127D
37D
restricted Obrechkoff
method (4)
C + 33D
C
C t 55D
C
+ +
22D 33D
Table I : The number of multiplications required by the various methods.
the SOR method for I - ahM converges asymptotically faster by a factor of %Jmax{q,o} than the corresponding method for 1 I - 4hM + l;?h2M2. References Golub, G., Direct methods for solving elliptic difference equations, in: Morris J.L. (ed.), Symposium on the Theory of Numerical Analysis (Lecture Notes in Math. 193, Springer Verlag, Berlin, 1971). Henrici, P., Applied and Computational Complex Analysis (Wiley, New York, 1976). Lambert, J.D., Computational Methods in Ordinary Differential Equations (Wiley, London, 1973). Makinson, G.J., Stable high order implicit methods for the numerical solution of systems of differential equations, The Computer Journal 3 (1968) 305-310. Makela, M., Nevanlinna, 0. and Sipila, A.H., On the concepts of convergence, consistency and stability in connection with some numerical methods, Numer. Math. 22 (1974) 261-274. N$rsett, S . P . , One-step methods of Hermite type for numerical integration of stiff systems, BIT 14 (1974) 63-17.
326
A . ISERLES
[7] Obrechkoff, N., Sur les quadraturcs mecaniques (Bulgarian, French summary), Spisanie Bulgar. Akad. Nauk 65 (1942) 191-289. [81
Wanner, G . , Hairer, E. and Ndrsett, S.P., Order stars and stability theorems, BIT 18 (1978) 475-489.
191
Young, D . M . , Iterative Solution of Large Linear Systems (Academic Press, New York, 1971).
A N A L , Y T I C A L A N D N U N E R I C A I . APPRlIACHE'S TO A S Y M P T O T I C PROBLEMS I N A N A L P S I S S. A x i ' l s s o n , L.S. F r d n k , A . vdii d e r S ! t i i s @ N ~ ~ - t l i - - i lland ~l F u b l i s h i n y corn pin^), 1 9 8 1
ids.)
ESTIMATING THE DISCRETIZATION ERROR IN THREE POINT DIFFERENCE SCHEMES
FOR SECOND ORDER LINEAR SINGULARLY PERTURBED
nvp
R.M.M. Mattheij Mathematisch Instituut University of Nijmegen Toernooiveld, 6525 ED Nijmegen The Netherlands
ABSTRACT Consider the problem (1)
+ py' + gy
-sy"
= r,
y(-l),y(l) given,
and a difference scheme on a set of nodalpoints t . , i = 0,...,N
+ b.u. + c , u .
a.u.
(2)
1 i+1
1 1
1
1-1
=
r
i
(where u. denotes an approximant to
y(ti)). By using a technique developped in 151 for estimating homogeneous solutions of (2), we show how we can obtain useful estimates for the local Green's functions { G . ( j ) } of (2) ({G,(j)} is a solution of ( 2 ) with B C G (j) = G (j) = 0, r , = $7 ) . Once the local discretization error is computed, 0 N 1 ij an estimate for the global error simply follows using these {G.(j)j . This method
provides for a good insight into the error propagation. It is also a useful alternative for positivity arguments (notably when ( 2 ) is not positive). We give several examples of applications
1.
Introduction. Consider the ODE
+ p(x)y' + q(x)y
-'y"
(1.1)
=
r(x),
-1
5
x 5 1
and the boundary conditions (1.21
y(-l)
=
A
, y(1)
= 1,.
Choose a set of equidistant nodal points to,..., tN, with t = -1, tN = 1, and let 0 V . t . .-t,= h. On this mesh we consider a general three point (difference) scheme 1
l+l
1
a,u.
(1.3) (wliere r ,
1 1+1
=
r(ti)
,
+ b.u, + 1 1
c.U. 1
1-1
= r. 1
but may be as well some approximation).
The purpose of th s paper is to give insight in the influence of local errors on the global error. A useful mean for this is the so called local Green's function,
328
R.M . M .
which will be defined explicitely in
§
MATTHEIJ
3. We do not pay any special attention to
the local errors; Their derivation is a straightforward, though sometimes very fedious work (cf C 3 1 ) . Moreover a complete analysis would be too much for a paper like this. On the other hand we are convinced that a suitable insight in these local Green's functions is at least sufficient to get a qualitative idea of the global error. Central in the present estimation is a paper of the author [ 5 1 , where some tools are given to estimate basis solutions of three term recursions. Since these last solutions play a paramount role in the Green's functions we thus are able to give a number of (interesting) estimates. A short review of [51 is
I 2. For higher dimensional problems (cf 161) it seems likely that estimates like the ones derived in 171 may do the same.
given in
The estimation theorems are demonstrated by three examples in 5 4. First it is shown how error estimates for Il'in's scheme might be found. A second example, dealing with a turning point problem shows to which sharp results the estimates may lead. And finally we consider the central difference scheme, some merits (especially if care is taken for the layer) and some problems. 2.
Basis solutions and their estimates. Since the homogeneous equation plays a fundamental role regardin9 the propa-
gation of errors, it makes sense to look for a suitable basis of those solutions. I4oreover we need estimates for them to estimate the global error. Let us denote a sequence {x,} by x Definition 2.1 go = 0
,
g1
Property 2.2
=
Let g be a solution of the homogeneous part of (1.3) defined by
1 and f a solution defined by fN
=
0
,
fN-l = 1.
The discrete boundary value problem has a unique solution iff f and
g are independent, i.e. form a basis. We can write the recursion as a system AU = B, where U = ( u l , . . . , u N- 1 ) I , and similarly B the vector of inhomogeneous terms. If f and g would be linearly
Proof: ___
dependent then fo
=
f = 0, so AF = 0, with F N
=
(fl,...,fN-l)l,thus implying that
A is singular etc.
0
We nowassume that only well posed problems (i.e. with nonsingular A like in the proof of 2.2) are considered, so f and g form a basis. In order to estimate these solutions we use the results of 151, where a number of estimation properties is given, based on both simple qeometrical and algebraical insights. We shall sketch some main results first: Regarding (1.3) we introduce the so called
local characteristic polynomial
329
ESTIMATING THE D I S C R E T I Z A T I O N ERROR
Definition
2.4
The recursion is called of Type I if both roots of (2.3) are
positive and of Type I1 if the roots of (2.3) have opposite sign for all i. Denote the roots of (2.3) by ai,Di; let lail 2 113. 1
.
In the setting of singular
perturbation problems it is no real restriction to assume that a , , Ci. are real, 1
1
which will be done therefore. Remark 2.5
In the usual meaning of positivity of a method, viz. that the corres-
ponding matrix A (see proof of 2.2) is an M-matrix, the recursion is of Type I. From (1.3) we can now deduce the following important nonlinear recursion bi - i_ ' a. a.
U.
'i+l
(2.6)
(NB we allow
6 - 1+1 U.
to be
p .
1
c)
or
1
1
_1 'i
m)
Corresponding to (2.6) define (2.7)
bi $i(x) = - a. 1
-
51 a. x 1
If 4 , is the same for a11 i then we can give a well known graphical interpretation 1
of the "path" of the
{p,},
see fig. 2.1 and fig. 2.2 for Type I and Type I1 res-
pectively. Note that if a , and 13. are well separated, w'e find convergence to a , in forward direction and to D . in backward direction. The estimates below are based on similar arguments, but for using a different graph (or 0.) for each step.
monolonic 10 t e r a Lion
fig.2.1.
Type I
330
R . M . M . MATTHEIJ
not monotonic iteration
,......... +
--bi
a.
\I
x = Oi(X)
fig. 2.2. Type I1
Especially, we denote the growth factors for the basis sulutions as
(2.8)
Ti =
fi f .
i'
I
i
1-1
=--
'i- 1
The following bounds for them can e.g. be given then (cf C5l) (N.B. By [ p ;...;pkl 1 we denote the smallest closed interval that contains p l,. . . ,Pi) . Theorem 2.9
Let the recursion be of Type I. Let
v,It1
ai 2
u,
I-g-
,
-b
E
8. 1
1
(b) Vi
ai t 13
,
a. 1
(1
2
.
j'
Then
0 ,
1
0
V.
E
min j
.L +2 ,aj
1
i
€r-
i-1 a. 1-1
+ -'i,--1 bi-l
-b . 1-1 a. 1-1 -c.
,
1 1 1
a. > 1 1
Theorem 2.11
b1 .
a. 1-1
6 and a . nonincreasing; Then -b
(c)
(i 2 2)
Let the recursion be of Type I and
,
I1
Then
a ;...;a 1 1 i- 1
;
1
Corollary 2.10 (a) Vi
(> 0 ) .
2 8 . : Then
1
T. 6 1
bi
,
-c,
,1 1 . ai+bi
Let the recursion be of Type 11, then
331
ESTIMATING THE DISCRETIZATION ERROR
Sketch of proofs for 2.10, 2.11: If e.g. we have
Type I, then is
fig. 2.1); on the other hand a . cannot become smaller than R . , -b . -a. always larger than 1, hence :i+l 2 -& - "i "i . i' a. ai bi Remark 2.12
-
Apart from 2.9
-b .
i+l
< -& a.
(see
which in turn is
2.11 one can find many other estimation theorems.
However we shall only consider applications for the above ones 3.
Discrete local Green's functions. In order to indicate how the local errors influence the global error and also
for estimating them, it is useful to have the following so called (discrete) local N Green's functions {G.(j)}i=O = G(j). First let 1 < j 5 N-1. Then they satisfy a , G . (j) + biGi(j) + ciGiWl(j),= 6 . . 1 1+1 13'
(3.1)
.
1
5
i
N-1
,
(where 6 , . is the usual Kronecker symbol) 13
and moreover the boundary condition G (j) = GN(j) = 0 .
(3.2)
0
We can formally find an expression for G.(j) in terms of basis solutions, making 1
a distinction between the cases i 5 j and i obtain
K;
i '
(a) Gi(j) =
K ,
-,
i
5
j
(b) Gi(j) =
K.
fi -, fi
i
2
j,
(3.3)
where
2
j and then matching them. So we
I
3 qj
3
satisfies
The expression in (3.4) needs further simplification. To this end define (3.5)
$ 3. = 9j-lfj - gjfj-1
Then by simple manipulation (NB in fact an order reduction "trick") we see (3.6)
.
332
R M. M. MATTHEIJ
Using (3.7) in (3.4) we obtain
Substituting this finally in (3.3)(a) we get (3.9) The same expression (3.0) for (3.3)(b) is less interesting regarding the estimation problem (cf 16,531). However we can also find a "backward" expression instead of (3.7). This then gives the following result (which can also be checked by simple substitution). (3.10)
fi N-1 C i' P. Jl F, G. (1) = - - 9N f N - ~aj .t=j+l P.
i Z j
Finally G(0) and G(N) are just defined by (3.11)
G .( 0 ) =
f.
fO (3.12)
G. (N) =
q.
9N The general solution of the problem (1.2), (1.3) is then given by N- 1 (3.13)
u.
=
C
Gi(j)ri
j=1
+
Gi(0)h
+
Gi(N)p
Now if the local truncation error 6 , is defined by (3.14)
aiy(titl) + b.y(til t ciy(ti-l) = ri + 6
i
Then we find for the global error e. defined by I'
(3.15)
e.
=
y(t.1-u. 1
I'
the following relation (3.16)
a.e. + b.e. + ciei-l = 6i 1 1+1 1 1
and eo = eN = 0. Hence we can write N- 1
This last expression can be used to estimate ei, once the 6 . are computed. The 3
importance of (3.17) for singular perturbation problems is that it allows for a
333
ESTIMATING THE DISCRETIZATION ERROR
splitting in errors arising from the layer and from the smooth part. 4. Applications In the following three subsections we give possible applications of the use of local Green's functions, estimated with the help of the results of 52. 4.1 Positive schemes and p > 0, q 2 0 If the recursion is of Type I (positive scheme, see 2 . 5 ) ,
then we find from
2.10, (3.9) and (3.10 Let the recursion be of Type I1 and V B . 1
Property 4.1
/i-1
.I
b,
lGi(j)
, /j-1
-a,
i-
1
5
u
~ <...I- ~ al,then
,
We now apply 4.1 to Il'in's scheme, viz
Property 4.2
Let p. be nonincreasing. Define )I = max exp\-,,.
Pih\
Then we have for
Il'in's scheme
N-&-1 G. (j) S
Proof:
i
-(-)
Pj I-$
1
I-, i f i > j
If pi is nonincreasing, ai is nonincreasing
-a
straightforward estimation. E . g . Pih 2E ,this is estimated where s a. = -. Corollary 4.4
If p has no monotonicity property then an estimate like in 4.3 holds
with pL replaced by n = min p , (cf 2 . 1 0 ( a ) ) . 4
J
Remark 4.5
3
h Note that the estimates for G.(j) in 4.3 - apart from the factor -
Pj
.
R M.M. MATTHEIJ
334
1 11-1 1 N-11-1 of (-1 which is 1 -J, 1-$ the toll paid for estimating inaccurately - resemble the singular solution (for
which arises from discretization and the factors i
S j)
and regular solution (for i
2
j)
(-)
. For the latter one we have a backward
Euler approximation in fact. As a whole they give rise to a nice approximation of the continuous local Green's function (cf [21) see fig.4.1.
local Green's f u n c t i o n
N
1
0
fig. 4.1 E.g. for the case h >
E,
it is fairly simple now to give an estimate for the global
error. Of course, this does not give any new result since it can e.g. be found in 131. However, our main intention is not to give a number of ready made estimates but rather a recipe for them. 4.2 A turning point p r e m Consider the equation (4.6)
-EY"
+
axy' = 0 ,
a < 0
Let y(-1) and y(1) be given again. At x = 0 we have a turning point. Denote the corresponding mesh point by tm. Suppose again that Il'in's scheme is used for discretization. It immediately follows that 1 is always a local characteristic root. We have therefore Property 4.7
If i < m, then
8,
=
1 < ai. If i > m, then
Bi
<
1
= a i.
If i = m,
335
ESTIMATING THE DISCRETIZATION ERROR
then a . 1
=
a 1.
=
1. The “non 1-root’‘equals exp
Rather than estimating G(j) for 1
I: j
S
N-1, we now use this information to esti-
mate the discrete solution itself, viz with the help of G(0) and G(N). Indeed, for the approximant u, there holds
Using the fact that the roots have a nice monotonic behaviour we now get, using
2.10, the following nice result Property 4.8 y(1) + y(-l),if i
5
m
~ ( - 1 )+ y(l), if i 2 m The result is also graphically represented in fig. 4.2.
t u r n i n g p o i x t problem . . a *
0
-1
*.,..,._..-.........
a . , .
,
+1
fig. 4.2 Remark 4.9
In fact we used a bit more in 4.8. For, if we have monotonicity of
local characteristic roots, then the growth factor is lying in between two consecutive roots (cf C5,551). Remark 4.10
The result of 4.8 is also sharp in comparison with [ 4 , CH81 where
one gets for the continuous case e.g.
336
R.M.M.
MATTHEIJ
where 6 is any positive number. 4.3 On some well and ill behaving of the central difference scheme As a final example of application of our estimates, we consider a Type I1
recursion: -2~+p.h (4.11)
2 2h
'i+l
4ct2qih u +
t
2h2
-2€-pih ~1i-1 = 2h2
rj
Just for simplicity, take q , p constant and assume both are positive. Then indeed (4.11) is a Type I1 recursion for
F.
small enough.
From 2.11 we then find the following estimates for the local Green's functions: Property 4.12
Let
E
2 be small with respect to h q and hp then
2qh is . This shows a very interesting feature of the central difference scheme: If -
P
really larger than 1, then the local Green's functions have an almost peak shape like is shown in fig. 4.3.
locaZ Green's functiori
fig. 4.3
337
ESTIMATING THE DISCRETIZATION ERROR
This means that errors damp out very quickly, and in fact
-
for smooth solutions -
the global error almost equals the local error (cf C6,§41). On the other hand the damping is not of the proper character regarding the singular solution. This then explains once more why this scheme is not appropriate for l a y a problems. However, if this boundary layer is treated differently (e.g. upwinding) then the merits of the central difference scheme for the rest of the interval are obvious (cf also 111).
-*
If q would be very small, then 4.12 does not make sense. In fact we then have
-*
local characteristic roots which are almost +1 and -1. More precisely a =: -1
B z 1
P
P'
(still 6 very small). The discrete solution is then given by (cf (3.11),
(3,12), (3.13)) (4.13)
u. =
1 a -B
I (BiaN-aiBN)A + (ai-&
Ll1
Hence for N even we may expect cancellation effects, in other words an ill conditioned problem, whereas for N odd everything is well conditioned! (cf C6, 6.61).
References
111
Axelsson,
O.,
Stability and quasi optimality of Galerkin finite element appro-
ximations for convection-diffusion equations, Report 8005, Mathematisch Instituut, Nijmegen, 1980. 121 Hemker, P.W.,
A numerical study of stiff two-point boundary value problem,
Thesis, Amsterdam 1977. C31 Kellog, R.B. and Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 (19781, 1025-1039. [41 O'Malley, R.E., Introduction to singular perturbations, Academic Press, New York, 1974.
[51 Mattheij, R.M.M., Accurate estimates of solutions of second order equations, Lin. Alg. Applics., 12 (1975), 29-54. [6l Mattheij, R.M.M.,
Estimates for the errors in the solution of linear boundary
value problems, due to perturbations. Report 8013, Mathematisch Instituut, Nijmegen, 1980. 171 Van der Sluis, A . ,
Estimating the solutions of slowly varying recursions,
SIAM J. Math. Anal. 7 (1976), 662-695.
This Page Intentionally Left Blank
ANALYTICAL AND NUMERICAL APPROACHES T O ASYMPTOTIC PROELEMS I N A N A L Y S I S 5 . A x e l s s o n , L . S . F r a n k , A . v a n der S l u i s ( e d s . ) 0 N o r t h - H o l l a n d P u b l i s h i n g C o m p a n y , 1981
FAILURE OF NERVE IMPULSE PXOPAGATION FOR NONUNIFORM NERVE AXONS 3 . P. PAUWELUSSEN Mathematical Centre Kruislaan 413 1098 S J Amsterdam The Netherlands
In this paper we shall discuss a nonlinear diffusion problem on an unbounded domain, arising in neurophysiology. We shall show that, depending on the choice of parameters, this problem may have two, one or no stationary solution which satisfy certain boundary conditions. In all of these situations we shall discuss the large time behaviour of solutions to the diffusion problem and interpret our results. I.
INTRODUCTION
If somewhere along a uniform unmyelinated nerve axon, the transmembrane potential is raised above a certain threshold, a potential wave may begin to travel with constant speed, down the axon. This threshold is an increasing function of the diameter and therefore, a sudden increase in this diameter at some point may have the effect that waves, travelling towards this point may fail to propagate any further. For the study of these phenonena, several reaction diffusion systems have been proposed by Goldstein and Rall r 5 1 , Rinzel [ I 0 1 and Pauwelussen C 7 1 C81. In this paper we shall consider the most simple model: i t leads to a nonlinear diffusion equation on an unbounded domain, which is treated in detail in C71. Let the variable t denote time and let x measure the distance along the axon. Suppose that the axon radius r(x) at x = 0 changes from 1 for x < 0 to the constant value r 1 0. At x = 0, the membrane potential u as well as the internal current which is proportional to r(x)'uX, should be continuous. Hence the gradient ux has a jump at x = 0. To remove this discontinuity we replace x b y x/r2 when x 0. This results in the following problem C 7 1 (1.1)
[ u t = eE(x)uxx
I
(1.2) (I) U(X,O)
where a [O,ll,
E
(O,I/Z),
x
x
u(~-u)(u-a),
x(x),
=
IR-= (-in),
+
x
E
IR+ u IR-,
t > O
IR,
is a continuous function, taking on v lues in the in erval = ( 0 , ~ )and e,(x) is given by
IR'
, x < o , (1.3)
e (x)
=
{I IE - = I -
r2
,x>o.
I n c81 the existence and uniqueness of a solution of a more general reaction-diffusion system is treated and as a special case it follows that there exists exactly one function u(x,t), which, together with its first spatial derivative u is continuous on R x R ' , which has continuous derivatives ut and uxx on IR- x R 3 and on IR' x IR', and which satisfies ( 1 . 1 ) and (1.2). To emphasize the dependence of u on 339
.
340
x
J P
. PAUWELUSSEN
we shall write this solution as u(x,t;x).
In the special situation that E = I and thus e,(x) 5 I , equation ( 1 . 1 ) allows travelling wave solutions, i.e. there exist solutions u(x,t) = w(z), z = x - ct, for some c E R such that w(-m) = 1 and w(+m) = 0. Thus w, and any translate of w as well, satisfies the equation w" + cw' + f(w)
=
0,
R
2 E
,
where f(w) = w(1-w)(w-a). It can be proved that except for the freedom of translations w is unique and strictly decreasing. Noreover the wave speed c is unique and c > 0, see Fife and Ilcleod [ 2 1 , Hadeler and Rothe 161 and Aronson and IJeinberger [ 1 1. Fife and Elcleod also proved that under the conditions (1.4)
lim inf x(x) > x 3 -m
a,
(1.5)
lim sup x -f +m
a,
x(x)
<
the solution u(x,t) converges to a travelling wave w(x-ct+xo) exponentially, for some xo, as t 3 a. If E < 1 but 1 - E small, one might expect that the solution of Problem I, with x still satisfying (1.4) and (1,5), behaves as a wave, travelling from the region where eE(x) = 1 towards the region where eE(x) = E . On the other hand, in view of the observations in the beginning of this Section, one might expect that for small E > 0 (i.e, r large) and initial function x which satisfies ( 1 . 4 ) and (1.5), the solution of Problem I is blocked in some sense at the point x = 0 . In fact, there exists an E * E ( 0 , l ) such that for E t (O,E*), two stationary solutions q(x) (i.e. qt:O) of equation ( 1 . 1 ) exist, one above the other, Both solutions are strictly decreasing, and for a certain class of initial functions x, u(x,t;)o converges towards the lower of these equilibrium solutions.
In the next Section we shall deal with the existence of these stationary s o l u tions while Section 3 will be devoted to their stability properties. 2. STATIONARY SOLUTIONS In this Section we shall consider the problem (2.1) (2.2)
(11)
{
eE(x) qxx q(-m)
= 1,
+
f(q) = 0, q(+m)
x
E
R\ {Ol,
t
0,
= 0.
For constant eE(x), eE(x) :E say, the points (q,q,) = ( 0 , O ) and q,qx) = (1,O) are critical points in the (q,qx)-plane and are saddle points. Moreover in the region W = {(q,qx):q, < 0, 0 < q < 11 there is only one stable manifold ending in (0,O) and one unstable manifold leaving ( 1 , O ) , We shall denote these manifolds by E-manifolds. Obviously, solutions of Problem I1 correspond to the stable E-manifold for x > 0 and the unstable l-manifold for x < 0, where these manifolds match for x = 0 .
FAILURE OF NERVE IMPULSE PROPAGATION
341
Then, the above manifolds intersect if E E (O,E*) whilst they do not intersect for E > E* and we have the following Theorem. THEOREM 2.1. >
there e x i s t no s o l u t i o n s of P r o b l e m II.
(i)
If
(ii)
If E = E*, there e x i s t s a u n i q u e s o l u t i o n q(x) and i t i s s t r i c t l y d e c r e a s i n g .
E
E*,
of P r o b l e m 11 w i t h q(0) = a
(iii) If 0
< E < E*, P r o b l e m II h a s e x a c t l y t w o s o l u t i o n s q+ a n d q-. d e c r e a s i n g , w h e r e q-(x) < q+(x), x E R a n d q-(O) < a < q ' ( 0 ) .
In fact, q+ and q- coincide if
E
They a r e
= E*.
RENARK. Equation (2.1) has also been considered by Fife and Peletier C41. They extended part (i) in Theorem 2.1 to the situation that eE(x) is nonincreasing and bounded away from zero.
A sketch of the corresponding bifureation diagram of q(0)vs.e 2.1. below.
E
Fig. 2.1.: q(0)
is given in fig.
E
VS.
E
3 . STABILITY AND PROPAGATION
and give some results on In this Section we shall first assume that c E (O,E*] stability and instability of the stationary solutions q+(x) and q-(x). Then for E E ( ~ * , 1 ) we shall give a result on the convergence of the solution of Problem I for x > 0 towards a travelling wave as t + m , To begin with we consider for where ho i s defined by q'(h0)
=
E
< E*
the function qh(X) z q+(x+h) for 0
5
h
ho
a.
From the fact that q+(x) is a translate of q-(x) for q'(x) > q'(0) as well as for q*(x) < q-(o) i t follows that qh(x) > q-(x) if h < ho, for all x E R , For x r: 0 we have qh(x) q"(x) h
=
-
> a and therefore (qh)
v
h)
+
f(qh)
< 0.
Hence for x # 0 (3.1)
eE(x)q;ll(x)
=
CeE(x)-eE(x+h)lq~(x)
i.e. qh(x) is a supersolution of ( 1 . 1 )
0,
and as a consequence, if x(x)
5
qh(X)
for
342
some h result.
J.P.
f:
I0,h ) then u(x,t:x)
5
0
PAWELUSSEN
C71 f o r t h i s c o m p a r i s o n
ql l (x) f o r a l l t 2 0 , s e e
I f 11 > 0 , qt1 i s a s t r i c t s u p e r s o l u t i o n o n a s u b i n t e r v a l o f IR. Hence by a modif i c a t i o n o f a r e s u l t d u e t o Aronson and l i e i n b e r g e r [ I ; P r o p o s i t i o n 2. 21 i t f o l l o w s t h a t u ( x , t ; q l l ) t e n d s t o t h e maximal s o l u t i o n o f e q u a t i o n ( 2 . 1 ) l y i n g be low q + , u n i f o r m l y o n bounded i n r e r v a l s . T h i s maximal s t a t i o n a r y s o l u t i o n i s q- a nd t h e r e f ore l i m sup u ( x , t ; x )
(3.2)
5
q-(x).
t - f m
I n o r d e r t o e x c l u d e t h e p o s s i b i l i t y o f u ( x , t : ) o t e n d i n g t o 0 u n i f o r m l y on IR a s m , we a ssu me t h a t t h e F i f e and Plcleod - c o n d i t i o n (1.4) h o l d s . Using s i m i l a r a r g u m e n t s a s i n 1 2 1 one c a n c o n s t r u c t a s u b s o l u t i o n Q(x) e q u a t i o n ( 1 . 1 ) (s ubs o l u t i o n s s a t i s f y ( 3 . 1 ) w i t h t h e i n e q u a l i t y - s i g n r e v e r s e d ) b e l o w x ( x ) a nd w i t h q - ( x ) a s o n l y s t a t i o n a r y s o l u t i o n o f P robl em ( 1 . 1 ) b e t w e e n g ( x ) and x ( x ) ( s e e L71 f o r d e t a i l s ) . C o n s e q u e n t l y , u ( x , t ) : Q ) f q- a nd t h e r e f o r e t
-f
(3.3)
l i m inf u(x,t:x) t
THEOREM 3.1.
(3 . 3 )
> q-(x).
m
-f
Let
E
< c* a n d s u p p o s e t h a t for
x ( x ) < qh(X)'
x
E
IR
and t h a t ( 1 . 4 ) i s s a t i s f i e d . Then u ( x , t ; x ) intervals
.
some
11
c (O,hO)
-
-f
q ( x ) as t
-f
m,
u n i f o r m l y on c l o s e d
A t t h e e x p e n s e o f much more c o m p l i c a t e d a n a l y s i s , t h i s r e s u l t c a n b e e x t e n d e d t o c = E* a n d h = 0 .
Next s u p p o s e t h a t h < 0. T hus q h i s o b t a i n e d from q+ b y s h i f t i n g i t t o t h e r i g h t and one c a n o v e r a d i s t a n c e I h l . Then q h i s a s u b s o l u t i o n f o r e q u a t i o n ( 1 . 1 ) p r o v e , u s i n g t h e same t e c h n i q u e s a s i n Theorem 3.1, t h a t i f x ( x ) > qkl(x) f o r h < 0 t h e n u ( x , t ; x ) a p p r o a c h e s 1 as t + m, M o r e o v e r , o n e e x p e c t s t h a t i n t h i s case u " t r a v e l s away" from t h e p o i n t x = 0. I n f a c t t h e f o l l o w i n g r e s u l t h o l d s C71. THEOKEM 3.2. (3 . 5 )
Let
E
E*
a n d suppose t h a t f o r some 11 < 0
' qh(X)
x(x)
a n d t h a t c o n d i t i o n ( 1 . 5 ) i s s a t i s f i e d . T h e n , w i t h w *(z ) = w(z&) there e x i s t n u m b e r s K, u > 0 a n d x o s u c h t h a t
l i i ( x , t ; x ) - w*(x-c*t+xo)
I
_i
Ke-pt,
a n d c* = c/;,
t t 0,
u n i f o r m l y on IR.
Recall t h a t w i s t h e t r a v e l l i n g wave s o l u t i o n u ( x , t ) = w ( x - c t ) of ( 1 . 1 ) E = I a n d n o t e t h a t t h e f u n c t i o n w*(z) : w(z&) s a t i s f i e s the equation EW"
+
CV'FW'
for
+ f(W) = 0 ,
i.e. w* i s t h e t r a v e l l i n g wave w i t h s p e e d c* = c a , c o r r e s p o n d i n g t o a u n i f o r m n e r v e a x o n o f r a d i u s r = c - I l 2 w h i c h i s a l s o t h e r a d i u s f o r x > 0 of t h e nonuni form n e r v e axon under c o n s i d e r a t i o n . Ag a in Theorem 3.2 c a n be e x t e n d e d i n t h e s e n s e t h a t ( 3 . 5 ) c a n b e r e p l a c e d by x ( x ) > q + ( x ) . I t f o l l o w s f r o m t h e a b o v e t h e o r e m s t h a t q-(x) i s s t a b l e a nd q + ( x ) s h o u l d b e r e g a r d e d as u n s t a b l e .
FAILURE OF NERVE IMPULSE PROPAGATION
343
The conditions (1.4) and (l.5), ensure if c = 1 that u(x,t;X) tends to w(x-ct+xO) for some x0 c IR as t + a, see c21. One can think of this result as that the convergence of u towards w is due to the fact that, initially, u resembles the wave w sufficiently strongly. This result can be extended over the interval ( € * , I 1 in the following way. a n d l e t c o n d i t i o n s (1.4) and (1.5) be s a t i s f i e d . THEOREII 3.3. L e t E E ( € * , I ] T h e n there e x i s t n u m b e r s K, LI 0 a n d xo s u c h t h a t
lu(x,t;)o u n i f o r m l y on R
.
- w*(x-c*t+xO)l
5
Ke-L! t ,
t -, 0,
RENARK. The demonstration of Theorem 3.2 - 3.3 i s stronp;ly based on the paper by Fife and Plcleod mentioned before, and a recent paper by Veling r 1 I l who has considered equation ( 1 . 1 ) for E = 1 within the framework of an initial boundary value problem on a half line, and he gave conditions under which the solution of this problem tends to a travelling wave for t +
-.
We conclude this Section with an estimate of the rate of convergence of u(x,t;~) towards q-. The proof which is given in 171, u s e s the techniques introduced in the proof of Theorem 5 in Fife and Peletier 131 (see also Theorem 4.3 in Pauwelusen and Peletier C9l). THEOREM 3.4. L e t
E < E*.
T h e n there e x i s t p o s i t i v e c o a s t a n t s 6 , a~n d K s u c h
that
w h e r e 11 -11
d e r i o t e s t h e supremum n o r m on C(IR).
REFEWNCES
[ I 1 Aronson, D.G., IJeinberger, H.F.: Nonlinear diffusion in population genetics, combustion and nerve impulse propagation. Lecture Notes in Yathematics 446, 5-49, Springer, New York 1975.
121 Fife, P.C., llcleod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions, Archive Rat. Elech. An, 65, 333-361 (1977). [3] Fife, P.C., Peletier, L.A.: Nonlinear diffusion in population genetics. Archive Rat. ?tech. An. 64, 93-109 (1977). r 4 1 Fife, P.C., Peletier, L.A.: Clines induced by variable migration. To appear in the proceedings of the "Conference on models of biological growth and spread." Heidelberg, 1978, Lecture Notes in Biomathematics, New York.
151 Goldstein, S.S.,
Rall, IJ.: Change of action potential shape and velocity for changing core conductor geometry. Biophys. J. 14, 731-757 (1974).
161 Iiadeler, K.P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251-263 (1975). C71 Pauwelussen, J . P . : Nerve impulse propagation in a branching nerve system: a simple model. (Preprint). Mathematical Centre Report (1980). [ S l Pauwelussen, J.P., One way traffic of pulses in a neuron. in preparation,
191 Pauwelussen, J.P., Peletier, L.A.: Clines in the presence of asymmetric migration. To appear in J. Math. Biol.
344
J. P. PAUWELUSSEN
[I01 Rinzel, J . : Repetitive nerve impulse propagation: numerical results and methods. In: Fitzgibbon, TJ.E., Tlalker, M.F. (eds) Nonlinear diffusion. Pitman. London, San Francisco, Ilelbourne (1977).
[I11 Veling, E.J,M.:
Travelling waves in an initial-boundary value problem. To appear i n : Proc. Ed, Math. SOC.
A N A L Y T I C A L AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS I N A N A L Y S I S S. A x e l s s O n , L . S . F r a n k , A . v a n d e r Sluis ( e d s . ) @ N o r t h - H o l l a n d P u b l i s h i n g C o m p a n y , 1981
THE PENALTY METHOD FOR THE V I B R A T I N G STKLNC WITH AN OBSTACLE
M i c h e l l e Schatzman C . N . R . S . e t U n i v e r s i t 6 P . e t M.Curie L a b o r a t o i r e d ' h n a l y s e NumErique Paris,France
We c o n s i d e r t h e f o l l o w i n g problem which modelizes t h e movement o f a s t r i n g v i b r a t i n g a g a i n s t an o b s t a c l e @ w i t h o u t l o s s of energy : 0 u : u t t - uxx 2 0 , u ( x , t ) 2 @ ( x ) , s u p p n u c I ( x , t ) / u(x,t) = @(x)l
,
(u;
+ U:)t
-
2(UxUt)x
=
0.
We prove t h a t i f u i s s u p p o r t e d by d i s c r e t e s p a c e - l i k e c u r v e s , t h e n t h e s o l u t i o n of t h e Cauchy problem i s unique.Under t h e same h y p o t h e s i s , t h e s o l u t i o n of t h e p e n a l i z e d problem (u'
0u'-
)-/'
-@
= 0
converges t o t h e s o l u t i o n of t h e t r u e problem. 1 . I N T K O D U C T I O N AND SUMMARY OF PREVIOUS RESULTS. We c o n s i d e r t h e s m a l l t r a n s v e r s e v i b r a t i o n s of an i n f i n i t e s t r i n g which i s c o n s t r a i n e d t o s t a y on one s i d e of an o b s t a c l e . T h e d i s p l a c e m e n t of t h e p o i n t of c o o r d i n a t e x a t t h e time t i s denoted by u ( x , t ) and t h e o b s t a c l e by @ ( x ) . T h e mat h e m a t i c a l model we propose f o r t h i s phenomenon i s D u i ut t
(1)
-
2
uxx
c I(x,t) / 2 @(x),
(2)
suppou
(3)
u(x,t)
(4)
(u:
0,
u(x,t) = @(x)l,
+ u p t - 2 ( uXu t ) X
= 0.
C o n d i t i o n ( I ) e x p r e s s e s t h e f a c t t h a t t h e s t r i n g does n o t s t i c k t o t h e o b s t a c l e ; c o n d i t i o n ( 2 ) means t h a t whenever t h e s t r i n g i s s t r i c t l y above t h e o b s t a c l e , i t s a t i s f i e s t h e wave e q u a t i o n ; ( 3 ) h a s an obvious meaning,and ( 4 ) i s t h e c o n s e r v a t i o n of e n e r g y ; t h i s form of c o n s e r v a t i o n of energy i s t h e most convenient o n e . I t i m p l i e s indeed t h a t t h e v e l o c i t y of a m a t e r i a l p o i n t of t h e s t r i n g i s r e v e r s e d through a s h o c k ; t o w r i t e t h i s r e s u l t under a mathematical form,we need a t r a c e theorem which w i l l be n e c e s s a r y anyway i n t h e s e q u e l of t h i s p a p e r , a n d which we proceed now t o d e s c r i b e . Let
6
and
n
be t h e c h a r a c t e r i s t i c c o o r d i n a t e s
and l e t u s denote
(6) Let
(7)
6-ri
s+rl
G(5,rl)
= u(-,-).
V = {u
E Hjoc(R x
R+) / V A > 0,W B > 0,W t
+ u:)(x,t)
dx
J-B(ui B
nfi
I t h a s been proved i n
2
O,A
C(A,B) < + m}.
, c h a p t e r V , t h a t i f u i s i n V,then 345
,
346
M. SCHATZMAN
i s i n c r e a s i n g f r o m [-a,")
F.
+
i s i n c r e a s i n g from We d e f i n e
dur
Functions N
5
at
-
t o L*(a,b),Va,Vb
a; %(LO)
[-c,m)
and
t o L2(c,d),Vc,Vd
a 3
aE are d e f i n e d o n
/ 5
>
- r l , n E Nq! w i t h N
0
> c.
set
-
is negligible;similarly,funcLions
{(E,n)
d
> a,and
and ari negligible.
{(c,q)
&?' an
/ 6 E Nt,,ri
-CJ
Definition l.A function a which is Lipschitz continwus with Lipschitz constant 1 will be said to define a non the-like curve;such a function will be called non time-like too. function o ,proposition
F o r a non t i m e - l i k e t e l l us t h a t
$c (.,a(.))
V . 2 and c o r o l l a r y V . 4 of
E Lloc (R,(l+U')dx)
(.,o(.)) E Lfoc (R,(I-o')dx) -
(.,(I(.))
E Lloc ( { x / O ( x ) > O ! , ( l + o ' ) d x )
( . , o ( . ) ) E Lioc (tx/o(x) > 0 ,(l-a')dx) a n d a l m o s t e v e r y w h e r e on ( x / l u ' ( x ) l -aCu (x,o(x)) at
=
,
< 1)
l i mu ( x , o ( x ) + h ) - u ( x , o ( x ) ) 11 hi0
e x i s t s ,and
a l m o s t e v e r y w h e r e on {x / l o ' ( x ) I Similarly-, the l i m i t
-
-a(axU t, o ( x ) )
=
limu ( x , a ( x ) - h )
e x i s t s a l m o s t e v e r y w h e r e on
-
< I!.
- U(X,O(X))
-I1
hJ.0 (x
/ lo'(x)I
I,o(x) > 0 ) and,
a l m o s t e v e r y w h e r e on { x / l o ' ( x ) l < I , a ( x ) > 01. L e t u s now s t a t e p r o p o s i t i o n v . 5 o f
CZI:
where
tzl
347
VIBRATING STRING WITH AN OBSTACLE
Promsition 2.Let u he in V,such that 41 2 0. Suppse that (12)
- 2 (ux u t ) x = 0,in the sense of distributions on 1R Then,for any non time-like u, 2
2
(ux + u t )
x
R+.
P r o p o s i t i o n 2 i m p l i e s indeed t h a t a l m o s t everywhere t h e v e l o c i t y of a m a t e r i a l p o i n t i s r e v e r s e d through a shock.The " n a t u r a l c c v d i t i o n "
-2: a+ at
-
-__
on t h e c o n t a c t s e t
{u =
$1.
has nu p r e c i s e mathematical meaning,as l o n g a s t h e s t r u c t u r e o f t h e c o n t a c t s e t i s unknown. I t was proved i n [ 2 1
@"
(14)
2
t h a t under t h e h y p o t h e s i s
0
and t h e c o m p a t i b i l i t y c o n d i t i o n s
(15)
u o ( x ) = U(X,O)
(16)
u , ( x ) = ut(x,O)
2 $(XI 2
O a.e.
on { x / u(x,O) = @ ( x ) }
assuming t h a t u ( . , 0 ) i s i n Kioc(LR) and u t ( . , 0 ) i s i n LIOc(lR),then t h e Cauchy problem f o r ( I ) , ( 2 ) , ( 3 ) and ( 4 ) p o s s e s s e s a unique s o l u t i o n . T h i s Cauchy problem w i l l b e c a l l e d problem Pm.
Definition 3.We shall call an o-n region d& of the x - t plane lens shaped if there exists two non time-like functions p and u such that p ( a ) = o ( a ) , p(b) = o ( b ) ; P ( X ) 2 u ( x ) on ( a , b ) ,
and
fi
= { ( x , t ) / a < x < b and p(x) < t < ~ ( x ) } .
'Ihe r e s t r i c t i o n o f Problem Pm t o a l e n s - l i k e r e g i o n u ( x , P ( x ) ) = u,,(x)
,
ut(x,p(x)) = u ~ ( x )
6&
,with i n i t i a l data
x t(a,b) a.e.
on { x E ( a , b ) / I u ' ( x ) l < 1
1 ,
w i l l b e c a l l e d Problem P R,
The s o l u t i o n of Problem P ws c o n s t r u c t e d a s f o l l o w s : L e t w b e t h e f r e e s o l u t i o n of t h e wave e q u a t i o n i . e .
(17)
{
ow
= 0 w(x,O) = U O ( X ) Wt(X,O) = Ul(X)
and l e t
-
(18)
T
(19)
T+
x, t
x, t
2
-
=
{(x',tl) / 0
=
{(x',t') / t' 5 - t + Ix
t' 5 t
-
Ix
x'I
-
X'II
1
b e r e s p e c t i v e l y t h e forward and backward wave cones. We s e t (20)
E=C(x,t)xw(x,Zm
,
,
348
M . SCHATZMAN
and we s h a l l assume from now on t h a t E i s not empty. Let
I = c) IT+
(21)
XI
t
/ (x,t) E E l
I
Then.the boundary of I i s a non time-like curve t = -c(x),as i t i s an envelope of cones with c h a r a c t e r i s t i c s i d e s . Let be the elementary s o l u t i o n supported by t h e forward l i g h t cone T+ ODo + on T 0.0 = (22) 0 elsewhere.
8
E {
It has been proved i n /2J t h a t the s o l u t i o n of problem
+#hP
P, is given by
(23) u = w I where p i s the measure defined by
-
(24)
US
- T''(X))
$(xDT(x))
&D
r e c a l l now some r e s u l t s on the penalized problem
nuE-
(UE
- $I-/€
uE(x,O) = uo(x) { a %X,O) t
5
0,
D
= u,(x)
where r = sup(-r,0). I t has been proved i n [ I J , s e c t i o n s I V and V t h a t t h e r e e x i s t s a subsequence uE which converges weakly
i n V t o a f u n c t i o n u such t h a t
au
(32)
x ( x D 0 ) = U1(x) a.e. on {x/u,(x) > @ ( x ) }
(33)
x ( x D O )( u l ( x )
aU
.
a.e. on { x / u o ( x ) = @ ( X I }
MOreover,Lemma 14 of [Z] can be s t a t e d a s follows
(34)
The l i m i t u of a subsequence u
E
s a t i s f i e s t h e energy condition (4)
if and only i f uE converges t o u i n Hioc@
x
lR+) strongly.
349
VIBRATING STRING WITH AN OBSTACLE
1 I . A UNIQUENESS THEOREM FOR A CLASS OF SOLUTIONS.
later :
(35) (36) (37)
We begin this section by the following result which will prove useful
Promsition 4 . L e t @J be l e n s - s h a d , a n d hunW bv p , and p2 , and suaaOse t u and v are twn functions of V which s a t i s f y (26)-(29),and (30) mdified i n an obvious wav when t h e i n i t i a l conc?iitions are aiven on p , .Assme moreover that OUl& = 0, u(x,P,(x)) = v(x,P,(x)) ,v x E [a.b], a+u a+u a.e. on Cx E [a,bl/lp;(x)l ~ ( x s p l ( x ) )= ~(x,P,(x))
Then u
=
v on
<
11.
&.
Proof.This result uses the same ideas as the proof of uniqueness up to the first =of influence,which was given in [2],section V.2;therefore we shall not give too many details. Let (38) v = o v , (39) F = suppVl& Suppose that F is not empty,and define
.
.
The set where a
As F
(40) For any so that (41)
J = U{T+ /(x,t) E F) XSt J is a union of forward characteristic cones,so that J = {(x,t)/t 2 a(x)) , is not time-like. is not empty, u = C(x.t)/a(x) < t < P2(X)1 # pj E > 0,and x in R,
-
T X,U (X)+E
(v
-
F #
0,
u)lu 2 0.
On the other handsas v is a solution of (26)-(29),and relations (35)-(37) imply (42) 6 - U)I& 2 0. Moreover,arguing as in [ 21 ,section 11, that (43)
so
I (XIa (XI11/ 1 a'(x)l
(30) modified, vIa
< 1.t C F
@.
< 11 = "i~x,u~x~~/~.'(x~l Therefore,as u is a solution ,relations (33)-(36)
give
which has as a consequence
to the line asthen we first note that,as
2
0;
350
SO
M. SCHATZMAN
that
which we square and substract from (45) to get
through the same computations as in the proof of Proposition V.5 of L21. Relation (46) with (44) and (10),(11) prove that
+
a(x,o(x))
o
=
at
> p,(x),~o'(x)l
a.e. on {x/u(x)
<
11.
But,by a computation similar to that of C21,Proposition II.3,if $ is supported in&.
because supp vIaC i(x,t)/t = o(x) > pl(x),lo'(x)I U v & , = 0,and we obtain a contradiction.R
< 11,thanks to (4l).Thus
We now turn to proving the uniqueness theorem whose statement follows. Theorem 5.Let & be lens shawd and bounded by p 1 and p, .Let w be given in V such that C3w = 0,
W(X,P1 (x)) 2 @(x), wt(x,pl(x)) 2 0 a.e. on {x/\p:(x)\ < 1 and w(x,P,(x)) = $(x)l. Let u and v satisfy (1)-(4) on ,with the same initial conditions
a
= V(X,P,(X))
U(X,P,(X)) + a u --(xspl(x)) at
=
+
W(X,P,(X)),
a v
= ~(X.P~(X)) =
w,(x,P,(x))
a,
a.e. on tx/lp;(x)l
< 11;
assume finally that in O u is suprted in a discrete union of mn tim-like curves.Then
uh
= VI&.
Proof .Let with
Supp
0
and
5
T1
=
5 -
inf(Ti+,
{(XsTn(X))/X
... <
Tn
-
I
=
T ~ )
2
a,b
...
R)
> 0, 'di.
s
351
VIBRATING STRING WITH AN OBSTACLE
UIRo
= VIL,
I
as (4) is a strower condition than ( 3 0 ) . The end of the argument is by induction: suppose that on n- 1 = {(Xst)/Tn-l(X)
< t < Tn(X)l
u is equal to v.Then,by Proposition 1 .
-(x,T~(x)) a +v
=
at
+
-(~,~~(x)),a.e. a u
on {X/\T;(X)]
< I}.
at
We may apply Lemma 4 to the regionkn+l,and this completes the proof of Theorem 5 . 1 Remark 6.The above argument can be applied when the string is finite with fixed ends;in this case.one considers data u o in Hi(O,L) and u 1 in L'(0.L); the space V is replaced by V, = {v E Hioc( [O,L] x IR+)/VA,Vt 5 A,!~(u: + u:) dx 2 C(A) < +m}. (47) Remark 7.111 the cases studied in C21,i.e. under the hypothesis 41" 0. when the string is infinite,and under the hypotheses @" 2 O , @ ( O ) < O , @ ( L ) < 8 when the string is finite with fixed ends,it has been shown that there exists a solution u which satisfies the assumptions of Theorem 5.An example of accumulation of singular lines can be given C31,when the obstacle is not concave,i.e. when @'I is not everywhere non-negative.Then,the above theorem shows uniqueness up to a (possib1e)accumulation of non time-like lines supporting nu.
111,CONVER&NCE
Let
(25)
{
uE
OF THE PENALIZED PWBLEFf.
be the solution of the penalized problem,i.e. (u' @j/E = 0
UU" -
-
UE(X,O)
= uo(x)
$(x,o)
u,(x).
=
The purpose of this section is to prove the following result : Theorem 8.Let u o and u 1 satisfving the c a w a t i b i l i t y conditions (15) and (16) be given.let be a lens shaped region bounded by zero and same non time-like curve.Supmse t h a t nroblem P , adnits
a
a solution such t h a t supp time-like curves.Then u
E
la
uIR
is a discrete union of non
converyes to u
strongly i n Hioc( 03 )
.
The first thing to be noticed is that we can compute explicitly the solution of the penalized problem with zero obstacle.and initial conditions given by UE(X,mx)
0,
=
auE r t(x,~x)
=
b < 0.
We seek a solution under the form u"(x,t)
=
t ( ' f
- mx).
352
M . SCHATZMAN
and a rapid computation yields
(uE(x,t) Clearly,as
=
,
-b(t - mx -)- iT
-
t
mx
2
TTJE(1
-
m').
tends to O,uE converges to
E
u(x,t) =-b(t - mx) which is the true solution. The idea of the proof of Theorem 8 consists in demonstrating that uE behaves will be first in a neighborhood of a singular line T as in formulae (@).This i performed in a region R which contains only one singular line,and,then,extended by induction to the whole region in which QI is supported by discrete non timelike curves. Thus,we shall consider first a lens shaped region R bounded by p, and p, and assume that R is relatively compact;we denote by (a,b) the projection of R on the x-axis.Let w satisfy in R
ow=o, and let
E
{(x,t) E R/w(x,t)
=
<
$1,
where the obstacle @ is such that $" is a measure.We assume that the set E is not empty and define
=
T(X) so
inf {t' + Ix-x'I /(xl,tl) E E},
that
I = {(X,t)/t
> T(X)}.
We suppose that problem P6< with data U(X,P,(X))
w(x,P,(x))
=
u t (x,P,(x))
wt(x,p,(x)) a.e. on {x/l~:tx)I admits a solution such that 0 u is supported in {(x,t) theorme 5,this solution is unique. =
<
11
E
6/t
=
T(x)},so
that,by
Next,we define a set B
B
=
(x
E
(a,b)/l+(x)l
< I,w,(x,T(x))
< 01.
We remark that B cannot be negligib1e;to see this,note that if u is the solution of Pm,then u =
w
+&*!J,
and
= -2 /wt(x,T(x)) ( 1 - T(x)') $(x,T(x)) dx which is zero if B is negligible.But then,u must be equal to w,which therefore is larger than $.This contradicts the assumption that E is not empty. Let now wE be a solution of WE. =
Q in R ,
such that WE
-+
w in H' ( R )
.
VIBRATING STRING WITH AN OBSTACLE
The functions w and w' w(x,t)
(50)
can be written respectively as
f(x+t) + g(x-t)
=
WE(X,t)
353
=
+ gE(x-t),
f?x+t)
and (49) implies that,if J, =
{x+t/(x,t)
@I,
J, =
{x-t/(x,t)
R},
then f E f in H 1 ( J , ) and g E + g in H 1 ( J L ) . (51) In order to exhibit a non time-like curve r'which approaches c w approaches w ,we let DE be the closure of the set +
t
T
and on which
t
{(x,t)
@/w"(x,t)
E
for a l l t'
$(x),and
=
< $(x)),
such that w'(x,t") and
E
{(x,t)/x E B and
'=
t,there exists t" E ( t , t ' )
I
It-T(x)
=
inf
E
E'I.
lfor (x,t') in D'}.
It'-T(x)
Define =
T'IX)
infIt' + Ix-x~l/(xl,tt)
It is rather easy to see that (52)
lim
all x in B,
~ ~ ( x= )T(x),for
C+O
and by the Ascoli-Arzela theorem,this convergence is uniform on
z.
Prowsition 9.Under the above hypotheses on w and wE,;'(x) W€(X,T'(X)) converge respectively to ;(x) and w~(x,T(x)) t alrost everywhere on B.
*r
Proof.It is enough to prove that w (.,T'(.)) t everywhere on
B"
{x
=
I-a and w (x,T(x))
(a,b)/l:(x)l
E
t
and
and T - converge almost < -a),
where a is any small positive number. For x in B ,the following relations hold
t*
f(X + i(X
so that
f
and
+
T(X))(l
+ T(X))
-
+
q(X))
i(X
-
b(X
<
T(X))
-
T(X))(l
-
= i(X)
;(X))
-Cl,
satisfy the following inequalities
(53)
i(x + T(x))
(54)
i(x - ~(x)) < ( a 2 + &(x))/2,on
( - a 2 + $(x))/2,on
B",
B".
Thanks to Lemma 11.6 of [2],on any connected component of the open set UE,the line of influence '1 is made o u t of at most two segments of characteristic,the first one being of slope +],and the second one of slope -1. Call UE
=
~
(
~
~ the , set x ~of )points of ' U
the set of points of U E where L
m 10.The limit as
.E
T = E
-I.
'E
where T = I,and UE ri
=
E -il
v(xk,xk)
goes to zero of the m e a s u r e of UEn BCL is zero.
354
M . SCHATZMAN
Proof.In order to get our result,we shall evaluate in two different ways ___
-
I (2;€(x+TE(x)) B%Ji
(55)
E
$(x))
dx
E
There is no component (c ,d ) of U E such that
% meas((cE,dE)
\ Ba) > 0;
5
if this were the case,one could extract converging subsequences,still denoted by c E and dE,with respective =
limits c and d,which would satisfy
I on (c,d) and meas((c,d)
n)'B
> 0;
This is a contradiction to the definition of.'B Let now be an arbitrary positive number,and 0 an open set containing B",with meas(0 \)'B
< 8;O
=
(ci,d.),
Then
E
E
-
(2iE<x+TE(x))
I
(xk,xk) "(c i,di)
+(x))
- @(x)],
dx =[wE(x+TE(x))
inf (xz,di) ffk
which is non-negative,according to the definition of U ' t;' There is at most one k(i) such that ci (f;(i) 'Xi(i)) * According to the remark made at the beginning of this proof,
-
~
lim (xE k(i) Finally , (56)
c.) (meas((ci,di) 1
-
1 (2iE(x+TE(x)) B%Ji
lim 13,E-fO
n.)'B
$(x))
dx
2 0.
On the other hand,let us extract a subsequence such that 1
+
h
and
dx
B'WE
-f
k
m
in L weak-*.
(I+TE) (Barn:)
B%JE We have the identity d Jc 1
1
d+TE(d) JC+TE(C)
=
I
(I+TE) (B'nUi)
dy/2,
for arbitrary c and d.In the limit, d Ic h
dx
(d) 1d+T k dy/2. C+? (c)
=
If we pass to the limit in (55),and take into account (53),there obtains (57)
lim
1
- i(x))
(~;'(X+T~(X))
BanUE
dx < -a2/k(y)
dy
=
-20,' !h(y)
dy.
5
The comparison of (56) and (57) Droves that lim m e a s ( U y ) C Y implies that lim meas(U EnB )
ri
For any 6 2 0,let
=
0..
=
0;a similar argument
355
VIBRATING STRING WITH AN OBSTACLE
RE'&
B?UE
= {X E
2
/;'(x)
We have the identity a.e. on {x/w
E
- 6).
1
(x,T
E
(x))
@(X)}
=
-
E
'E
-f (x+T (x)) + &(x)/2 = (iE(x-TE(x)) $(x)/2)(1 - ;'(x))/(] +;€(X)). (58) Relation (58) will prove useful in estimating the measure o f RE'6;it implies
1
<
. -.
-
IiE(x-TE(x))
~ ( ~ ) / 2 ~ ( I - ~ ' ( x ) ) / ( l + ~ dx. ~(x))
The right hand side of (59) is equal to
E
- 4((I-TE)-'z)/2
liE(z)
E,6
(1-7 )R
~/(I+;E((I-~E)"z))dz
which i s estimated from above by (60)
(6meas
IiEILL +I6II,m(6xneas REy6)o's /2)/(2-6).
(
In order to estimate from below the ldft hand side of (59),we consider weakly convergent subsequences
If we perform the change o f variable y
11:" k'dy
-
which converges,as
E
1;
=
x+~~(x),the left hand side of (59) becomes
hE dx/21
goes to zero to
11; k dy - 16 h dx/2/ = \J(?(x+T(x)) (61) as we have the identity for arbitrary c and d i'cd hE dx
/d*Tr(d'
=
-
&(x)/2)
dxl,
k E dx.
c+T (c)
Finally,by comparing (60) and (61),we get lim meas
(62)
~€9'
5~
6 2 ,
E+O
where K does not depend o n 6. End
of
t h e proof
of Prowsition 9.
We shall construct a set L and a sequence
a
E
such that
L c Ba;rneas (B \L) i s arbitrarily small;
(63)
E
E
f n(x+~ "(x))
-+
f(x+r(x)),for
all x in L .
For any arbitrary positive @,there exists a compact set L B included in Jl,such that,thanks to Egorov's theorem
iEIL
B
converges to
Let u s evaluate meas {x
E
?I,
a € B / X + T (x)
B
in the Co(L,,,)
topology;mes(J1\L ) < B.
B 7
E J,\LB}.
For any positive B,the above quantity is estimated by + ( 0 1 6 ) + p, 2 K&'+ (B/U + qE, to zero as E goes to zero. + K ) 2 y2-",where y is arbitrary. such that &:(I &:,and
meas R€ 9 6 where p
E
and q
We choose
:6
go =
Subsequently,we choose
E
n
so
small that
3 56
M . SCHATZMAN
If we set
c
L = ( n (I+T n)-l ( J , \ L ~ ) ) n B',
n then,it is rather easy to show that the set L,and the sequence We can find a similar set L',such that
t
F
satisfy (63).
F
meas (L'\B")
is arbitrarily small;; "(I+T n)c+ L ( I + T ) on L ' . * n converges to With the help of identity (58),it is now clear that T everywhere on L f' L',and therefore,the proof is complete..
T
almost
Prcof of Theorem 8. Under the same hypotheses as those of Proposition 9,we consider the penalized problem (u" - @ ) - t E
OUF-
0
=
= wF(x,p,(x)) a.e. on {x/i~!~,(x) 1 ~ 1 1 . t < ~(x)} t o the As F tends to zero,this sequence converges on IR' = {x/p,(x) solution u of problem P@,and this convergence is strong in H'(6l');note that,on d < ' ,
u'(x,p,(x))
u = w.Define now a set
and
a
w'(x,pl(x));u~(x,p,(x))
=
b"
from ut,as E c is defined from wr,between (51) and ( 5 2 ) ,
new space like curve
oE= inf {t'+Ix-x'l/(x',t') E I.']. There exists a function v'defined on 6x such that Qt
O;v'(x,o'(x))
=
vE(x,oE(x))
=
u"(x,oE(x));
uE(x,af-(x)) a.e. on {x/(At (x)I < I ) .
=
It is fairly obvious that vc converges to u apply Proposition 9 to vE and 0 ' .
=
w on @',in H'(@')
strong;thus,we may
Define functions of the new vnriahles X and T and of x by
Then
;'(x,X,T)
=
u ~ ~ ( x + X ~ ~ , ~ ~ - /&, ~;:)tT~~
;'(x,X,T)
=
v~(x+X~,a'(x)+TJ;)/~~.
iE satisfies ;'(x,X,T)=
iE(x,X,T)
+
T
which we shall write as GE(x,X,T) The operator A'
=
-1
(lF(x,X',T'))-
dX'dT'/2
X,T
(AE;€) (x,X,T).
is obviously decreasing,so that,for any integer p and q,
5 E; 5
(AE)2p+'(0)
(AcIZq(O).
-
The limit of A',Ao is associated to the function w"which exists if w is differentiable at (x,T(x)),and x is in B : ;'(x,X,T) The fixed point
=
;" of
w (x,r(x))T t
A'
+ w (x,T(x))X;
is given by formulae (48),with
E=
1,m = ;(x),b
=
wt(x,'r(x)).
Thus,for almost all x in B, uE(x,t)
%
;O(O,(t-T(x))/&-),as
E
tends to zero.
The remainder of the proof is a simple induction on the regions which contain only one singular line..
VIBRATING STRING WITH AN OBSTACLE
357
REFERENCES Bamberger,A. and Schatzman,M.,New results on the vibrating string with a continuous obstacle,V.R.C. Technical Summary Report,Madison,Wisconsin, 1980. Schatzman,M.,A Hyperbolic Problem of Second Order with Unilateral Constraints:the Vibrating String with a concave Obstacle,J.Math.Anal. Appl.,73(1980)138-191. Schatzman,M.,in preparation.
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ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N ANALYSIS S . A x e l s s o n , L.S. F r a n k , A . v a n d e r S l u i s ( e d s . ) 0 North-Holland P u b l i s h i n g Cornpcwu, 1981
ON THE ASYMPTOTIC BEHAVIOR OF THE SOLUTION OF A NONLINEAR VOLTERRA INTEGRAL EQUATION K. Soni and R.P. Soni Department o f Mathematics U n i v e r s i t y o f Tennessee Knoxvi 11e, Tennessee 3791 6 U.S.A.
We determine t h e a s y m p t o t i c b e h a v i o r o f t h e s o l u t i o n o f t h e n o n l i n e a r Vol t e r r a i n t e g r a l e q u a t i o n g(t)
f
+ .-li2
0
(t
-
s ) - l 1 2 $'(s)ds
-
=
71
-'I2j;
(t
-
s)-l12f(s)ds
when t h e b e h a v i o r o f f ( t ) as t + i s p r e s c r i b e d and g i v e some examples t o i n d i c a t e t h e range o f a p p l i c a b i l i t y o f t h e results. INTRODUCTION I n t h i s paper we i n v e s t i g a t e t h e a s y m p t o t i c b e h a v i o r o f t h e s o l u t i o n o f t h e non1 i n e a r Vol t e r r a i n t e g r a l e q u a t i o n
when f ( t ) i s l o c a l l y i n t e g r a b l e , bounded and n o n n e g a t i v e i n [0, -) but only i t s b e h a v i o r f o r l a r g e t i s known. T h i s i n t e g r a l e q u a t i o n a r i s e s i n t h e s t u d y o f t h e f o l l o w i n g boundary v a l u e problem:
t) ,
Tt(xY t ) = Tx,(xy
T,(o,
t ) = a ~ " ( 0 ,t )
T ( x , 0) = 0 , T ( x , t) -+ 0 as
-
x z o , t,O, f(t)
,
t z
o ,
x > o , x
t >- O .
+ m
I n 1951, Mann and Wolf [ll]d i s c u s s e d t h i s problem i n a s l i g h t l y d i f f e r e n t form. S i n c e t h e n many r e l a t e d as w e l l as more g e n e r a l i n t e g r a l e q u a t i o n s have been i n v e s t i g a t e d . I n p a r t i c u l a r , we m e n t i o n Padmavally (1958), Levinson (1960), Friedman (1963), L e v i n (1962), Londen (1972), K e l l e r and Olmstead (1972), Handelsman and Olmstead (1972, 1973), Olmstead and Handelsman (1976). I t i s known t h a t i f f ( t ) i s n o n n e g a t i v e and bounded, t h e n (1.1) has a u n i q u e s o l u t i o n which i s bounded and nonnegative. Furthermore, K e l l e r and Olmstead proved t h a t i f m) , t h e n t h e s o l u t i o n $ ( t ) s a t i s f i e s $ ( t ) 'L IT'/*E(m) t-l" , t + m where E ( m ) > 0 f o r n > 3 and E ( m ) = 0 f o r n 5 2 (E(m) i s t h e n e t energy f l u x i n t o t h e s o l i d throu-gh t h e s u r f a c e which r a d i a t e s n o n l i n e a r l y ) . Handelsman and Olmstead o b t a i n more d e t a i l e d i n f o r m a t i o n about t h e a s y m p t o t i c b e h a v i o r under t h e assumption
f ( t ) E L[O,
f(t)
%
vo
t-ao +
v1 t
-a 1 +
... ,
T h e i r t e c h n i q u e t o some e x t e n t i s f o r m a l . 359
t +
-.
They a n t i c i p a t e t h e a s y m p t o t i c b e h a v i o r
360
K. SONI and R . P . SONI
o f t h e s o l u t i o n by assuming t h a t +(t)
Q
c o tmb0 (log t)m
t
... ,
co > 0
, bo 5
0
,m
-1, 0, 1
=
,... .
Then t h e y use t h e M e l l i n c o n v o l u t i o n t e c h n i q u e t o o b t a i n t h e a s y m p t o t i c b e h a v i o r o f the f r a c t i o n a l i n t e g r a l t r a n s f o r m s and r e p l a c e ( 1 . l ) by t h e c o r r e s p o n d i n g a s y m p t o t i c expansions. The unknown c o e f f i c i e n t s c o and t h e exponents bo, m a r e o b t a i n e d by comparing t h e e x p r e s s i o n s on t h e two s i d e s . R e c e n t l y Soni and Soni [16] used a s i m i l a r t e c h n i q u e t o extend t h e r e s u l t s o f Handelsman and Olmstead. They assume t h a t f ( t ) belongs t o t h e c l a s s of bounded,
-
nonnegative f u n c t i o n s which s a t i s f y f ( t ) Q t-a L ( t ) as t -t , C( 5 0 ( L ( t ) i s a f u n c t i o n which v a r i e s s l o w l y i n t h e sense o f Karamata [2]) and t h e y seek a which belongs t o t h e same c l a s s and s a t i s f i e s (1.1) a s y m p t o t i f u n c t i o n $,(t) cally, that is, t (1.2) $,(t) + -’I2 ( t s)-”* +!(s)ds Q H ( t - s)-’/*f(s)ds, t + a
j‘,
-’”
-
0
L i k e Handelsman and Olmstead, t h e y o b t a i n t h e b e h a v i o r o f
.
$,(t)
for all
C(
5
.
0,
n 1. 1 I n case L ( t ) + c as t + m , c > 0 , t h e i r r e s u l t s agree w i t h those g i v e n by Handelsman and Olmstead. However, t h e y do n o t c l a i m t h a t t h e unique s o l u t i o n o f (1.1) i s a s y m p t o t i c t o $,(t) , I n f a c t , f r o m (1.2), i t i s c l e a r t h a t
where
Thus, if E ( t ) i s t h e e r r o r i n v o l v e d i n e s t i m a t i n g t h e b e h a v i o r o f t h e a c t u a l s o l u t i o n o f (1.1) by t h a t o f $,(t) , we can o n l y c l a i m t h a t E ( t ) = o ( F ( t ) ) t m where F ( t ) i s d e f i n e d by (1.4). I f +,(t) i s o f t h e same o r d e r as -f
, the e r r o r i s o f a lower order.
B u t i f $,(t) i s of a l o w e r o r d e r t h a n i t i s c o n c e i v a b l e t h a t t h e e r r o r may be o f t h e same o r d e r as, o r even o f F(t) I n t h i s paper we determine t h e bea h i g h e r o r d e r than t h e o r d e r o f $,(t) y 0 l a < 1 h a v i u r o f t h e a c t u a l s o l u t i o n o f (1.1) when f ( t ) % t - ” L ( t ) , t + and examine t h e e x t e n t t o which i t agrees w i t h t h e b e h a v i o r o f $,(t) F(t)
.
-
.
The main r e s u l t s a r e g i v e n i n t h e f o l l o w i n g s e c t i o n . I n t h e l a s t s e c t i o n we comsatisfying pare t h e b e h a v i o r o f t h e a c t u a l s o l u t i o n w i t h t h e b e h a v i o r o f $,(t) (1.2) and g i v e some examples. These examples p r o v i d e f u r t h e r i n s i g h t i n t o t h e n a t u r e o f t h e s o l u t i o n and show t h a t , t o a c e r t a i n e x t e n t , t h e r e s u l t s a r e t h e best possible. M A I N RESULTS Unless s t a t e d o t h e r w i s e , we assume t h a t f ( t ) i s nonnegative, bounded and l o c a l l y i n t e g r a b l e i n [0, m ) ; $ ( t ) i s t h e unique, bounded, nonnegative s o l u t i o n o f (1.1); L ( t ) i s s l o w l y v a r y i n g i n t h e sense o f Karamata [2, pp. 1 31 and F ( t ) i s d e f i n e d by (1.4). The symbols 0 o a r e t h e Landau o r d e r symbols. Two
-
I
361
NONLINEAR VOLTERRA I N T E G R A L EQUATION
functions h(t)/g(t)
.
g ( t ) and h ( t ) a r e o f t h e same o r d e r a t i n f i n i t y i f b o t h a r e bounded as t 3 a
,
g(t)/h(t)
and
We prove t h e f o l l o w i n g : THEOREM. (a)
Let
If
CY
-
n 1 >
1 !Y
,05
or if
CY
<
CY
-
1
,
L2 =
n
and
L(t) + 0
as
as
,
t +
-
, then
imp1 i e s $(t)
(2.2)
k t-at1/2 L ( t ) , t
%
-f
m
,
R > 0
where
k = r(1
(2.3)
(b)
If
-
3
-
a)/r(z and
a - ; = X
L(t) + R ,
where
c1
(2.4)
x
(c)
Let
.
a)
t
+
t
-f
m
t h e n (2.1) i m p l i e s
m
i s t h e u n i q u e p o s i t i v e s o l u t i o n of n
r(1
-
F(t) Q(t)
If a - - 1c E 2 n same o r d e r as
=
U)
t
3
x r(s - a )
= i? r ( 1
-
.
.)
be d e f i n e d by (1.4) and 1e t t
-ut1/2 L ( t ) , 0
orif Q(t)
t <
C Y - ~ = .and
L t) 2 n a t i n f i n i t y implies
-
-f
m
as
t
+
-
then
We m e n t i o n h e r e t h a t i n t h e above theorem p a r t ( c ) i f , f o r some
F(t)
C
> 0
i s o f the
,
F ( t ) Q C t-CLt1/2 L ( t ) as t , then F ( t ) and Q ( t ) a r e o f t h e same o r d e r a t i n f i n i t y and (2.5) h o l d s . I n p a r t i c u l a r , t h e c o n c l u s i o n h o l d s under t h e same c o n d i t i o n s on CY and L ( t ) when f ( t ) s a t i s f i e s ( 2 . 1 ) because i n t h a t case F ( t ) s a t i s f i e s such a r e l a t i o n (see (2.6) b e l o w ) . -f
The p r o o f of t h e theorem depends o n t h e f o l l o w i n g r e s u l t , s t a t e d as a Lemma, which can be o b t a i n e d by u s i n g a theorem o f B o j a n i c and Karamata [ 2 , Thm. 51. See a l s o [16, Lemma 1 3
.
362
K. SONI and R.P. SONI
LEMMA. g(t)
’L
Let
g(t)
be l o c a l l y i n t e g r a b l e and bounded i n [0,
t-’ N ( t ) , t
-f
, where
m
If
i s s l o w l y v a r y i n g a t i n f i n i t y , then as
N(t)
t - t m ,
.
m)
’I2 t-”+l12N ( t )
r(1
- v ) / r ( z3
- v), v < 1
,
v = l , (t-’I2
,
g(s)ds
0
v > l .
PROOF OF THE THEOREM. (a)
By (2.1)
k
where
i s d e f i n e d by ( 2 . 3 ) .
-
lim t-
(2.8)
and ( 2 . 6 ) ,
4*
ta-l/2
Since
Ik
$(t)
i s nonnegative, by (1.1)
.
I f f o r t h e moment we assume t h a t
1
t
(2.9)
0
(t
- s)-”*
@‘(s)ds = o ( t -
then by (1.1) and (2.7)
at112
L(t))
t
I
+ m
,
,
t-t-
t-t-
-
-112
lim t-t-
Thus t o p r o v e ( 2 . 2 ) , we o n l y have t o show t h a t (2.9) h o l d s . (2.8), we can f i n d N = N ( ~ ) such t h a t (2.11) For
Since
$ ( t ) < (1 +
t > N
$(t)
E)
k L ( t ) t-”t1’2 , t 2N > 1
$
Let
n
k.
(s)ds E >
0
.
By
.
,
i s bounded,
I1 = U(t’1/2),
t +
m
.
Therefore, we o n l y have t o show
NONLINEAR VOLTERRA INTEGRAL EQUATION
363
that
(2.12)
I2 = o(t
-a+l/2
1)
v <
Now l e t
CY
-
I
l
a
z
.
-
,
t
-f
.
=
ccn s o t h a t -;~+1/2 n L (t)) Ip = U(t
F i r s t suppose t h a t
((2.61,
L(t))
ci
.
n
=
I n t h i s case
n(a
>
,t 1 -$
1 -f
and
.
m
L(t)
+
0
as
t
+ =
.
By
T h e r e f o r e (2.12) i s s a t i s f i e d .
may be g r e a t e r than, equal t o o r
By u s i n g ( 2 . 6 ) and some w e l l known p r o p e r t i e s o f t h e s l o w l y v a r y i n g l e s s than 1 f u n c t i o n s (see, f o r example [ l , Thm. l ] ) , we f i n d t h a t (2.12) i s s a t i s f i e d i n each case. T h i s completes t h e p r o o f o f (2.2). (b).
Suppose t h a t
grable i n (1.1).
[ O , =)
Then
f,(t)
and
and
(+y(t)
-
+l(t)
f2(t) and
+;(t))/(q(t)
except f o r t h e s e t where [14, p. 5431, we o b t a i n
Ql(t)
a r e bounded, n o n n e g a t i v e and l o c a l l y i n t e -
+,(t)
-
are t h e corresponding s o l u t i o n s o f
$2(t))
= +,(t)
.
i s bounded u n i f o r m l y i n
[0, =)
Using Padmavally's approach
Now d e f i n e
where [0, -)
(2.15) where
$ ( t ) i s any bounded, n o n n e g a t i v e f u n c t i o n which i s l o c a l l y i n t e g r a b l e i n and s a t i s f i e s $ ( t )*
c1
II
-ll2
t-"
r(z 3 -
a)/r(l
- a) ,
i s t h e u n i q u e p o s i t i v e s o l u t i o n o f (2.4).
Since
~1
< 1
and
F,
=-:
by (2.61,
Note t h a t
$l(t)
i s bounded, nonnegative, l o c a l l y i n t e g r a b l e i n
satisfies
(2.17)
+,(t)
+
TI-^'^
i',
(t
-
s)-'l2
+;(S)
ds =
TI
[
-'I2( t
-
s)-"*
[0, =)
and
fl(s)ds
,
K. S O N I and R.P. SON1
364
where (2.18)
fl(t)
But
f(t)
f(t)
Q
-
fl(t)
=
,t
1 t-a
-f
(c)
.
,05t
$(t)
,t
.
+ =
<
m
TO prove t h i s a s s e r t i o n , we observe t h a t
The a s y m p t o t i c b e h a v i o r o f
fl ( t )
can be o b t a i n e d by
-.
I f we use (2.4) t o s i m p l i f y t h a t expression, we a g a i n o b t a i n , Since f ( t ) - f l ( t ) i s bounded, by (2.19), $ ] ( t ) l = o ( t -a+'/') as t and t h e c o n c l u s i o n f o l l o w s .
,t
i? t-"
%
-
I#(t)
T"'
o(t-")
~1
(2.15) and (2.16). fl(t)
+
$y(t)
=
-f
-f
Let
(2.20)
1:
P(t) =
f(s)ds
,
t
>0
.
By ( l . l ) , u s i n g f r a c t i o n a l i n t e g r a l o f o r d e r one h a l f , we o b t a i n
(2.21) Since
t
0
(t
F(t)
-
s)-'"
$ ( s ) ds
t
71
'I2
i s o f t h e same o r d e r as
1
t
0
@'(s)ds
=
71'"
.
p(t)
t-"'l12 L ( t ) as t
+
and
m
.
by u s i n g (2.6) we conclude t h a t P ( t ) i s o f t h e 'same o r d e r as F(t) = o(P(t)) , t + m I n case n = 1 , by (1.1) ,
T h e r e f o r e , t h e c o n c l u s i o n f o l l o w s by (2.21).
By Hb'l d e r ' s i n e q u a l it y
(2.24)
i" 0
(t
-
s ) - l 1 2 #(s)ds 5
(r 0
Now l e t
ds)'/p(r ( t 0
-
n > 1
.
tl-' L ( t )
.
Thus,
By (2.21),
S)-q'2ds)''q(jt
+n(s)ds)'/n
,
365
NONLINEAR VOLTERRA INTEGRAL EQUATION
-
.
If n / ( n 1 ) < 2 , we may choose q = n / ( n - 1 ) and p-' = 0 Otherwise we may choose q t o be any number l e s s t h a n 2 . By (2.23) and (2.24),
=
,
O(P(t))
t
-f m
.
The l a s t r e l a t i o n f o l l o w s f r o m t h e f a c t t h a t P ( t ) i s o f t h e o r d e r o f tl-' L ( t ) L ( t ) ) , t + m under t h e g i v e n as t -f m and t h a t t 1/2-a/n ( L ( t ) ) ' l n = o ( t ' - ' c o n d i t i o n s . The c o n c l u s i o n now f o l l o w s f r o m (2.21). COMPARISION RESULTS AND EXAMPLES We g i v e below t h e b e h a v i o r o f
+,(t)
As s t a t e d e a r l i e r ,
as g i v e n i n [16].
@,(t) i s an a s y m p t o t i c s o l u t i o n o f (1.1) i n t h e sense o f (1.2)
i s assumed t h a t
(2.4)
s a t i s f i e s ( 2 . 1 ) and
f(t)
, (t-CL'n (L(t))lln,
We n o t e t h a t i f s o l u t i o n o f (1.1)
If
and
c1
and ( 1 . 3 ) .
It
a r e d e f i n e d by (2.3) and
respectively. L(t) , n
when
k
0 ( a < 1
$(t)
: <
C(
- '1i
or if
C(
ci
-
l
a
- or
->
u - 2 ~ n= and K L t)
- L2 = 5n
and
L(t)
3
- 2l = a
and
L(t)
-f
and
L(t)
= a
-
has t h e same b e h a v i o r as
$,(t)
as
-f
0,
,
il
-
or
-
CL
l --<
2
a
n-
.
tends t o a l i m i t , t h e t
-+
m
.
I n a l l o t h e r cases
, even i f (2.1) i s s a t i s f i e d , we o n l y o b t a i n
is u l t i m a t e l y monotone, (3.2) i m p l i e s
$(t)
%
$,(t)
,t
-f
m
(see, f o r
example [15, Lemma 71). I n g e n e r a l , t h i s c o n d i t i o n may n o t be s a t i s f i e d . Some r e s u l t s c o n c e r n i n g t h e m o n o t o n i c i t y o f t h e s o l u t i o n a r e g i v e n by Friedman (1963) b u t t h e y - a r e n o t a p p l i c a b l e here. Suppose t h a t f l ( t ) and f 2 ( t ) d i f f e r o n l y i n a f i n i t e interval
(a, b)
,
0 (a
<
b
,
By (2.13),
as t 3 m. The r i g h t s i d e i n t h e above i n e q u a l i t y i s o f t h e same o r d e r as tm1l2 T h e r e f o r e i t seems p l a u s i b l e t h a t under t h e c o n d i t i o n (2.1), t h e a c t u a l s o l u t i o n i n c e r t a i n cases may n o t be a s y m p t o t i c t o $,(t)
.
The c o n c l u s i o n (2.5) i s c o m p a r a t i v e l y weak b u t i t p r o v i d e s some i n f o r m a t i o n even when (2.1) is n o t s a t i s f i e d . Consider t h e f o l l o w i n g example.
366
K. SONI and R.P. SONI
Example 1.
and l e t
Let
E(t)
n(t)
be t h e f r a c t i o n a l i n t e g r a l o f
o f o r d e r one h a l f ,
n , 0 -< t -< 1 (3.4)
E(t) =
( nt
(2/J;;)
m ) - l
,t
>
1
.
Define
For
n = 1
and
$(t) =
f(t)
4;;s i n
as d e f i n e d above, (1.1) has t h e s o l u t i o n t Jo(t)
+
24;;
E(t)
C l e a r l y , f ( t ) does n o t s a t i s f y (2.1), t h e y do s a t i s f y (2.5).
.
,t
0
+(t)
i s not asymptotic t o
f(t)
but
Now we g i v e some examples t o p r o v e some a s s e r t i o n s o f t h e n e g a t i v e t y p e . I n t h e s e examples n ( t ) and E ( t ) denote f u n c t i o n s d e f i n e d by (3.3) and (3.4) r e s p e c t i v e l y . From (1.1) i t may appear t h a t i f F ( t ) d e f i n e d by (1.4) i s nonnegative, t h e s o l u t i o n would be nonnegative. T h i s i s false. Example 2.
Let
$ ( t ) = (l/v‘f)
J,(fi) +
4;;n ( t ) , t
0
and
Then (3.6)
$(t) +(t)
satisfies
+
lo t
T - ~ / ~
(t
-
s ) - ’ j 2 q ( s ) d s = F(t)
t2O.
C l e a r l y , F ( t ) i s nonnegative b u t q ( t ) i s n o t . F o r t h e p o s i t i v i t y o f t h e s o l u t i o n , F ( t ) must s a t i s f y some a d d i t i o n a l cond t i o n . One such c o n d i t i o n is g i v e n by Friedman [4, Thm. 21.
367
NONLINEAR VOLTERRA INTEGRAL EQUATION
Next we show t h a t t h e a s y m p t o t i c b e h a v i o r o f F ( t ) asymptotic behavior o f the s o l u t i o n completely. Example 3.
Let
$,(t) =
(3.7)
does n o t d e t e r m i n e t h e
(Z/Jx)s i n 2 ( J t / 2 )
t
4;
E(t)
,t 50
and
Then
$1 ( t )
Clearly,
satisfies
Fl(t)
$ ( t )t i s g i v e n by
17,
1 t
71-1/2
4;; , t
i' 0
$ ( t )= ( &
(t
+
m
.
$(t) of
But t h e s o l u t i o n
-
t 1 ) ElI2
(-d)where
Ea(-)
F i n a l l y we g i v e an example t o show t h a t when f ( t ) f r o m (1.1) t h a t $ ( t ) 5 f ( t ) This i s f a l s e .
.
Example 4.
$,(t)
> f(t)
Remark.
whenever
whereas
i s nonnegative, i t may appear
a s i n example 3 and l e t
i s t h e s o l u t i o n o f (1.1) when
q(t) +(t)
Define
-
i s t h e w e l l known
M i t t a g - L e f f l e r f u n c t i o n . T h e r e f o r e , $ ( t )% (1 t &)/V'Z, t + O l ( t ) i s o s c i l l a t o r y . However, (2.5) i s s a t i s f i e d .
J,(n)
f.(t)
i s d e f i n e d as above.
For
t > 1
,
i s negative.
I n a l l o u r examples we have assumed n = 1 because t h i s i s t h e s i m p l e s t l u < F o r t h e c a l c u l a t i o n o f t h e f r a c t i o n a l i n t e g r a l s , we
case f o r w h i c h a
-
.
have used [3] and some s i m p l e p r o p e r t i e s o f t h e Bessel f u n c t i o n s . REFERENCES
[l]
Adamovic, D.D., Sur quelques p r o p r i & s des f o n c t i o n s a c r o i s s a n c e l e n t e 136. de Karamata I, Mat. Vesnik 3(1966) 123
[2]
B o j a n i c , R. and Karamata, J . , On s l o w l y v a r y i n g f u n c t i o n s and a s y m p t o t i c r e l , a t i o n s , MRC T e c h n i c a l Summary R e p o r t 432, Wisconsin, 1963.
[3]
Erdelyi, 1954)
-
.
A,, Tables o f I n t e g r a l Transforms, V o l . Z(McGraw-Hill , New York,
368
K. SONI and R . P .
SONI
[41
Friedman, A., On i n t e g r a l e q u a t i o n s o f V o l t e r r a type, J . ,4nalyse Math. l l ( 1 9 6 3 ) 381 413.
PI
Handelsman, R.A. and Olmstead, W.E., Asymptotic s o l u t i o n t o a c l a s s o f nonl i n e a r Vol t e r r a i n t e g r a l e q u a t i o n s , S I A M J . Appl Math. 22( 1972) 384. 373
-
-
.
Handelsman, R.A. and Olmstead, W.E., Asymptotic A n a l y s i s o f a c l a s s o f n o n l i n e a r i n t e g r a l e q u a t i o n s , i n : N o n l i n e a r Problems i n P h y s i c a l Sciences and B i o l o g y , L e c t u r e Notes i n Math. V o l . 322 ( S p r i n g e r - V e r l a g , B e r l i n , 1973). Temperature o f a n o n l i n e a r l y r a d i a t i n g K e l l e r , J.B. and Olmstead, W.E., s e m i - f i n i t e s o l i d , Q u a r t . Appl. Math. 29(1972) 485 - 476. L e v i n , J . J . , On a n o n l i n e a r V o l t e r r a e q u a t i o n , J . Math. Anal. Appl. 39(1972) 458 - 476. Levinson, N., A n o n l i n e a r V o l t e r r a e q u a t i o n a r i s i n g i n t h e t h e o r y o f s u p e r f l u i d i t y , J . Math. Anal. Appl. l ( 1 9 6 0 ) 1 - 11. Londen, S., Anal. Appl
the solutions o f a nonlinear Volterra . On39(1972) 564 - 573.
e q u a t i o n , J . Math.
Mann, W.R. and Wolf, F., Heat t r a n s f e r between s o l i d s and gases under n o n l i n e a r boundary c o n d i t i o n s , Q u a r t . Appl. Math. 9(1951) 163 - 184. Olrnstead, W.E. and Handelsman, R.A., A s y m p t o t i c s o l u t i o n t o a c l a s s o f n o n l i n e a r V o l t e r r a i n t e g r a l e q u a t i o n s 11, S I A M J . Appl. Math. 30(1976) 189. 180
-
Diffusion i n a semi-infinite region Olmstead, W.E. and Handelsman, R.A., w i t h n o n l i n e a r s u r f a c e d i s s i p a t i o n , S I A M R E V I E W 18(1976) 275 - 291. Padmavally, K., On a n o n l i n e a r i n t e g r a l e q u a t i o n , J . Math. and Mech. 7(1958) 533 - 555. Soni, K. and Soni, R.P., S l o w l y v a r y i n g f u n c t i o n s and a s y m p t o t i c b e h a v i o r o f a c l a s s o f i n t e g r a l t r a n s f o r m s 11, J . Math. Anal. Appl. 49(1975) 495. 477
-
Soni, K. and Soni, R.P., Asymptotic s o l u t i o n t o a c l a s s o f n o n l i n e a r V o l t e r r a i n t e g r a l e q u a t i o n s ( t o appear i n : J . I n t . Eq.).
A N A L Y T I C A L AND N U N E R I C A L APPRUACIIES TO A S Y M P T O T I C PROBLEMS I N AWALI'SIS S . Axelsson, L . S . F r a n k , A . 5vdn d e r S l u ? s ( e d . 5 . ) @ North-Hol l a n d P u b 1 i s h i n y Company, 1 9 8 1
CONSERVATIVE UPWIND F I N I T E ELEMENT APPROXIMATION AND I T S APPLICATIONS Masahisa Tabata Department o f Mathematics, Kyoto U n i v e r s i t y K y o t o 606, J a p a n and A n a l y s e Numerique, U n i v e r s i t e P i e r r e e t M a r i e C u r i e 75230 P a r i s , F r a n c e We p r e s e n t a f i n i t e e l e m e n t a p p r o x i m a t i o n w h i c h i s e f f e c t i v e f o r d i f f u s i o n problems w i t h dominated n o n - l i n e a r convectioti t e r m s . T h i s a p p r o x i m a t i o n i s c o n s e r v a t i v e and of u p w i n d t y p e . We a n a l y s e i t i n c o n n e c t i o n w i t h L 2 - t h e o r y , L 1- c o n t r a c t i o n and m o n o t o n i c i t y . As an a p p l i c a t i o n we c o n s i d e r a n o n - l i n e a r e l l i p t i c p r o b l e m and g i v e e r r o r a n a l y s i s . I n t h e p r o c e s s t h e e x i s t e n c e o f t h e e x a c t s o l u t i o n i s a l s o proved.
INTRODUCTION I n t h i s p a p e r we s t u d y f i n i t e e l e m e n t a p p r o x i m a t i o n t o d i f f u s i o n p r o b l e m s w i t h dominated l i n e a r o r n o n - l i n e a r convection terms. It i s well-known t h a t t h e c o n v e n t i o n a l f i n i t e e l e m e n t a p p r o x i m a t i o n does n o t g i v e good n u m e r i c a l s o l u t i o n s f o r t h e s e p r o b l e m s e v e n i n t h e l i n e a r case. I n o r d e r t o overcome t h i s d i f f i c u l t y v a r i o u s methods ( P e t r o v - G a l e r k i n method, u p w i n d method, a r t i f i c i a l v i s c o s i t y method, e t c . ) h a v e been p r o p o s e d , s e e t h e s u r v e y p a p e r of H e i n r i c h and Z i e n k i e w i c z ( 7 1 . M o s t o f them a r e c o n c e r n e d w i t h l i n e a r p r o b l e m s . F o r p r o b l e m s w i t h nonl i n e a r c o n v e c t i o n t e r m s t h e r e a r e f e w r e s u l t s . I k e d a ( 8 1 t r e a t e d them b y u s i n g a r t i f i c i a l v i s c o s i t y method. A b o u t t h e o t h e r methods, f o r example, P e t r o v G a l e r k i n method o r u p w i n d method, i t i s n o t s o e a s y t o e x t e n d them t o t h e nonl i n e a r case, s i n c e t h e s e methods u s e t h e d i r e c t i o n o f f l o w , w h i c h i s a l s o unkown i n t h i s case. The p u r p o s e o f t h i s p a p e r i s t o p r e s e n t a c o n s e r v a t i v e u p w i n d f i n i t e e l e m e n t a p p r o x i m a t i o n t o d i f f u s i o n p r o b l e m s w i t h d o m i n a t e d n o n - l i n e a r c o n v e c t i o n and t o 2 1 a n a l y s e i t i n c o n n e c t i o n w i t h L - t h e o r y , L - c o n t r a c t i o n and m o n o t o n i c i t y . T h i s a p p r o x i m a t i o n i s an e x t e n s i o n o f t h e r e s u l t f o r l i n e a r p r o b l e m s o b t a i n e d b y BabaT a b a t a c l l . The d i f f i c u l t y on t h e unknown f l o w d i r e c t i o n i s overcome b y t h e u s e of " s p l i t t i n g technique" o f n o n - l i n e a r f u n c t i o n proposed by O s h e r ( l 7 ) . "Conservative" means t h a t i n each d u a l p o l y g o n ( s e e F i g s . 1 and 2 ) t h e mass-balance i s p r e s e r v e d and t h a t as a r e s u l t a d i s c r e t e mass c o n s e r v a t i o n l a w (1.19) i s r e a l i z e d . T h i s 1 p r o p e r t y i s i m p o r t a n t a l s o i n n o n - l i n e a r problems i n connection w i t h L c o n t r a c t i o n (Remark 1 . 4 ) . I n s e c t i o n 1 we o b s e r v e some b a s i c p r o p e r t i e s o f c o n s e r v a t i v e u p w i n d f i n i t e It i s noted t h a t e l e m e n t a p p r o x i m a t i o n t o l i n e a r o p e r a t o r L v = -EAV + V . [ b v l . d u a l d e c o m p o s i t i o n p l a y s an i m p o r t a n t r o l e t o c o n s t r u c t t h e c o n s e r v a t i v e a p p r o x i m a t i o n . The d e r i v a t i o n o f c o n s e r v a t i v e u p w i n d a p p r o x i m a t i o n i s a l s o e x p l a i ned
.
Fh(vh) t o n o n - l i n e a r i s embedded i n H-'(Q) by
I n s e c t i o n 2 we p r e s e n t a c o n s e r v a t i v e u p w i n d a p p r o x i m a t i o n convection term
F(v) = V-[f(x,v)l.
After
Fh(vh)
u s i n g an a v e r a g e o p e r a t o r , we show t h a t F ( v ) h h
approximates
F(vh)
with order
h
370
M.
i n H -l(O ).
TABATA
T h i s e s t i m a t e i s used i n t h e f o l l o w i n g s e c t i o n .
I n s e c t i o n 3 as an a p p l i c a t i o n o f t h e c o n s e r v a t i v e upwind approximat ion we con s ider a n o n - l i n e a r e l l i p t i c problem, - € A V f V * [ f ( x , v ) l + g(X,V) = 0 i n a bounded domain o f R2 w i t h homogeneous boundary c o n d i t i o n . A f t e r showing t h a t our f i n i t e element scheme has a unique s o l u t i o n Vh, we prove t h a t IIvh v I [ 5~
-
cEh.
I n t h i s process t h e e x i s t e n c e o f t h e e x act s o l u t i o n
~
i s a l s o proved.
V
Proofs of t h e l a s t two theorems a r e o m i t t e d o r o n l y sketched. The complete proofs as w e l l as f u r t h e r d i s c u s s i o n s w i t h r e s p e c t t o t h e above n o n - l i n e a r e l l i p t i c problem w i l l be g i v e n i n t h e f o r t h c o m i n g paper (221. $1. DUAL DECOMPOSITION AND CONSERVATIVE UPWIND FINITE ELEMENT APPROXIMATION I n t h i s s e c t i o n we r e v i e w c o n s e r v a t i v e upwind f i n i t e element approximat ion f o r l i n e a r problems o b t a i n e d by Baba-Tabatall) and Kanayama(ll1. Some b a s i c p r o p e r t i e s o f t h i s a p p r o x i m a t i o n a r e n o t i c e d (Lemma 1.1 and P r o p o s i t i o n 1 . 2 ) . I n t h e f o l l o w i n g s e c t i o n corresponding p r o p e r t i e s w i l l be examined f o r approximat ion t o n o n - l i n e a r c on v e c t i o n terms. We r e s t r i c t ourselves t o two dimensional case. F o r t h e genera l dimensional case we r e f e r t o [ll. L e t s1 be a convex p o l y g o n a l domain i n R2 1inear o p e r a t o r d e f i n e d by
w i t h boundary
r.
Let L
be a
LV = -€AV t V * c b v l ,
(1.1)
a+
where E i s a s m a l l p o s i t i v e number, b: R2, i s a g i v e n smooth f u n c t i o n and V i s t h e divergence o p e r a t o r w i t h r e s p e c t t o x. As was mentioned i n t h e i n t r o d u c t i o n , when E i s s m a l l , t h e c o n v e n t i onal f i n i t e element scheme does n o t g i v e good approximate s o l u t i o n s . Our a p p r o x i m at ion i s based on an i n t e g r a l formula on t h e dual polygon o f each nodal p o i n t . Bef ore d e s c r i b i n g i t we tra n s f o rm (1 . 1 ) i n t o t h e weak form. We c o n s i der t h e boundary c o n d i t i o n (1.2) where
dv/dn = 0 n
such t h a t
on
rl
and
v = 0
= E
f
= =
and
for
where
rl
and To are p o r t i o n s of Then t h e week f orm o f (1.1) i s ,
i s t h e u n i t o u t e r normal t o
rln ro = 4 , r 1 U r 0 = r.
r
ro,
on
v, $
E
H:
3
r
c
I g r a d vograd Q dx, n -1 v b - g r a d $ dx + 1
vben $ ds. rl We d i v i d e n i n t o t h e u n i o n o f ( c l o s e d ) t r i a n g l e s , K = U{ Tk; k=l, NE). We assume t h a t t h e t r i a n g u l a t i o n i s r e g u l a r and o f (weakly) acut e type, i . e . , t h e r e e x i s t s a p o s i t i v e c o n s t a n t eo independent o f d i v i s i o n such t h a t eo 5
n
...,
e5
IT/^ f o r every an g l e e o f any t r i a n g l e Tk. Denote by h t h e maximum s i d e l e n g t h o f a l l t h e t r i a n g l e s . L e t Pi, i=l,...,N, be a l l t h e nodal p o i n t s i n W,
...,
which i s d i v i d e d i n t o t h r e e p a r t s , Pi, i = l , No, nodal p o i n t s i n n, Pi, i = Notl,. ,N1, nodal p o i n t s on r l , Pi, i=N tl,.. ,N, nodal p o i n t s on 1 ro. With each nodal p o i n t Pi we a s s o c i a t e a f u n c t i o n $ihy continuous i n C(G), l i n e a r on
..
.
371
CONSERVATIVE UPWIND F I N I T E ELEMENT APPROXIMATION
We denote by Vh (resp. Vhl ) the l i n e a r each t r i a n g l e , s a t i s f y i n g Qih(Pj) = Aij. span o f +ihy i=l, N, ( r e s p . N1), which i s a f i n i t e dimensional subspace o f 1 H 1(Q) ( r e s p . HE). We a l s o use a d u a l decomposition o f 0 ,h = U1: Di, i = l , . ,NI, where Di i s a dual p o l y g o n o f Pi, whose v e r t i c e s a r e o r t h o c e n t e r s of t r i a n g l e s By v i r t u e o f t h e d i v i s i o n o f a c u t e t y p e t h e o r t h o c e n t e r o f each surrounding Pi.
...,
..
triangle l i e s within i t s triangle. Let Pj We denote by Pij the midpoint o f the side
be an a d j a c e n t nodal p o i n t t o Pi# P.P and b y rij t h e s i d e o f aDi 1
j
which passes Pij. (When Pi i s a boundary nodal p o i n t we t a k e as Di a polygon of whose v e r t i c e s a r e o r t h o c e n t e r s o f t r i a n g l e s surrounding Pi, m i d p o i n t s Pij Pi and t h e a d j a c e n t boundary nodal p o i n t s P j y and t h e nodal p o i n t Pi. We I'ij
denote by
the portion
Fig.1.
where yij
(1.5 ) (1.6 )
t
Bij
Bij nij
= max(nij*b(P.
F ig. 2.
Boundary nodal p o i n t Pi and t h e dual polygon Di
rij,
.), 0),
1J
= min(nij*b(P..), 1J
0),
the u n i t vector w i t h d i r e c t i o n
A; = Ai
r.)
1 5 j ( # i )6 N, i s a d j a c e n t t o PiIY
= the length o f
=
o f boundary
I n t e r i o r nodal p o i n t Pi and t h e dual polygon Di
{ j; P j ,
Ai
Pipij
n {No+1,Not2,. ..,NI,
y i j = t h e l e n g t h of
rij,
qj,
M.
372
(1.5))
B i tj = max(n;j*b(P..),
0),
(1.6’)
8;; = m i n ( n i j * b ( P . . ) ,
o),
1J
TABATA
1J n i j = t h e o u t e r normal v e c t o r t o
r
at
Pij.
The d i f f u s i o n term A ‘ i s approximated i n t h e convent ional way. F o r t h e conv ec t io n t e r m we have used an upwind technique. We now e x p l a i n how we have d e r i v e d Bh and how t h e upwind t e c h n i q u e has been a p p l i e d . The dual product i s approximated as f o l l o w s : (1.7)
= =
O e [ b v l @ dx s2 $(Pi)/D
i= 1 N
O*(bvldx i
z1 @(Pi) / nebv ds i=1 aD, I = N1 c $(Pi).c nij-bv ds t C @(Pi) X 1 n i j * b v ds i=1 J E A ~ rij PiErl jd; rij 2 N1 c $(Pi) X y . .{P.+ .V(Pi) t B;jv(Pj)l. =
I
i=1
+ c
piErl
jeAi
@(Pi)
lJ
C
jd;
.y! . { B ! t.V(Pi) ’J
lJ
+ B;jv(Pj)l
*
thus we a r r i v e a t t h e d e f i n i t i o n o f Bh, I n t h e above t r a n s f o r m a t i o n t h e r e are two key p o i n t s . The one i s t o use t h e i n t e g r a l formula on t h e dual polygon Di ( i n t h e t h i r d l i n e ) , and t h e o t h e r i s t o use t h e upwind t echnique ( i n t h e l a s t two l i n e s ) . The f o r m e r was used i n t h e f i n i t e d i f f e r e n c e method by N o g i ( l 6 1 , see a l s o Roache[l91. The l a t t e r i s e x p l a i n e d as f o l l o w s . I f nij*b(P. . ) > 0 (resp. < 0 ) , Pi
1J
( r e s p . P.) l i e s i n t h e upwind d i r e c t i o n s i n c e t h e f l o w goes o u t o f (resp. J
e n t e r s i n t o ) Di: Therefore we use c o n s t a n t l y t h e v a l u e of p o i n t i n approximating nij*bv.
v
a t t h e upwind nodal
Here we not e t h a t t h e way o f c o n s t r u c t i n g dual decomposition i s n o t unique. I n f a c t Baba-Tabata(l1 proposed t h e use of dual polygons Di whose v e r t i c e s a r e ba ry c ent e rs of t r i a n g l e s s u r r o u n d i n g Pi. I n t h e two dimensional case Kanayama [111 used t h e dual polygon Di discussed above. (See a l s o MacNealtl31 , where o r t h o c e n t e r s were used t o c o n s t r u c t an asymmetric f i n i t e d i f f e r e n c e n e t work.) Although t h i s way cannot be a p p l i c a b l e i n t h e h i g h e r dimensional case ( s i n c e t h e o r t h o c e n t e r of n-simplex does n o t always l i e i n i t even i f t h e t r i a n g u l a t i o n i s o f acute type; f o r example, c o n s i d e r 3-simplex w i t h v e r t i c e s Ao(O,O,O), Al(l,O,O), A2(0,1,0) I A3(0,0,1)), t h e use o f o r t h o c e n t e r s leads t o an i n t e r e s t i n g
(1.8)
=
-I A V N
n
$ dx
-1 $(Pi i=1
1
+
1 dv/dn r
4 ds
Av dx t z $(Pi)/ Di piEr aD
nr
dv/dn ds
373
CONSERVATIVE UPWIND F I N I T E ELEMENT APPROXIMATION
=
2
N
- c
$(Pi)( dv/dn ds t C @(pi)/ i=1 aDi PiEr aDi N - C @(Pi) c y . . ( v ( P . ) - V ( P i ) ) / P i p j . i=l j E h i lJ
nr
dv/dn ds
I t l o o k s as i f t h e l a s t l i n e were a new a p p r o x i m a t i o n of . I t i s , however, shown by I w a k i l 9 1 (see a l s o (111) t h a t t h e l a s t l i n e i s n o t h i n g b u t h,
vh
where
@h a r e i n t e r p o l a t i n g f u n c t i o n s such t h a t
and
vh(Pi)
= v(Pi),
@,(Pi)
= $(Pi) a t a l l nodal p o i n t s . T h i s f a c t i m p l i e s t h a t if we use d u a l polygons decided by o r t h o c e n t e r s , t h e d e r i v e d a p p r o x i m a t i o n Bh i s w e l l f i t t e d t o Ah, t h e
a p p r o x i m a t i o n o f d i f f u s i o n term. F o r an a r t i f i c i a l v i s c o s i t y t e c h n i q u e based on t h i s o b s e r v a t i o n we r e f e r t o I k e d a [ 8 1 . Now
t
Bij
and
By. B;~,
d e f i n e d by ( 1 . 5 ) and ( 1 . 6 ) have t h e f o l l o w i n g p r o p e r t i e s .
1J
Lemma 1.1. t
- B+i j ’
(i)
~j~=
-
(ii)
6:.
~f~
= nij.b(P..),
0,
e i j 5 0.
(iii)
@tj2 t
1J
P r o p o s i t i o n 1.2.
where Proof.
1
E
Vlh
Bji
for
=
1J
vh
E
Vlh
we have
i s c o n s i d e r e d as
N1 @it,’ i=1
The f i r s t a s s e r t i o n i s proved by u s i n g t h e f a c t
Lemma 1.1.
We now show ( i i ) .
Substituting
I$h = 1
E
6;J 2 0
By ( i ) o f Lemma 1.1 we have f o r
i n t o (1.9) we o b t a i n ( i i ) .
Vlh
By v i r t u e o f t h e d i v i s i o n o f a c u t e t y p e t h e a p p r o x i m a t i o n Ah properties
.
P r o p o s i t i o n 1.3.(Ciarlet-Raviart(41) (i)
ah/av
(ii)
h=
where
aij
=
h
- c
and ( i i i ) o f vh, @ h Vl~h,
(P.) (=a..) J 1J
c
p . E r iEAj,Piiro J O
I R g r a d I$ih*grad
For
5
0,
vh
@ j h dx.
Vlh
we have
...,N1,
i#j,l,
a. .v (Pi), ’J
E
has t h e f o l l o w i n g
374
M.
TABATA
F o r an a p p l i c a t i o n we c o n s i d e r t h e problem u tt Lu = 0 i n Qx(0,T) c o n d i t i o n u o ( x ) . The boundary c o n d i t i o n i s g i v e n by (1.2) w i t h
s a t i s f i e s b - n = 0 on r , ( o r i f t h e boundary c o n d i t i o n i s t h e s o l u t i o n u s a t i s f i e s t h e mass c o n s e r v a t ion law (1.10)
R
-
w i t h an i n i t i a l If b
rl =.'l
Edu/dn
t
b * n u = O),
u ( x , t ) dx = const.(=/ u o ( x ) d x ) . R
Furthermore if u0 i s non-negative, so i s u. We t a k e n o t i c e of t hese two p r o p e r t i e s o f t h e s o l u t i o n U . The f i n i t e element scheme we t a k e i s :
(1.11)
where
DT
(1.12)
I
ui(P i)
h
t
B
E
(1'14)
i=l,
...,N ,
h'
mi = t h e area o f
and vh
i n Vi,
uO(x)dx/mi,
=
such t h a t
Di i s t h e f o r w a r d d i f f e r e n c e o p e r a t o r d e f i n e d by k DTUh = (uhk+l_ u ~ ) / T w i t h a time step T , Lh = EA
(1.13)
.. , N T c V h
F i n d I u hk ) , k=l,2,. k k DTuh t Lhuh = 0
Di
,
i s defined by N h i & v h ( P i
Vhc V i
for
Note t h a t i n (1.14) a lumping t e c h n i q u e i s used. (1.15) = T( u h k),
up1
where (1.16)
T
i s an o p e r a t o r f r o m V,, T(Vh) = vh
-
into
VA
Qh E Vh.
Scheme (1.11) i s e q u i v a l e n t t o
defined by
TL~V~.
By v i r t u e o f P r o p o s i t i o n s 1 . 2 and 1.3 i t i s n o t d i f f i c u l t t o show t h a t f o r every h and E w i t h a s u i t a b l e c h o i c e o f 'I ( s t a b i l i t y c o n d i t i o n ) (1.17)
%T(v,),
Qih>h/avh(Pj)
(1.18)
< T (v h), l > h =
0,
i,j=l,
...,N,
Therefore t h e s o l u t i o n uh o f (1.11) i s non-negative i f so i s uo, s i n c e (1.16) i s a non-negative ( o r monotone) scheme. Fu r t h e r m ore uh s a t i s f i e s t h e d i s c r e t e v e r s i o n o f (l,lO), (1.19)
I
uo(x)dx).
n
We r e f e r t o (11 f o r t h e d i s c u s s i o n on convergence. tends t o zero we r e f e r t o K i k u c h i - U s h i j i m a [ l Z I .
F o r t h e behavior when
E
( i ) I n o r d e r t o s a t i s f y t h e c o n d i t i o n (1.17) i n t h e convent ional Remark 1.4. f i n i t e element method, h m u s t b e s m a l l i n p r o p o r t i o n t o E and e ( a n g l e o f t r i a n g l e ) must be s t r i c t l y l e s s than 1r/2 ( ( 2 0 1). ( i i ) The c o n s e r v a t i v e p r o p e r t y (1.18) i s i m p o r t a n t a l s o i n n o n - l i n e a r h y p e r b o l i c problems W a t t N(u) = 0. L e t us c o n s i d e r t h e scheme (1.15) w i t h
CONSERVATIVE UPWIND F I N I T E ELEMENT APPROXIMATION
(1.20)
-
T (v h ) = vh
375
.rNh(vh)’
where Nh i s an a p p r o x i m a t i o n o f N. Then, C r a n d a l l - T a r t a r ’ s Lemma (see C r a n d a l l Majda(51) says t h a t under t h e c o n d i t i o n (1.18) t h e m o n o t o n i c i t y (1.17) i s 1 e q u i v a l e n t t o t h e L - c o n t r a c t i o n o f T,
-
T(wh)( Y l > h
5 ‘ I V h - Whl l’h. Y
The ex pre s s ion (3.7), which w i l l be g i v e n i n s e c t i o n 3, has a c l o s e connect ion with this. 52. APPROXIMATION OF NON-LINEAR CONVECTION TERMS I n t h i s s e c t i o n we e x t e n d t h e a p p r o x i m a t i o n p r e sent ed i n t h e p r e v i o u s s e c t i o n t o and B f j we use t h e problems w i t h n o n - l i n e a r c o n v e c t i o n terms. I n p l a c e of 8’ lj s p l i t t i n g t e c hniqu e o f n o n - l i n e a r f u n c t i o n ( i n t o monotone i n c r e a s i n g p a r t ( 2 . 6 ) and monotone decreasing p a r t ( 2 . 7 ) ) proposed by O s h e r ( l 7 ) . We preserve t h e n o t a t i o n g i v e n i n s e c t i o n 1. Let
be a n o n - l i n e a r o p e r a t o r d e f i n e d by N”(v) = -EAV
(2.1) where (2.2 )
f:
5
x
R
-P
t
V*If(x,v)l,
R2, i s a smooth f u n c t i o n s a t i s f y i n g
f (x , O ) = 0.
Sub jec t t o t h e homogeneous boundary c o n d i t i o n (2.3 )
v = 0
on
r,
t h e weak f o r m o f ( 2 . 1 ) ’ i s y f o r (2.4 )
=
€1g r a d n
v, @
E
1
Ho(Q),
v - g r a d @ dx
-
j’ f ( x , v ) * g r a d @ dx R
= E t < F ( v ) , @>.
Let
1 voh ( = V hf lHo ( Q ) ) b e t h e l i n e a r span of
Qih, i = l y . . . ’ N 0 . We denote by Ah 1 and Fh (voh v;~) a p p r o x i m a t i o n s o f A and F ( H +~ H-’)’ r e s p e c t i v e l y . is defined by (1.3). Fh i s d e f i n e d by, f o r vh, $h E Voh, -f
@h’h where
+
=
‘@ ’ ‘
NO
i=l
(p’)
jEAi
t
Yijcfij(vh(Pi)) t ffj(vh(Pj))}S
V
(2.6)
f..(v) =
j max(0, ni * a f / a v ( P i ,v) )dv,
(2.7)
ffj(v) =
1 min(0,
1J
0 V
0
nij-af/av(Pij,v))dv.
The f o l l o w i n g lemma and P r o p o s i t i o n correspond t o Lemma 1.1 and P r o p o s i t i o n 1.2
376
(ii)
M . TABATA
h
=
We now observe how Fh
C
P.Er J
y. . f t . ( V h(Pi)).
C
lJ lJ
iEAj,Pi&
approximates
i n t o VAh. F o r t h e comparison w i t h Let o p e r a t o r f rom VOh i n t o H-’. Voh s a t i s f y i n g (2 . 8)
F.
H i t h e r t o Fh
i s t h e o p e r a t o r from
Voh
F and f o r l a t e r use we extend Fh t o an Qh be an average o p e r a t o r from Hfi i n t o
Qh = I
where Si i s t h e s u p p o r t o f @ihy Sij = Si U S and hi i s t h e maximum sidej l e n g t h of t r i a n g l e s i n Si. For an example o f such an average o p e r a t o r we r e f e r t o [221. We e x t e n d Fh(vh) i n t o a f u n c t i o n a l on H i by (2.13)
< F h (v h ) , @> = ‘Fh(vh)s
Qh@>h
for
Vh
E
voh,
@
E
1
Ho.
F o r any p o s i t i v e number M we p u t (2.14)
EM = C @h E voh; -M
5 @h 5
The f o l owing lemma assures t h a t
H-?
MI.
Fh ( v h )
approximates
F (vh)
w i t h order
h
in
.
Lemma 2 . 1 F o r any p o s i t i v e number M t h e r e e x i s t two p o s i t i v e constants ci(M,f), i=O,1, such t h a t f o r vh E. EM, (2’15)
IIFh(Vh)llL2
(2 *1 6)
IIF(’h)
Proof.
L e t vh
-
CIIIVhll 1
Fh(vh)llH-l
H
’
5 { c l l l V h l l H 1 ‘ O1h*
be any f u n c t i o n i n EM
and
$
E
&(n) be any t e s t f u n c t i o n .
t h e d e f i n i t i o n (2.13) we have CFh(Vh)9 4’ = NO ( Q h @ ) ( P i ) 1 v i j { f t j ( v h ( P i ) ) i=l jEAi Using Lemma 2 . 1 and n o t i n g t h a t
’fT*(vh(Pj))}* 1J
By
371
CONSERVATIVE UPWIND F I N I T E ELEMENT APPROXIMATION
C
yijnij*f(Pi,
vh(Pi))
1
v lJ . . C - * * ) = jEhi c
j €Ai
= 0,
we o b t a i n jEhi
+
C
@>I
yijnij*Cf(Pij,
N
i=1
J €Ai
We now proceed t h e p r o o f o f (2.16).
I t i s n o t d i f f i c u l t t o see t h a t (2.10) i m p l i e s
1I
A p p l y i n g Lemma 2 . 1 , we have
Since i t h o l d s t h a t
we o b t a i n
+ cOlhll@ll 1.
1 h H1
f(Pi,
vh(Pi))l.
Y i j { l v h ( P i ) - v h Pj)l+lPij-Pil
T h i s proves ( 2 @ = vh. The d e f i n i t i o n (2.13) leads
= I1 t 12'
< { c Ilv
-
' cg}ll@/lL 2'
h e r e we have used ( 2 . 9 ) and ( 2 . 1 2 ) w i t h
=
f T j ( V h (P j ) ) I vh(Pi))
2 ~ ( M B C~o )I ( Q h @ ) ( P i ) I 5 {clllVhllH1
1'11
t
j €Ai
T h e r e f o r e we g e t I
y .1J . t - f T 1J . ( v h (P i) )
H
1
370
M . TABATA
her e we have used (2.12) and (2.11) w i t h
Thus we g e t (2.16).
0 = vh.
( i ) Ikeda[81 s t u d i e d t h e f o l l o w i n g approximat ion i n p l a c e of (2.5),
Remark 2.4. N
T h i s a ppro x ima t io n does n o t s a t i s f y (i) o f P r o p o s i t i o n 2.2. The use of a r t i f i c i a l v i s c o s i t y , however, can r e c o v e r t h i s d e f e c t . i n t o an o p e r a t o r from (11) As w i l l be seen i n s e c t i o n 3, t h e e x t e n s i o n o f Fh Voh i n t o H - l i s s u f f i c i e n t f o r o u r purpose. However, i n some cases ( f o r emample, when one uses t h e a n a l y s i s o f Brezzi-Rappaz-Raviart[21), i t i s r e q u i r e d t h a t Fh i s an o p e r a t o r f r o m H i i n t o H-'. T h i s i s done by d e f i n i n g f o r v , 1 4 E Ho'
h.
I t i s n o t d i f f i c u l t t o see t h a t Lemma 2.3 i s v a l i d n o t o n l y f o r
also f o r
v
E
1 Ho.
vh
E
Voh
but
53. NON-LINEAR ELLIPTIC PROBLEMS
I n t h i s s e c t i o n we a p p l y t h e c o n s e r v a t i v e upwind t echnique t o t h e f o l l o w i n g nonl i n e a r e l l i p t i c problem,
.
{
(3.1)
-EAV t V - ( f ( x , v ) l
t g(x,v)
v
= 0
in
= 0
on
a;
r,
2 R , g:
where f : 3 x R + 8 x R + R, a r e smooth f u n c t i o n s . By u s i n g t h e f i n i t e dif f e re nc e method O s h e r [ l 7 ] s t u d i e d t h i s problem. He showed t h a t t h e s o l u t i o n of a n o n - l i n e a r f i n i t e d i f f e r e n c e scheme f o r (3.1) i s obt ained as t h e L 1 - l i m i t o f s o l u t i o n of t h e corresponding e v o l u t i o n a l f i n i t e d i f f e r e n c e scheme. T h i s i s a l s o r e a l i z e d i n o u r f i n i t e element scheme (Theorem 3.4). E r r o r estimate i s derived i n t h e frame work o f L 2 - t h e o r y . We b e g i n by t r a n s f o r m i n g ( 3 . 1 ) i n t o t h e e q u i v a l e n t weak form:
{ where (3.2)
Find v
E
Hip)
such t h a t
i n H-l(Q),
N (v ) :EAV t F ( v ) t G(v) = 0
A
and F
a r e d e f i n e d i n (2.4) and G = j' g(x,v)4 dx
for
i s d e f i n e d by
v, 4
E
Ho(Q). 1
a We assume t h a t (Hl)
There e x i s t s a p o s i t i v e number ag/av(x,v)
(H2)
When
IvI
2
6
for
x
6 E
such t h a t
n, v
E
R;
tends t o i n f i n i t y , i t h o l d s t h a t
supCID:f(x,v)l;
XE~,
la[=l)/infIlg(x,v)l;
XEQ
1
+
0.
379
CONSERVATIVE UPWIND F I N I T E ELEMENT APPROXIMATION
We c o ns ider t h e f o l l o w i n g f i n i t e element scheme f o r ( P o ) :
‘[
(‘oh) where V&,
F i n d vh
such t h a t
Voh
E
i n VAh,
Nh(vh) E €Ahvh t Fh(vh) t Gh(vh) = 0
Ah and Fh i s d e f i n e d by
a r e d e f i n e d by (1.3) and (2. 5), r e s p e c t i v e l y and
(3. 3 )
+h’h
= i~~ig(Pi.vh(Pi))Ph(Pi)
here mi
i s t h e area of
Di
for
VhD @h voh,
introduced i n (1.13)-
I n o r d e r t o e s t i m a t e t h e d i f f e r e n c e between Gh
and G we ext end Gh
o p e r a t o r f rom Voh i n t o H-’ by u s i n g t h e average o p e r a t o r w i t h r e s p e c t t o Fh i n s e c t i o n 2,
Gh: VOh+
h
for
vh
E
Voh,
Q,
t o an
as was done
1 0 E Ho.
S i m i l a r l y t o Lemma 2.3 we have .Lemma 3.1. F o r any p o s i t i v e number M t h e r e e x i s t two p o s i t i v e const ant s ci(M,g), i=O,1, such t h a t f o r vh E EM, IlGh(vh)l/L2 2 ‘0’
-
G(vh)ll 5 { c l l \ V h l l H 1 colh’ HIn o r d e r t o s o l v e t h e n o n - l i n e a r problem (Poh) t h e corresponding e v o l u t i o n a l problem i s i n t r o d u c e d : k c Voh such t h a t F i n d { u h l , k=1,2, IIGh(’h)
(plh)
1
...
k DrUh
t
k Nh(uh) = 0
i n VAhs
u i : given,
where
DT
i s t h e f o r w a r d d i f f e r e n c e o p e r a t o r def ined i n (1.12).
By (1.20) we d e f i n e an o p e r a t o r
LMma 3.7,
T
f r o m Voh
5M
and
T
T
2
i s monotone on T(vh)
where
such t h a t
M
t h e r e e x i s t s a p o s i t i v e c o n s t a n t ro(f,g,h,
Toy
EM, i . e . ,
5 T(wh)
is
T(-M) 2 -M.
Lemma 3.3, F o r any p o s i t i v e number M) such t h a t under t h e c o n d i t i o n 0 <
Then (Plh)
VAh.
There e x i s t s a p o s i t i v e c o n s t a n t M ( f ,g) T(M)
(3. 4)
into
By v i r t u e o f (H2) we have
r e w r i t t e n i n t h e f o r m (1.15).
if
vh
5 Wh
EM i s t h e s e t d e f i n e d i n (2.14).
and
vh I Wh
E
EM,
380
Pro of .
TABATA
M.
I t s u f f i c e s t o show t h a t f o r
vh
ah/av h ( PJo ) -: 0,
(3.5)
under t h e c o n d i t i o n ( 3 . 4 ) .
E
EM,
i,j=l,...,N
0’
For j # i,we have by P r o p o s i t i o n s 1.3 and 2.2,
a h / a v h ( P j ) = - T { a i j
ah/ avh(Pj)}
- 0. ?
F or j = i we have a h / a v h ( P i ) = m i
(3.6)
t
-
‘{
miag/av(Pi,
$ih>h
’ a
I
$ih’h/ avh(Pi)
vh(Pi))>.
Since af / av and ag/av a r e bounded on % [ - M , M ) , there exists a positive such t h a t under t h e c o n d i t i o n (3. 4) t h e r i g h t - h a n d s i d e of cons t a nt T,,(f,g,h,M) (3.6) i s non-negative. T h i s completes t h e proof. We now f i x t h e p o s i t i v e constants M and T~ by Lemmas 3.2 and 3.3. an o p e r a t o r f rom EM i n t o EM s i n c e i t h o l d s t h a t f o r vh€ E M ,
Then T
is
-M 5 - T(-M) 5 - T( v h ) 5 T(M) 5 M . Theorem 3.4. (i) There e x i s t s a u n i q u e s o l u t i o n vh o f (Poh). ( i i ) The s o l u t i o n uh of (Plh) w i t h any i n i t i a l v a l u e i n as k tends t o i n f i n i t y , more p r e c i s e l y ,
k
(3.7)
-
Vhl, l > h
EM
converges t o
vh
5 e-k6T < ( U0h - VhlS 1 >hi k=1,2,...
Theorem 3 , 5 . (i) There e x i s t s a unique s o l u t i o n
v
E
2
H
(a)
o f (Po), which i s i s o l a t e d , i . e . 1 Ho o n t o H-’.
t h e l i n e a r i z e d o p e r a t o r of N a t v i s isomorphism f rom ( i i ) When h tends t o z e r o , we have
where
vh
i s t he s o l u t i o n of ( P o h ) .
O u t l i n e o f t h e p r o o f o f Theorem 3.5. By t h e f a c t t h a t vh E EM and by Lemmas 2.3 and 3.1, { v h I h i s a bounded sequence i n H1(n). T h e r e f o r e t h e r e i s a subsequence 1 2 converging t o an element v i n L (n) and weakly i n H (a),which i s a s o l u t i o n of (Po). It f o l l o w s from ( H l ) t h a t v i s an i s o l a t e d unique s o l u t i o n . I n o r d e r t o p ro v e t h e second a s s e r t i o n we use Mock’s met hodll51.
Let
0
E
Hi(Q)
be t h e s o l u t i o n of (3. 5 )
EAO = -Fh(vh)
-
Gh(vh)
in
H-’.
The d i f f e r e n c e between 0 and vh i s e s t i m a t ed by t h e c l a s s i c a l t h e o r y s i n c e vh can be c o ns idere d as t h e f i n i t e element s o l u t i o n o f ( 3 . 8 ) , whose r i g h t - h a n d s i d e i s e s t imat e d by Lemmas 2.3 and 3.1. F o r e s t i m a t i n g t h e d i f f e r e n c e between 0 and v t h e f o l l o w i n g i n e q u a l i t y i s e s s e n t i a l ,
38 1
where
M1,
c(M~,E) are p o s i t i v e constants.
11' - VllHl 5 c'uF(')
-
F h ( v h ) l l -1 H
Thus we have +
w h i c h i s e s t i m a t e d b y Lemmas 2.3 and 3.1 and
IIG(')
-
Gh(vh
(10 - vhllH1.
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