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= completion of the
cm
.
F
The space L is the P functions in R with respect to the norm
We will discuss this problem later but let me first remark that when F and aR are sufficiently smooth Problem I is just the classical problem
The second problem is the classical Dirichlet problem Problem 11:
Given f
E
c'(~R)
, find
u
E
c'(R)
such that
Au = 0 in R
We will refer to well known results (eg. on regularity) for this problem when needed. We can immediately state the following theorems. Theorem 1: There exists a unique solution to problem I. Theorem 2: If
aR is such that at each point of
aR there exists
a barrier then problem I1 has a unique solution. Theorem 2 is classical. As we will see Theorem 1 (existence part) can be proved by means of a difference method. Let us now formulate corresponding difference problems and investigate their properties. Let ENh be the set of mesh points in E
N
form (ilh,
', iNh) , for h
> 0 and
il,
, i.e., points of the
. ., iN
integers.
.
We make some definitions. a) Kh = R r) ENh
( ei
is the vector with 1 in the i s position and 0 in the others ) N
We can immeidately state the following. Lemmal: in E Nh
Suppose
.
- A V 3 0 in R h , V 2 0 in EN,, h
- R,, .
Then V 5 0
From this it immediately follows that the discrete problem
Ah V(x) = F(x)
, x E R,,
has always one and only one solution for any given F and' g can introduce the discrete Green's function Gh(x,y)
.
defined as
Thus we
We can now make some statements about G Lemma2: Lemma 3:
.
.
Gh(x,y)30
F or any V defined on
\
This follows from uniqueness in the discrete problem, We next introduce the following function. Let
L Then we can prove Lemma 4:
For suitably chosen yN
Gh(x,y) .' V(x-y)
,
Y
E
Rh
.
, a,
and do
(independent of h )
The proof of lemma 4 consists i n showing t h a t it i s possible t o choose yN
,a
and d
0
i n such a way that f o r y
A
h,x
A
n>x
E
-N ~(x-y)=h
,
x = y
V(x-y)aO
,
X # Y
V(x-y)
all x
0
Then i t follows from lemma 1 'applied t o V(x-y) true
- Gh(x,y)
t h a t lemma 4 is
From lemma 4 we can prove
Lemma 5: of
.
Let
h and x
N
1$ p < ~ _ 2
.
Then there i s a constant
C
P
independent
such that
After using lemma 4 the sum i s estimated by comparing the sum with corresponding analogou~i n t e g r a l s using the f a c t that X
# Yo
.
, for
For the case N 2 3
x
-
y
is subharmonic,
6 1 N-2,
example, we can obtain the estimate
where S i s a s u f f i c i e n t l y l a r g e sphere containing R and having center a t an a r b i t r a r y point y
o
E
R
.
The integral i s convergent i f
p <
N N-2 '
-
By taking V Lemma 6:
YE^\
1 in lema 3 we obtain
Gh(x,y) = 1
•
Now it is a simple matter to give a convergence theorem. First we define the discrete problem. Problem 111:
A u (x) h h
Of course
u,,
c2m .
0
xERh
exists and is unique. Now we have
Theorem 3: u r
=
Let u be the solution of problem I1 and suppose that
Then u,,
f
u uniformly as h
+
0
.
We apply lemma 3 to
0
.
Hence by lemmas 5 and 6,
Clearly the right hand side tends to zero as h Immediately from (7) we can deduce
+
Theorem 4 :
If
u
E
c*'~(K) 0
The proof is obvious. Since u is harmonic,
inu ul
a 6 Ch
in
and
.
- u r n ( E Ch
lu(y)
, then
< a ,< 1
Now no matter how smooth u is in the closure of R the best possible result is with a = 1
.
The last term on the right of (7) prevents us
from obtaining a higher order estimate. As we will see, however, theorems
3 and 4 can both be improved in the sense that if we place some restrictions on aR then we can obtain similar theorems with u less regular up to the boundary. Thesrem 5: Suppose aR is such that u can be approximated uniformly
- .
by a sequence of functions each of which is harmonic in R uniformly as h cube Ch(x) = Remark. Brelot
+
IY
0
.
( Gh
Then i h
-t
u
is the extension of uh as a constant on each
[xi - h/2 < yi ,< xi
+ h/2 , i = 1 , ... , N}
)
Conditions on aR have been given by several authors,(c.f.
[8] and Walsa [ll] such
that every
function
on aR can be approximated uniformly by functions harmonic in
continuous
K
.
This is not necessary for our theorem as we can see by the following example. Let R be star-shaped with respect to some point which, without loss, we call the origin. Then if u exists, it can be uniformly approximated by a
-
sequence each member of which is harmonic in R
.
For, define
where pn f 1 as n
.
0
+
Now
Au(pnx) = 0 in
since u r c09;) ,cn -+ u uniformly in K
.
.
if
Clearly
Thus starehapedness is a
sufficient condition for the convergence of uh to u whenever u exists. The proof of theorem 5 is immediate. We have only to estimate u,
- un for h small and n large. But from lennna 3
which we can clearly make as small as we wish by first taking n large and then h small. We also can prove the following. Let aR
Theorem 6:
E
cL
(piecewise, with no reentrant cusps if 1
.
R and u but not on h
.
N = 2 ). Suppose that u is Hijlder continuous with exponent 0 < X Then, for every
E >
0
sup luR
where K(E)
6
sh1
L K(E)
is a constant which depends on
E,
The proof of this is based on the following two lemmas. Lemma 7:
Let aR
.
Let d(x)
in 'ii
E
cL
and u be harmonic in R and h
be the distance from the point x
E
R to the boundary
(which is well defined in a strip S6 of fixed width 6 ) every
E
> 0
there is a K(E)
for x
E
Rhfl
Ss
.
such that
- B6lder continuous
.
Then for
This lemma follows from the mean value theorem for harmonic functions. One obtains an estimate for the HBlder continuity of the second derivatives which depends on the distance to the boundary. The next lema is crucial. Lemma 8:
For every
E > 0
there exists K(E)
such that, if aR E 'C
The proof of this.lemma is tedeous, long and involved, but the motivation is the following. We consider formally
where G is the continuous Green's function and we have assumed that d has been suitably extended to R
.
Then, formally,
It is not difficult to see using the maximum principle that for aR E C2
$(XI
a c dE(x)
Hence the procedure for proving lemma 8 is based on constructing a suitable comparison function and using the maximum principle, lemma 1. We shall now turn our attention, for a time to problem I. First I will briefly sketch a proof of Theorem 1. For uniqueness it suffices to
IVA~= 0 , V ~ E V
show that if cm
$ E CQ
then there exists a 4
E
then v = 0
.
But one can show that if
V such that A4 = $
.
Uniqueness follows.
The existence follows once the inequality
is established.
(One used the Hahn-Banach theorem.)
Since we shall be interested in the convergence properties of related difference schemes, we shall sketch the proof of (8) by the difference method. For any integrable function f we define
where C (x) is the cube with center x h axes, defined previously. We let
4
h
be the solution of
, side h and sides parallel to
Then it follows from lemmas 3 and 5 and HBlder's inequality that
where
h
is the extension of
Oh as a constant in each cube Ch '
What we would like to show is Lemma 9:
Let
l
.
4
E
V
.
Then
mh
+
4 as h
+
0
, weakly in L
P
This can be done by using (9) and the fact that Lp
,1 <
p
,
<
is reflexive and hence bounded sets are weakly compact. We then choose
a sequence weakly as n
, hn + 0
{hn]
.
+
converges to 4
as n
+
We then show that
such that
mh
+$EL n P ' $ = 4 and that every sequence
.
Clearly (8) follows from lemma 9 and Theorem 1 is then proved. We next want to pose a discrete problem analogous to problem I
Problem I1 :
where \(x)
is the function of y for fixed x defined by
.
Note that i f
F
E
L1
then
It i s now rather easy t o prove Theorem 7: I and I1 h+O
Let F
, respectively.
.
E
and u and uh be t h e solutions t o problems
L1
Then ii + u weakly i n L h P
We consider the following i d e n t i t y for a r b i t r a r y
, 1f
@
p <
E
V
N-2
as
.
:
Again, using lemma 5 we can show t h a t
so that again we can take a sequence ii + ii (some element of L ) hn P weakly i n L 3 1 < p < Clearly the l e f t hand s i d e of (10) converges P N-2 to ti^$ a s n + NOW + $ weakly i n L for 1 < p < P 'hn
-. .
and by (8) and (9)
$ and @ hn
-
a r e bounded.
Thus it is a simple matter
t o conclude that
Hence ii is the solution of problem I o r ii = u converges t o u so that
ii h
-r
u
.
.
But every sub sequence
In order t o show the strong convergence we introduce a sequence Fn
E
lim n*
c O a ( ~ ) f o r each u and such that
R
IF-F
.
introduce the corresponding functions u and u nh n
.
case of t h e work of +
see that as h
+
u n
E
V
We also
Now we have
where C does L1 The middle term can be treated rather e a s i l y d i r e c t l y
but we can observe t h a t since Fn
I l ~ ~ ~ -L2 u ~ l 0l
.
IF-F 11
Now the f i r s t and l a s t terms a r e bounded by C not depend on h
o
=
CBa as
h
C (R)cL2 f o r each n 0
.
f o r fixed n
0
But since it i s easy to
1
+ 0 llii n d U n Lp
and hence i s bounded so t h a t i n fact
.
0 for a l l p <
a s a special
we conclude that
[9] +
a
E
Thus taking
n large and then h
small we
have proved. Theorem 8:
Let
F
E
L1
and uh and u be the solutions t o problem I
-
Then iih + u strongly i n L , 1 I p < as h + ~ . P N-2 The next two theorems give information when F is not necessarily i n L 1 '
and I respectively. 1
Theorem 9:
Let
F
E
continuous functions i n R
(L,)'
and l e t
.
Suppose t h a t
C
be the c l a s s of piecewise
P
F
, when
considered a s a
n
functional on C La has support Q contained i n R (i.e. 0 i s P closed and i f 9 E C nLa and 4 = 0 on Q then <$ ,F> = 0 ). Then P iih + u weakly i n L , 1 $ p < - a s h + 0 P N-2
.
We e a s i l y obtain
1t then must be shown that
iih A9 =and
+ <$,F>
,
llchll
L
as h
6 C
0
'
(Loo)
P
-+
Il~ll
.
.
The additional hypothesis is imposed now since F can not necessarily be approximated (strongly) by nice functions. However by using the results of Bramble and Hubbard [5] to (
we can show uniform convergence of
on compact subsets of R
.
6h
.
Instead of puting a condition on F we could impose one on aR Theorem 10: Let F
E
(L,)'
and aR
$h to
$
-+
u weakly
( E
V will belong to C1 (K) so
X = 1 can be applied to show the uniform convergence
that theorem 6 with of
Then 6 h
.
- .
as h - + ~ in L , 1 $ p < P N-2 If aR E c2 then in fact each
,
.
cL
E
in K
.
In fact using the estimate of theorem 6 (slightly
modified) we obtain a kind of estimate for the rate of weak convergence. Theorem 11: Let F
E
(L,)'
OR
and
where C depends on J, but not on h
E
c2
.
Then for any
J,
m E
.
We next observe that if F is the "delta function" when restricted to V for an arbitrary point x then the solution u of problem I will be the Green's function with singular point x existence and uniqueness of G(x,y)
(11)
((XI
=
-J
.
That is we have the
such that
G(X,Y) ~ ( y )dy
R
Now it is possible to prove the following.
,
v
4
E
v
.
Co
Lemma 10:
Let F
E
.
L1
Then
is the solution to problem I. is symmetric
This lemma is proved by first showing that G(x,y) and then applying the basic relation (11) enjoyed by G
.
The only
difficulty arrises in showing that the theorem of Fubini-Tonelli on interchange of integration can be applied. This can be done with the aid of the difference approximations. Having the representation of lema 10 we can show Theorem 12: Let F c L
q
pointwise in R as h
+
0
, for some
.
Clearly from lemma 3
and hence
Combining this with (12) we have
q > N/2
.
Then ti, n
+
u
But from theorem 9 and the continuity of $ in R it follows that for N each fixed x E R ch(xy.) Gfx,.) weakly in Lp , 1 1 p < N-2 '
-
-+
Since F
E
L
P
, for some
q > N/2 the theorem follows,
In this same spirit we can prove Theorem 13: Let F
E
L
9
aR
for some q > N/2 and
E
c2
.
.
Then 6 + u uniformly as h -+ 0 h To show this we simply approximate F strongly in L by a sequence 9 with Fn E c~(R) for each n Then F
.
so that
where C does not depend on x and $n
E
V
.
For large n the second
term is small and by the previous remarks
n'
Onh Finally, if F is smooth and
uniformly as h
-+
O
.
aR is smooth we have the analog of
theorem 6. Theorem 14: Suppose F we have for every
where K(E)
E
coy'
and aR
E > 0
is a constant independent of h
.
E
c2
.
Then for problem I
This is similar to theorem 6 for 11. Take u such that 1 u 1
E
"'c
,
Then set u2
=
- U1
.
Aul
-
F
Theorem 14 then follows from
theorems 4 and 6. In order to obtain rate of convergence estimates which are of higher order it is clearly necessary to modify the difference scheme near the boundary. We will for the time being still be considering problem I but with various assumptions on aR and F We shall now redefine Rt, and 3%
a)
c)
Nh(x)
3%
.
.
is the set of "neighbors" of x with respect to
ah
is the set of points on aR which lie on "mesh lines".
For V defined on
aRh
we define
a
,
where uih is the distance between adjacent points of direction 0 < ai
<1
in the xi
.
Now everything said so far remains true with this definition the minor redefinition of (F)h
and \(x)
, with
regarding problem I1.
Essentially, if one defines F as "zero" outside R then werything is the same. However we can give now an additional estimate for Gh which allows us to obtain second order convergence for smooth solutions. Lemma 11:
This is trivially obtained by taking for V in lemma 3, V = 0 on 3%
and V = 1 in
R,,
.
Then
Let us call the analog of problem 11, problem 12. Then we get immediately Theoren 15: Let u and uh be th; s~lutionsof I and I respectively and u
E
(m .
C4
2
Then
We simply note that
so that puting V = uh
-u
in lemma 3 we have the result. In fact we
note that even though the approximation locally near the boundary is of order h the contribution to the error is a term of order h3
.
This,
we will see, will lead to higher order approximations.' Before mentioning other approximations we shall see what can be said if certain more specific information is known about F
.
Thus we shall
consider briefly the case in which F is very smooth except at one point and 3R is also smooth. Again in this case the Green's function technique is fruitful. The following theorem gives information about the error in this case. Theorem 16: Let aR be smooth and u is an arbitrary point of of a mesh cube.) O < X f l and
E
4
C m-0) where the origin 0
(for convenience always chosen to be the center
Suppose that m
+X
> 2
-N
where m is an integer and
where k is a multi-index kl = (kl integers, lkl =
1 ki
- . , k.J
)
,
kl,
...,k.J non-negative
and
The fur u and uh the solutions of I and I2 respectively
In this case since we have quite specific knowledge of the behavior of the solution at the origin we obtain an estimate for the error which is point dependent. Thus even if the solution u has the form 2-Nt6 ~ ( x )= 1x1
t
uh
we get from the theorem that
+
regular function
,
6> 0
,
u uniformly on every compact subset
not containing the origin. Also note that under the assumption (13) if u
E
,6 > 0 (m=Z)
then the convergence is second order. This shows clearly that the usual sufficient condition that u
E
4 C (K) is far from necessary for 0(h2)
E
h C (m=0) we obtain a uniform rate of hh
convergence. Note also that when u or h
1-E
,
As might be imagined the theorem is proved by using the representation lemma 3 and estimating the resulting expressions. The details are long and technical and are found in
[7]
but I want to point out the crucial
points. First of all the essential ingredient is the majorant (lemma 4)
.
for Gh
This tells us that, as might be expected Gh behaves quite
like the fundamental solution for Laplace's equation in the neighborhood of the singularity. Having this we are led to proving a sum relation analogous to a well known integral expression Lemma 12:
If
-N < p ,q < 0
and x
,z E
Ix-yl .:ah , I Z - ~::ah, ~ a > O , V y ER,,
Rh
are such that
then
The proof is done by showing that the sum is majorized by C
I
R
Ix-yIP Iz-ylQdy For this formulation, however, we can get a sharper theorem than
either theorem 6 or 15. Theorem 17: Let aR
N
=
2).
E
C* (piecewise, with no reentrant cusps if
Suppose that u is the solution of I1 and uh is the solution
of 1I2 (the analog of I1 for the reformulation). 1 Then for
E
> 0
Let u
E
cPyA(K)
.
The proof is similar to that of theorem 6. Note that with u
E
c2"(F)
cl",
A
the theorem 4 would show only a rate of hA and for
< 1 we would conclude nothing. However, we get second order
-
convergence when the second derivatives are HBlder continuous in R , Clearly these methods are not restricted to these particular difference formulations for the Dirichlet problem for Poisson's equation, One can treat a)
More general operators (second order).
b)
Various boundary conditions.
c)
Eigenvalue problems.
d)
Various difference approximations.
I shall discuss an example of the last extension, since it brings out the fact that in the transition from the interior to a curved boundary one can (in the Dirichlet problem) take approximations which are of the order of accuracy (locally) worse by a factor of hL and still obtain as a global error that of the interior. For this example I choose N = 2 and consider the nine point approximation
so that Ah is now locally 4th order. Now we take A (14) in
R,,
at points of Rh
involved in (14). to
A
.
At
(say
$ = R,, - %
$
)
to be defined by
h
where only I$, points are
we take a second order approximation
aRh we take as in the second formulation. One can then show
by appropriate modifications of the previous Green's function method that if u,, is the solution of our new problem I1 (for Au = 0 ) then we have 3 Theorem 18: Let u be the solution of I1 and u uh
E
C6
(n .
Then if
is the solution of 113
But in fact we can lessen considerably the requirement that u
E
6 C (R)
and obtain by the methods of theorem 17 Theorem 19: Let aR r respectively. Then if u
E
c2
and u and u the solution of I1 and 113 h
CP" (T)
So far the approximations mentioned have all possessed a common property, i.e., that of being "of positive type." This means that if ~~j is the matrix of coefficients of the linear system then
t h e second condition possibly f a i l i n g near the boundary.
In f a c t ,
it is t h i s condition, together with
t h a t makes lemma 1 t r i v i a l . To show t h a t t h i s is j u s t a convenience I wish t o give another example,
Suppose instead of (15) we used the 9-point
It i s possible t o show t h a t the e r r o r is of the order h
and probably a theorem l i k e 19 i s also true.
4
O(h ) approximation
4
when u
E
6 C @)
Since the properties (15) and (16)
a r e not possessed by the resulting system the analysis is much more d i f f i c u l t . It i s i n t e r e s t i n g t o note, however, that a corresponding d i s c r e t e Green's
function w i l l s t i l l be positive, a f a c t which i s no longer completely t r i v i a l .
As i s evident the preceding discussion is i n many respects special f o r second order equations, since much use was made of the maximum principle o r , what is the same, the p o s i t i v i t y of the Green's function.
Thus i t appears
t h a t , i n attempting t o t r e a t higher order equations, we should work more with norms other than the maximum norm.
I would like first to sketch some results of Thornge[lO ] on higher order equations and difference approximations. Consider the differential operator
..
are multi-indices, i.e. B=(B1,. ,BN) , where the B 's j n The a are non-negative integers, ( 6 ( = Bj and similarly for y BY j=l are real constants and
where B and y
1
.
We assume that L is elliptic, i.e. for real 5 =
The Dirichlet problem (111)
has a smooth solution provided F and aR are sufficiently smooth.
Consider approximations of the form
where u = u(Sh) and the C 's are complex numbers defined for all a 5 a but zero except for a finite number of a's A point ([+a)h will be
.
called a neighbor of [h
if Ca f 0
.
This time Rh will be defined as those points of RflENh whose neighbors also lie in R
.
-
Define
\=
.
a\
The characteristic
polynomial of L is defined as the trigonometric polynomial h
where 0 = (O1,...,ON)
, (a,0)
=
1 aj 0j
.
Because of periodicity 0
can
be taken in the set
We say that Lh is consistent with L if at an a&trary
point (which we
take as the origin)
Lh uo
=
+
L u(0) (
Now it can be shown
o(1)
when h
+ O(hk) -
+
0
.
consistant of order k )
Lemma.13:
Lh is consistent with L if and only if
p(B) = L(0)
+
o
( [elzm)
when 0
+
0
.
Now we shall denote the set of complex valued mesh functions on
R,
by Dh and
The sum will always be finite since all functions considered will vanish outside some bounded set. Define
and
Now we call the difference operator Lh elliptic if
p(9) > 0 for 0 # 0
In particular if Lh is elliptic p(9)
E
S
p(9)
satisfies
.
is real so that C-a =
ra .
With this definition, Thom6e then gives two a priori inequalities which we state as the next two theorems. Theorem 20: Let Lh be consistent with L
.
Then Lh is elliptic
if and only if there is a constant C independent of u and h such that
The main tool in the proof is the "Fourier transform." For this reason only constant coefficients are treated.
<
If we define = Rh
-4
R;1
and also
Then it can be shown
that part of
R,,
whose neighbors are in
R,,
and
Theorem '21: Let
aR be sufficiently smooth and L consistent h
with L and elliptic. Then
This theorem will show that the difference approximation can be cruder near the boundary. Let uhg be the solution of problem 111.
From theorem 20, I11 has one and only one solution. We have the following 1 convergence estimate of ThomCe
.
Theorem 22: Let u and u be the solutions of I11 and 1111 h respectively. Suppose Lh consistent with L and elliptic and u Then if e h
=
u
- uh
E
, 2mtl C 0.
in Rh and 0 outside Rh we have
As interesting examples ThomCe gives a number of applications of his theorem to special cases. I want to discuss two of them since in these two one can obtain an additional inequality which together with theorem 22 shows that
(A) Let
and take,
.. (We remark that in this case we can also take ulJ to be variable and treat the self adjoint operator
The matrix aij is assumed symmetric and to satisfy
for real 5
.
In obtaining (17) the reason for the low power of h is that near the boundary the approximation gave rise to a lower order error term. Thus we want to try to estimate the lm-1) L2 norm in such a way that the approximations near the boundary are not so important. For motivation we consider a very simple way of obtaining an L2 inequality in the continuous problem. We suppose that aR is smooth and let J, be the smooth function satisfying
Then by the maximum principle (
:: 0 in R
.
Clearly since u and $
are zero on aR
Thus since
( $
0 in R
and hence
Now the importance of this estimate is that $ = 0 on aR and hence Lu is not so influential near the boundary. It is just this type of estimate that would give us something for the difference problem. Unfortunately (18) does not hold in the case of the present difference approximation. However what can be shown in the following. For any V such that V = 0 in
E~
-%
where C does not depend on h
.
The f a c t t o note is that although U8)
holds pointwise it was only used i n the aean.
Now one expects, because of
consistency, that an expression f o r the difference operator, analogous t o (18) w i l l hold t o within higher erder terms.
It turns out that these terms
can be estimated i n the means, hence giving us (20). Theorem 23:
Thus we conclude
In the case of example (A) we have
The one thing that must be used here i s an inequality given by Thomge. That is t h a t
(B)
For the second example i f we take
(N=2)
and
then we have the f i r s t boundary value problem f o r the biharmonic equation Thomee's r e s u l t is the f i r s t e r r o r estimate i n t h i s problem for a general domain. h1I2
He obtains according t o theorem 22 an estimate of the order of
f o r the m = 2 norm.
We want t o look a t the m
- 1= 1
obtain an order h estimate f o r t h e e r r o r i n t h i s norm.
norm and
Again we are motivated by a pointwise differential inequality. Miranda, in obtaining a maximum principle,made use of the fact that
(Unfortunately this is special for N = 2 ). Again we can obtain the analogus expression for the difference operator, which is
where the subscript x and
denote the usual forward and backward divided
differences. Once again we can introduce a function 4 such that
Then it is (almost) clear that the last term
can be estimated in the mean by
Thus we are led to the discrete a priori inequality
.
5 C h This together with theorem 22 leads to the estimate llehll h,1 Although it is not true that the maximum norm can be estimated by the
Dirichlet integral in the continuous case, we can obtain a meaningful estimate in the discrete case, N = 2 Lemma14:
Let, V = O in E N - R h
max
IvI Rn
. 112
I clln hl
,
Then
IlvI1
+
h,1
This can be obtained by using the discrete Green's function G for Ah From the representation lemma 3 and partial summation we have
and by Schwarz's inequality
But it is easy to see that
Hence we obtain the estimate
lehl I clln hl 112 h
.
Concerning second order approximations i n t h i s problem (B) I have given one i n Bramble
[ 4 ] and ~lkl, i n 1131 has given one.
To my knowledge no second order approximation has been proved i n general for
(A).
As regards the second order approximation given by Zl6mal f o r problem (B)
I wish to mention t h a t it i s simplier than t h e one given by me and also t h a t the technique used by Zl6mal holds f o r more general 4& order equations but for N = 2
.
He shows, e s s e n t i a l l y , t h a t a c e r t a i n second order interpolation
near the boundary i s s u f f i c i e n t t o increase the r a t e of convergence from h1I2
f o r the m = 2
-
h t o h2 f o r the m
norm t o h3I2 and i n the case of example (B)
- 1= 1-
norm.
from
BIBLIOGRAPHY
~abuzka,I., Prgger, M., and Vit6s$t,E. Numerical Processes in Differential Equations. Interscience publishers, New York (1966). Bramble, J. H. "On the convergence of difference schemes for classical and weak solutions of the Dirichlet problem." To appear in the proceedings on Differential Equations and Their Applications 11, Bratislava, Czechoslovakia (1966). Bramble, J. H. (editor) Numerical Solution of Partial Differential Equations. Academic Press, New York (1966)
.
Bramble, J. H. "A second order finite difference analog of the first biharmonic boundary value problem" Numerische Mathematik 9, 236-249 (1966). Bramble, J. H. and Hubbard, B. E. "Approximation of derivatives by finite difference methods in elliptic boundary value problems." Contributions to Differential Equations, Vol. 111, No. 4 (1964). Bramble, J. H., Hubbard, B. E. "Discretization error in the classical Dirichlet problem for Laplace's equation by finite difference methods." Univ. of Md. Tech. Note BN-484 (1967) (to appear, SIAM Series B). Bramble, J. H., Hubbard, B. E., and Zlgmal, M. "Discrete analogs of the Dirichlet problem with isolated singularities." Univ. of Md. Tech. Note BN-475 (1966) (in print). ~rglot,M, "Sur l'approximation et la convergence dans la theorie des fonctions harmoniques ou holomorphes .I1 Bull. Soc. Math. France 73, 55-70 (1945). 9.
Cga, J. "Sur l'approximation des problemes aux limites elljptiques .I1 Compte rendus 254, 1729-1731 (1962)
10.
~homge,V. "Elliptic difference operators and Dirichlet's problem." Contributions to Differential Equations, Vol. 111, No. 3 (1964).
11.
Walsh, J. L. "The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions."
12.
Zlgmal, M. "Asymptotic error estimates in solving elliptic equations of the fourth order by the method of finite differences." SIAM Series B2, 337-344 (1965).
13.
Zlhal, M. "Discretization and error estimates for elliptic boundary value problems of the fourth order." (in print).
.
CENTRO INTERNAZIONALE MATEMATIC0 ESTIVO (C. I. NI. E. )
G. CAPRIZ
THE NUMERICAL APPROACH TO HYDRODYNAMIC PROBLEMS
Corso tenuto ad Ispra dal 3-11 Luglio
1967
THE NUMERICAL APPROACB TO 1IYDRODYNAMIC PROBLUIS
by G. Capriz
(Centro Studi Calcolatrici Elettroniche iiel CIJR presso l'Universit2 d i Fisa, Pisa, I t a l y )
1. Introduction
I n t e r e s t i n the numerical solution of hydrodynamic problems for instance, has been ? l i v e for a long time; the book of r e f . bears the date 1322. The reasons of tile i n t e r e s t are obvious: so few e x p l i c i t solutions of the equations of hydrodynamics a r e known and g r e a t gaps s t i l l e x i s t i n the knowledge on the q u a l i t a t i v e behaviour of general solutions [A, 21 A s r e f . 4 already shows, attempts a t numerical integration were made before the age of computers: some references t o t h i s e a r l i e r work can be found i n the textbooks of Allen and Thom-Apelt , among others. Von ilcumann called a t t e n t i o n repeatedly t o t h i s f i e l d of research [e. g . 6 , p. 2361 , ~ u g g e s t i n gt h a t computers would be the r i g h t t o o l f o r iilquiry. Attention was devoted a t f i r s t t o studies of conpressible flows [6- A; 7 , vol. 4 1 ; sometimes through the integration of reciuced equations of the boundary layer type [ I , vol. 3; 8, and the papers quoted there] Pile h e u r i s t i c i n t e r e s t of tlie chlculations was soon pointed out and anong thc f i r s t problems tackled were those for which the c u r i o s i t y of thc experimenter had not y e t been s a t i s f i e d by the r e s u l t s of the theoretician [9, 103 Incompressible flows a r c studied now with g r e a t zest; there is i n t e r e s t i l l such flows for analysis of motions with a f r e e surface , 12, g ] , and of motions of natural convection 5 , for
A,
.
[il
.
.
l;,
wheathcr analysis and prediction k ] p c r h a p s using a "shallow fluid" approximation l8] , for forecasts of flood waves i n r i v e r s and many other questions.
[s,
b]
G. Capriz
Perhaps the moot ambitious goal is pursued by tliose r e s e a r ~ h workers who t r y t o decide, by a thorough numerical study, as t o what extent the Navier-Stokes equations ( i n a f i n i t e difference form) a r e able t o describe phenomena of fluid flow i n s t a b i l i t y and even of t r a n s i t i o n t o turbulence. Interesting r e s u l t s havo been already obtained i n the description of the formation of Karman vortices behind an obstacle , of the spike and bubble i n the Rayleigh-Taylor form of i n s t a b i l i t y f o r superposed f l u i d s 21 of the Taylor vortices a t high lieynolds number i n the Couette flow The calculations a r e so precise t h a t they can be used t o deduce values of the functionals of flow (suc;i a s heat transfer coefficients and viscous drag c o e f f i c i e n t s ) much nore s a t i s f a c t o r i l y than by other approximate means. The wealth of r e s u l t s obtained by IIarlow and collaborators a t the Los Alamos Laboratory a r e so spectacular t h a t they have found ,space i n S c i e n t i f i c A r i ~ e r i c a n u , Science , Datamation , Sciences Attempts have been made t o follow, i n a f l u i d flow, the production of s c ~ a l le d a e s from larger ones i n three dimensions but the work was hampered by the occurrence of numerical i n s t a b i l i t y , 292 Silrilar and otner d i f f i c u l t i e s have limited the range of r e s u l t s obtained i n studies on the development of perturbations i n laminar plane Poiseuille flow @, 21, 2?] Nore d e t a i l e d are the conclusions of another analysis of t r a n s i t i o n from laminar t o turbulent flow (for a flow over a f l a t p l a t e ) ; proposals have a l s o been made for a d i r b c t numerical study of turbulent flows.
E l4
E l-J
k].
.
.
E l 231 .
G. Capriz
2 . F i n i t e d i f f e r e n c e approximations f o r t h e Uavier-Stokes equations
I n almost have r e s o r t e d t o equations. So we consideration t o
&
=
-
a l l t h e r e s e a r c i ~ e squoted i n Section 1 the authors f i n i t e d i f f e r e n c e analogues of t h e Navier-Stokes introduce now those equations r e s t r i c t i n g our t h e incompressible case
(grad Y ) . 'I&grad Y t
9
,
here % i s speed, $ i s t h e r a t i o of pressure over (constant) d e n s i t y , 3 kinematic v i s c o s i t y and g applied f o r c e per u n i t mass. To eqns ( l ) , (2) t h e appropriate boundary conditions must be added, perhaps on unknown boundaries (flows with a f r e e s u r f a c e ) . For t h e purposes of a numerical study, d i s c r e t c equivalents t o ( I ) , (2) and t h e boundary conditions can be used, which a r e based on a n e t of p o i n t s where t h e r e l e v a n t q u a n t i t i e s must be determined. The d i s c r e t c e q u i v a l e n t must have a form which suggests f e a s i b l e numerical alqorithms ; they must be s u f f i c i e n t l y accurate without leading t o cul. ,crsome computations and not be s u b j e c t t o numerical i n s t a b i l i t y . Although t h e requirements a r e numerous and s t r i n g e n t t h e r e i s a v a r i e t y of procedures t h a t meets them; t h e choice depends on a n o t w e l l defined c r i t e r i o n of economy. To o b t a i n convenient numerical algorithms t h e d i f f e r e n t i a l eqns ( I ) , (2) a r e not t h e b e s t s t a r t i n g p o i n t s , f o r a number of reasons; f i r s t of a l l one must t r y t o s e p a r a t e t h e unknowns% and 'S I f t h e boundary conditions do not involve Y , t h i s unknown can be eliminated a l t o g e t h e r from (1) using (2). I n f a c t , t h i s equation s t a t e s t h a t 3 i s solenoidal; hence it can be expressed a s t h e c u r l of a vector p o t e n t i a l y which i s i t s e l f s o l e n o i d a l ; a t t h e same t i m e y c a n be eliminated from (1) by taking t h e c u r l of both
G. Capriz
sides :
X = c u r l 4-
,
$=
,
curl 3 %=curl
at
+
c u r l (grad
2
y ,
, )
=
-
V curl
X
t curl
-g
;
(4
-
here account was taken of the identity: c u r l c u r l _v = grad div 1 2. Thus, using eqns ( 3 ) ( 4 ) the conservation of mass is exactly verified. I f one wants t o work i n terms of the variables 5 and 9 d i r e c t l y , one can s u b s t i t u t e ( 2 ) with a consequence of ( l ) ,i n whose derivation ( 2 ) plays a r61e. Here d i f f i c u l t i e s are net because the very important eqn ( 2 ) would thus intervene only i n d i r e c t l y ; i n practice one finds t h a t g r e a t care must be taken i n the computation i f the approximate values of %have t o correspond t o values of D which a r e s u f f i c i e n t l y small t o be accepted. In theory one could r e l y on the following consequence of (1)
AJ= d i v -g
- div
(grad
3 2.)
.
(5)
In practice one finds t h a t the use of d i s c r e t e equivalents of ( I ) , (5) leads t o rapid accumulation of e r r o r s and t o large values of D, a t l e a s t where the d i s c r e t i z a t i o n i s based on a r e l a t i v e l y coarse net. I t i s more convenient to s u b s t i t u t e (1) with the equation e t d i v
at
(y
@
y ) = - g r a d p - Y curl curl y
and (5) by t h i s consequence of ( 6 ) :
t -g
(6)
G. Capriz
This equation implics t h a t D ( t + A t ) vanishes though t h e " s t a r t i n g value" D ( t ) may be d i f f e r e n t from zero; by such a device e r r o r s introduced a t one s t a g e tend t o be reduced i n the next ( k ) . The form given t o t h e non-linear term i n ( 6 ) is more convenient than t h e form of t h e corresponding term i n (1) f o r our purposes. I n f a c t , onc aims a t trasforming t h e d i f f e r e n t i a l equation i n t o a d i f f e r e n c e equation (spacewise)through t h e following s t e p s : i ) i n t e g r a t e over a mesh-elenent V , transforming a l l volume i n t e g r a l s
containing space d e r i v a t i v e s i n t o s u r f a c e i n t e g r a l s over t h e boundary S of V. i i ) approximate surface i n t e g r a l s using only t h e values of t h e
functions a t t h e meshpoints. For s t e p (1) eqn ( 6 ) is d i r e c t l y f i t ( l ) ] ; 2rccisoi.y
irc
get
(n,
so, of course, a r e eqns
(3),
u n i t vector of t h e e x t c r i o r normal)
S i m i l a r l y fro^. ( 7 ) it follows
--(*)
The "penalty method" or the "mctilod of a r t i f i c i a l d e r i v a t i v e s " described by Professor Lions could a l s o have been used.
G. Capriz
We quote here a l s o t h e i n t e g r a t e d v e r s i o n s of eqns ( 3 ) , ( 4 )
&
dd v
t
Js
( 3 - grad &1x 2 dS
=
tiumerical quadratures must now be introduced t o approximate the i n t e g r a l s i n ( 8 ) , ( 9 ) o r ( l o ) , (11). For t h e sake of s i m p l i c i t y we consider h e r e only t h e case of a r e g u l a r cubic mesh. I t i s easy t o r e a l i z e then, (although we n o t e n t e r h e r e i n t o d e t a i l s ) t h a t , f o r the simplest and r e l a t i v e l y most p r e c i s e approximation of eqns (S),
...
( l l ) ,one n u s t introduce a cubic l a t t i c e with t h e following
condition: I f f o r i n s t a n c e i s supposed t o be known on one l a t t i c e p o i n t Po , then %must be known on t h e s i x n e a r e s t p o i n t s P1... 6' and convcrscly. A s a vcry s i n ~ p l eexample, consider t h e second eqn (10) : we have (h, mcsh-size) 6
Then, t h e s t r u c t u r e of t h e system (8) , ( 9 ) and (lo), (11) i s such t h a t , t o achieve b e s t approximations, it i s convenient t o take V successively as coinciZcnt with different b u t overlapping cubic
G. Capriz
c e l l s . For instance, with referencc t o eqnc (101, (11) notice t h a t must be known i n the centres of the faccs of the f i r s t c e l l V1 whereasumust be known a t the centre of t!le c e l l i t s e l f . The c e l l s of type V2 must be such t h a t + i s known a t the ccntres of the faces, whereas f i s known a t the centre of the c e l l i t s e l f , and so on. The procedure can thus be arganizcd so t h a t no interpolation i s required except f o r the approxinate cxpression of the non-linear terns. We must introduce now a d i s c r e t i z a t i o n i n the :ime variable. Leaving without a superscript the values a t the end \ of the k-th time s t e p and using the superscript k t i for values a t the end tktl of the (kt1)-th step, tile simplest f i n i t e difference approximations t o (8), (9) are
Y
-
6(,1 -n .
grad
rg
t
[div @
-s]
dS
+
These approximations a r c very rough, but have the g r e a t advantage of leading t o exnLicit formulae. 'Ilie care taken i n writing the condition which implies conservation of mass and a l s o the special form given t o the term measuring the diffusion of momentum i s j u s t i f i e d now: it allows the acceptance of the rough formulae above. Leaving aside f o r the moment the question of the boundary conditions, the process t o follow is t h i s . Assume t h a t the i n i t i a l values of 3 be given a t t=O. The corrcsponding d i s t r i b u t i o n of [P i s determined through eqn (13). This s t e p can be
G, Capriz acconlplished through one of the many methods available for the integration of Laplace equation, for instance through an i t e r a t i v e overrelaxation procedure 121 Successively, the right-hand side
.
of eqn (12) is computed and new values of 2 are determined. The process is then repeated. Attention was confined so f a r t o time dependent flows. There i s i n t e r e s t of course a l s o i n the study of steady flows; f o r such a study some of the remarks s t i l l apply. The d i f f i c u l t i e s i n respect t o diffusion of d i l a t a t i o n cio not occur; we find instead problems of convergence i n the schcnes of successive approximation t h a t must be introCuceci t o deal with tile non-linear terms.
-
3. Boundary conditions
ijcundary concitions for the approximate analysis of our problems must not be l i g h t l y stated. For instance: i s the usual condition of no s l i p a t a wall always justifiecl? Only a reference t o physical circumstances allows one t o give a s a t i s f a c t o r y answer t o t h i s question. Whether or not slippage i s t o be alloweci depends upon the thic1:ncss of the boundary layer t h a t one would expect t o develop i n the t r u e fluid. I f t k i s is much l e s s than the ~ l i ~ e n s i o nofs a l a t t i c e c e l l and one i s not interested i n the u e t a i l of tile bounaary flow t!len a f r e e s l i p condition i s appropriate; i f the boundary layer i s much larger than one c c l l , then a no-slip condition i s required. For i n t e r r ~ c u i a t ecases, the proper condition t o use depends upon the exact circumstances, and i n some cases i t i s appropriate t o t r y both ways and compare the r e s u l t s . Another point one must emphasize: sometimes it is convenient for computational purposes t o introduce f i c t i t i a s mesh-points ogtside the boundary. I f such a device i s used, one must be sure t h a t the f i n i t e difference approximation t o D vanishes a l s o a t the exterior
G. Capriz
f i c t i t i o u s . c e l l s so t h a t no diffusion o£ D inside the boundary occurs. A l l these warnings a r e of course of an experimental
character and a r c connected not t o any inadequacy i n principle of the f i n i t e difference approximations, but rather t o the need t o operate with a r e l a t i v e l y small number of c e l l s . Even more d e l i c a t e is the question of writing adequate approximations t o conditions a t a free surface 1 Over 1 conditions on s t r e s s components must be s t a t e d ; for instance i f the externally applied s t r e s s is a pressure Ya we should use, i n a system of cartesian coordinates, the conditions
.
-
where N i s the exterior normal t o and T ( I ), T ( * ) a r e two orthogonal tangential vectors. These conditions arc very d i f f i c u l t t o s e t up s a t i s f a c t o r i l y on a computer and workers i n the f i e l d have resorted t o conditions such as D=O, 49 = (f t o balance equations a and unknowns. The f i r s t choice i s j u s t i f i e d on the grounds t h a t the gravest source of e r r o r s i s diffusion of d i l a t a t i o n D throuqh the boundaries. The second choice i s notivatcd by the renark t h a t often viscous e f f e c t s a r e small when compared with a d i r c c t l b imposed s t r e s s . On the other hand the l o c a l orientation of the surface can be usually determined only very roughly, so t h a t a more precise use of eqns ( 1 4 ) i s not j u s t i f i e d . I t remains t o follow the changes of the free surfacc w i t h time. This is accomplished by introducing marker p a r t i c l e s on the f r e e surface (actually i n the marker-and-cell method the rnarker
-
p a r t i c l e s a r e distributed throughout the f l u i d , though, f o r analytical
G. Capriz
purposes, they are e s s e n t i a l only a t the boundary). The speed of the p a r t i c l e s is determined by interpolation or extrapolation from nearest mesh points; f i n a l l y t h e i r movement i s followed s t e p by step. A p r a c t i c a l procedure is a s follows. One builds up i n the computer a picture of the f l u i d s e t i n a wider f i e l d of c e l l s where the f r e e surface can impinge. There are markers t o show which c e l l s are occupied ( a t l e a s t i n p a r t ) and which a r e free. Pressure and velocity f i e l d s a r e determined over a l l the occupied c e l l s , boundary conditions intervening i n the boundary c e l l s . To avoid ambiguities ( i . e . a wrong labeling of i n t e r n a l c e l l s a s empty) a t l e a s t four marker p a r t i c l e s for c e l l are d i s t r i b u t e d a t time t = O i n a l l occupied c e l l s with further provisions f o r exceptional cases.
G, Capriz
4. 14umerical i n s t a b i l i t y ;
accuracy
Phenomena of numerical i n s t a b i l i t y have been mentioned already; i t is well known t h a t t h e i r onset depends c r i t i c a l l y on meshsize, and time-step size. It i s a l s o comon experience t h a t e x p l i c i t algorithms such a s t h a t embodied i n formulae (12), (13) a r e usually nore prone t o the disease than more complex implicit algorithms. I f we reduce eqns (12), (13) t o a non-dimensional form by introducing a typical velocity U a typical body force per u n i t mass G , thc time s t e p 5 and the meshsize h (assuming for simplicity t h a t the l a t t i c e is cubic though phenometia of ins t a b i l i t y may be yucnchcd sometimes by introducing meshes with appropriatc side-ratio 1311 ) we see t h a t the solution of the i i n i t c difference equations Jepends locally on the following parameters
For those who a r e physically inclined we remark t h a t d can be 1 construed as a Strouhal n u d e r of the flow based on the numerical time s t e p and meshsize. ~ i r c i l a r l ? d and d can be conlbined with d l t o express nufi!erical Froude and Reynolds nunbers
Conditioils of numerical s t a b i l i t y can then be expressed throug!~ limitatiorls on rl , Fll , Ri! The choice of the ralcvant values of Nand C will depend on the problem i n hand, of course. ' can be taken bs the In the study of flows with a free surface U
.
G. Capriz
speed of surface waves: using shallow f l u i d theory %=
( 'i;t a n h k H )
1 Z
,
, wave
number; H, depth of f l u i d . I n t h e experiments of ilarlow and c o l l a b o r a t o r s t y p i c a l s t a b i l i t y conditions were found t o be, experimentally, [g, p. 28)
k
I n o t h e r cases t h e l o c a l v e l o c i t y intervenes
[i, p. 1371
i n s t u d i e s on t h e behaviour of a p e r t u r b a t i o n i n a steady flow t h e excess speed due t o t h e p e r t u r b a t i o n seems t o have relevance. I n a l l cases it was found by experiment o r was suggestcd by heuris t i c arguments t h a t RN must be of ordcr of unity i f i n s t a b i l i t y has t o be avoided. Although t h e value o f u t h a t must be used i s n o t known e x a c t l y i n advance, rough evaluations a r e u s u a l l y possible. Then t h e condition j u s t mentioned can give an i d e a of t h e s i z e of t h e problem i n hand from a computational p o i n t of view. I f conditions such a s (15) a r e s a t i s f i e d t h e r e s u l t s of a computation a r e l i k e l y t o look reasonable, i . e . not wildly wrong, but they may s t i l l be f a r from accurate. I t would be nice t o have some t e s t s f o r accuracy. A check on tile value of D must always be kept with automatic s t o p when D reaches an unacceptaL-lc l e v e l . I f t h e condition of incompressibility i s s a t i s f a c t o r i l y appro::inated t h e measure of t h e domain occupied by t h e f l u i d ( a s shown by nar1,;er p a r t i c l e s ) must be constant. I n t h e marker-and-cell method a chccl; can t h e r e f o r e be nade by comparing t h e a w ~ b e rof cell:; Ij sontaining a t l e a s t one p a r t i c l e with t h e n u ~ b e rof boundary c e l l s (since these c e l l s a r e constantly c o n t r o l l e d i n a program, t h e check i s simple).
G: Capriz
The experimental value of 5 can be compared with t h e o r e t i c a l estimates One such estimate f o r plane problems is
pq -
where A i s the constant area of a cross-section of the region occupied by the f l u i d , P is the length of the boundary of the crossthe r a t i o of p a r t i c l e spacing t o c e l l size. section and Further checks a r e sometimes made on the basis of evaluations of t ~ t a ki i n e t i c energy.
3. rlumerical analysis of hydrodynamic s t a b i l i t y of steady flows_, I mentioned already t h a t a good deal of research e f f o r t i s
applied t o the numerical study of s t a b i l i t y of c e r t a i n c l a s s i c a l flows: the Poiseuille flow, the Couette flow, the flow over a f l a t p l a t e , etc. In these cases the boundaries are fixed and one can make use conveniently of eqn. ( 3 ) , ( 4 ) ; the time-independent functions describing a fundamental flow a r e supposed t o be known:
w
-
-
A) N
-
= curl
at
-9
,
-
t curl [ g a d
= curl
-
N
N
'U 7 t o t
,
5)
X- - -X0 ' - 4; 4- do, V
and d e t a i l s on the behaviour of perturbations X s N a r e required. 2 :3 The equations a r e :
&
V
q= - -9
~ = c u r l y r ,
-
t grad U.
4.
curl
X
G. Capriz &
Eecause t h e f i e l d of
can be taken t o be solenoidal, t h i s
-
equation can be w r i t t e n i n t e r n of
aaY - c u r l a t
Lgrad c u r l
-
*(
3+
only
curl
ry
-
) t grad t e e c u r l
y]= N
-
I t i s convenient t o w r i t e inunediately t h i s equation a l s o i n a non-dimensional form using a t y p i c a l v e l o c i t y U and a t y p i c a l dimension of the domain L , introducing t h e notation
and a physical Reynolds n u d e r
and presuming now t h a t t h e operators non-dinensional space v a r i a b l e s
-h A I*-
az
R curl [grad curl
and c u r l a c t over
A
P .(? + c u r l I*)9
t grad s
-
curl
r'] -
Usually one wants t o know t h e s o l u t i o n of eqn (16) f o r a s u f f i c i e n t l y ample i n t e r v a l of time and over a domain f o r the space v a r i a b l e s which i s not bounded, though sometimes t h e expected phenomenon is periodic i n one o r more space v a r i a b l e s and a reduction t o a bounded domain ensues. $
On the p a r t of t h e boundary t h a t represents walls (fixed or i n steady notion) ??. = 0 ; o f t e n one can conclude from t h i s t h a t a l l components of \y and t h e i r normal d e r i v a t i v e s vanish. N
There may be conditions a t i n f i n i t y and, on o t h e r p a r t s of
=
G. Capriz
the boundary, periodicity conditions may apply; besides the i n i t i a l conditions must be known. When one is interested i n the decay of an istantaneaus disturbance or i n the s p a d i n g of s e l f amplified perturbation, these a r e the only conditions t h a t apply. In other cases perturbations may be Continuously fed from outside; then y and derivatives a r e assigned on portions of the boundary as known functions of time Because the choice ?' k' = 0 corresponds t o the fundamental solutions of eqn (16) the i n t e r e s t centres a t f i r s t on the small perturbations. Although a precise statement can be made only i n one special case, i t is generally presuned t h a t the behaviour of a perturbation of small amplitude can be q u a l i t a t i v e l y decided on the basis of the linearized equation
-
.
' r I*) 3
-
R c u r l [grad c u r l
1.
-
5 + grad?
curl
I*]=
We come thus t o a rather complex linear diffusion problem; i n the * has only one non-vanishing component eqn plane case, where ( 3 ) has been the object of many c l a s s i c a l studies, f o r instance those r e l a t e d t o the s t a b i l i t y of Poiseuille flo~v,o r the flow on a f l a t plate. Because the c o e f f i c i e n t s of eqn ( 1 7 ) a r e independent of time, the solutions can be written as l i n e a r combinations of functions of the type
Y
where A i s a function of the space variables only and 1; i s a complex constant. The equation t h a t follows f o r _A , from (17),
G. Capriz
kA&
-R
c u r l [grad c u r l
&
. 5 t grad 5 . c u r l ;]
=
and the associated boundary conditions add t o an eigenvalue I t i s essential problem depending on the positive parameter Rc t o decide which is the infimum Rc of the s e t of values of R f o r which one eigenvalue K(R) a t l e a s t has a positive r e a l part. I n t h i s f i e l d the early work of Thomas must be quoted [ g ] I t happens sometimes t h a t the value of k corresponding t o Rc vanishes; t h i s analytical f a c t i s r e l a t e d t o the physical existence of non-trivial steady flows. I n such cases the eigenvaluc problem ( l C ) i s further simplified. Cesides, a search for non-trivial solutions of the non l i n e a r problen:
.
.
Ah
*
- c u r l [grad c u r l Y * ( +~ c u r l y*) + - + grad 2 . c u r l \Y (19)
J =
R
*
J
e
with the associated boundary conditions, can be attempted. For a special case of t h i s problem we have d e f i n i t e r e s u l t s due n t o Velte, Kirchgassner and others research vorkcrs a t Freiburg c36, 37 3 8 1 The special case i s examiiled i n some d e t a i l l a t e r . Mention nust be b r i e f l y nade here of 'the numerical techniques used t o tuckle eqns (16) , (2); (g), (2)with the associated boundary conditions. A t r i v i a l e x p l i c i t method can be used i n connection with eqn ( 1 6 ) , ( 1 7 ) ; but more often, t o lessen phenomena of numerical i n s t a b i l i t y , it is more convenient t o evaluate the term under the biharmonic operator as the average of tile values a t t i m e s t a n d '2 t h Z , mantaining f o r the other terms the evalutation a t time
- .
I f such technique is adopted a matrix representing the
G. Capriz
d i s c r e t e equivalent of a l i n e a r combination of the operators A A and 4 must be inverted. Even when use i s made of an e x p l i c i t method a matrix inversion (although simpler) i s required. Techniques of d i r e c t inversion or i t e r a t i v e methods must t e called for. Direct inversion though cumbersone may bc a t t r a c t i v e because it i s needed only once for a l l time steps. The economy of the procedure i s much enhanced i n cases where the solution i s periodic i n one or more space variables because tile matrices involved are then c i r c u l a n t i n submatrices which may even be c i r c u l a n t i n t h e i r turn. Formulae f o r the inversion of c i r c u l a n t or block c i r c u l a n t matrices a r e quoted i n the next section [39, 403. For the solution of the problem ( 1 9 ) with the associated boundary conditions an i t e r a t i v e procedure is always called f o r , t o deal with the non-linear terms. Starting with a reasonable guess, one can make use of tlie d i s c r e t e equivalent of the i t e r a t i o n
A A Y * ( ~ )-= R
-
t grad
2
' (k-l).
curl
(5+ c u r l y (k-11
-
curl y s
-
Here again, i f the boundary conditions express periodicity a t l e a s t i n one variable, techniques of inversion of c i r c u l a n t matrices may be of use. Both i n the analysis of the time-dependent case and during the i t e r a t i o n (20) phenomena of numerical i n s t a b i l i t y may occur. A word of warning is liecessary here; a mild form of numerical i n s t a b i l i t y i n diffusion problems may be wrongly taken sometimes as indicative of hydrodynamic i n s t a b i l i t y . The study of the same problem with two d i f f e r e n t meshsizes (one rectangular and one square for instance, i n the plane case) is recommended. "Nmerical" eddies change then wavelength so as t o cover the same number of c e l l s (the typical wavelength of "numerical" eddies i s ten c e l l s ) .
We have mentioned tliat i n t:ic n m e r i c a l solution of our problems under pericdicity conditions bloc]:-circulant natriccr; appcar. To show t h a t sirnple deviccs can savc a t t i n e s a l o t of k~orl;, thc property of these matrices i s rccallcd hcrc, t h a t a l l o ~ ~ans cacicr inversion. Let ii0 f&-]-
n
=
I
no , nl ,
...nn-1 ) =
. . . . ?-.-.., . ... .
-*io
.
L J , , ~
L
nl
A '. *.. .Ao be a block-circulant matrix, wilcrc the Ai arc blocl;s of ordcr n. Let I be the i d e n t i t y matrix of ordcr n and $, bl , .bn-l the m-th roots of unity and put I.
.
..
..
v
=
Then I
v-l =
El
lo 1
and Y'AV=A =
f
:.-I
#(do)
. . a, I . ... . 1.
& "-1
m-lI. 0
0
. *bm-l1
...
0
:. :
#(
.
.,
;
b-m-l
C. Capriz where
I£ the matrices
(4
)
a r e not singular a l s o A
is son singular
and A-l= % A-'
"A
-1
.
is a l s o block-circulant; precisely
Therefore the inversion of the matrix A of ordcr man is reduced t o the invcrsion of n rnatrices of orclcr n I f , besides, the matrix A i s block-syrinetric (A1 = $-1 , A2 = , ) , then we need i n v e r t only [n/2] ~ a t r i c c s ;the nuccessivc algebraic manipulations a r e also simpler then because the inverse matrix i s a l s o bloc]:-synmetric.
.
...
7. A simple analytical scheme f o r the study of the s t a b i l i t y of Couette flow. To i l l u s t r a t e with one example the analytical and numerical problems t h a t a r i s e i n the study of thc s t a b i l i t y 02 a steady flow, wc examine now i n some d e t a i l the behaviour of perturbations introduced i n the circumferential flow between two concentric coaxial r o t a t i n g cylinders (Couette flow). An analysis of t h i s problem has i n t e r e s t for many reasons: (i) the Couette flow i s one of the very few steady flows of a viscous fluid for which one has a precise a n a l y t i c a l description. (ii) the s t a b i l i t y of the Couette flow can be studied i n the laboratory
through r e l a t i v e l y sir:ple cxperirients. There have Lcen precise
G, Capriz
experimental studies of Taylor, Conelly and more recently of coles 42. tlie Couette flow is subject t o a form of hydrodynamic insta(iii) b i l i t y t h a t lends i t s e l f t o an analytical treatment, through l i n c a r i s a t i o n of tlic perturbation equation, w i t i i forecasts anply con£irned Ly experiments tile special kype of hydrodynamic i n s t a b i l i t y lends i t s e l f (iv) t o a rigorous a n a l y t i c a l treatment also through a study of ti.e corilplete non-linear equations [37] I.,e fu1ldar:cntal reason f ~ (r i i ) ,(iii),( i v ) i s tilt f a c t t h a t i n s t a l i l i t - i s iri r.0s.l; cases due t o t r a n s i t i o n t o other forms of steady flor, (Taylor vortices for which a x i a l symmetry s t i l l holds or Coles vavy v o r t i c e s ) rathcr than t o t r a n s i t i o n towards turbulence (a:, ilapycns inctcac for Poiseuille fiow). Apart fror, i t s analytical-el:pcrimental i n t e r e s t , tlie Taylor vortcs £10;: ;;as ir;i~ortai~cei n prncticc a t l o a s t f o r tvo rcasons: the flcbi i i ~plair, Learirigs of larcjc r o t o r s (turLines, a l t e r n a t o r s ) i s roug,~l;. a Cosettc floi,, suLject a t high spceds t o Taylor ins t a l i l i t j r ; billci~Prylor vortices appear tlie viscous losses i n the lubricarlt Lecom c!~ciliiiglmr than is forecast on the assumption of Couctte flov; 1,encc the i n t e r e s t of a precisc understanding of t h e Taylor i n s t a L i l i t y . Therc have been also a t t e a p t s t o use the Taylor vorticcs a s s e a l s i l l Lcaringr;. Secondlj, boundary layers along concave rralls a r c subject t o i n s t a b i l i t y of a similar kind, t h a t can I*! s t u i i e i Ly s i ~ ! : i l a rmeans (Gdrtler vortices) The s t a r t i n g point f o r a nunerical anal-rjis of tile ( a x i s y m e t r i c ) Taylor vortices a r e equation (16) anci tilc appropriate boundary coniitions. iraturally tile p e c u l i a r i t i e s of tile problem allow C binplizication; 2 i s i n the circunferential direction, y i s assumed not t o iepcnd on 6 but only on t h c r a d i a l and G i a l coordinates r and z. i:e can use the gap r Z - rl between cvli?ler::
k, 21 .
[sf441 .
.
-
-
(4 .
G. Capriz
a s typical dinension and put =
(r2 - rl) = v t / (r2 - rllL.
Z/
5
rl) / (r2 - rl) , together
= (r
-
CI
with
T
Then the r a d i a l and a x i a l components of spccc: are r e l a t e d t o tile derivatives of the transverse cornpoilent of )Y and it would Le uneconomical t o introduce the other two compo~entsof y*simplyt o dg f i n e the transverse component of speed ; t h e simplest sc;leme der ivcs P from the use of second component y of y and the transverse conpollent of speeu. Reasons of simplicity connectcc! ~ i i t i ispecial features of the ,problem (such a s tile a x i a l ayzmetry) sucjcjcst a s l i g h t modification of the usual formulae and the adoption of t!ie following ones, which a r e s e l f explanatory , fi2angular speeds of i n t e r n a l and external cylinder respectively , 1 = /
*
-
(al
n2 nl)
In writing the two scalar equations a formal complication follows from the use of cylinqrical coordinates. To make t h i s exposition a s simple a s possible reference is made here only t o an asymptotic oase: t h a t of small clearance (r2 rl) / r 2 & 1 The equations valid i n t h a t case a r e
-
(see
2
f o r the derivation); here T i s a mean R e y n o l d s ' n d e r
.
G. Capriz
The boundary conditions express: vanishing of mean a x i a l flow, periodicity i n the a x i a l d i r e c t i o n (with a r b i t r a r y period 2 q), vanishing of the perturbation on tile cylinders
A f u r t i ~ c rsimplification can be obtained i n (21) by choosing t o
put 2 = 1 ; it may seem t h a t the s i r p i i f i c a t i o n denies physical significance t o the resulting prohlcn. In f a c t i t i s found t h a t the problem i s i n t e r e s t i n g and c e r t a i n conscquenccs (such as tile c r i t i c a l Taylor number) deduced i n the special case can bc applied with good approx'ination a l s o for 1 i n the closed i n t e r v a l ( 0 , l ) . \Je consider then i n the following sections t h i s problem: find i n the rectangle (06 Q 1 , - q i j 5 q , t 3 0 ) a v c c t d r ( Y / , V ) s a t i s f y i n g the equations
5
the boundary conditions (22) and given i n i t i a l conditions.
C. Capriz 3. -S . o ~ er e s u l t s regarding thc d i f f e r e n t i a l problem. 'ule Lcgin the study of our example with an analysis of
properties of Lounhry and i n i t i a l value problems related t o p r o l l e r ~(22), (23). To begin with, it is convenient t o consider from a partly formal point of view tile solution of the l i n e a r system
5 5
with the boundary conditions (22). Obviously i f ( , ) , Iv ( 5 ) i s n solution of this problem so i s a l s o )Ir ( J t b) t o be odd and V Y ( , t L) (b, any constant) ; we require of t o be even i n Then we separate variables, looking f o r solutions Vn of tlle type
5, 5 3
5,
ynl
The functions An
.
, Bn
s a t i s f y the system of equations (an = n ~ / q )
and the boundary conditions
Supposing t h a t an eigenvalue (
,)
A
Tn
and a corresponding eigenvector
e x i s t f o r problem (25). (26) , then multiplying both
s i d e s of the f i r s t eqn (25) by the complex conjugate A* An of
A
G. Capriz
and i n t e g r a t i n g over ( 0 , l ) one g e t s
where
Similarly from the second eqn (25) one g e t s
with
Cecause an i s a r e a l p o s i t i v e number, such a r e a l s o I1 , ana I2 ; these two numbers vanish only on t h e t r i v i a l s o l u t i o n of (25), (26). lience I = I i s r e a l negative (from eqn ( 2 8 ) ) , and t h e eigenvalue A i s r e a l p o s i t i v e (from eqn ( 2 7 ) ) . Tn As a consequence tihe associated cigenvector can be taken t o have r e a l components. From t h c formulae above it follows t h a t , i f A do not vanish, An , En
On t h e other hand
G.
Capriz
and f i n a l l y
The equation, which the eigenvalucs s a t i s f y , i s e a s i l y found. A A IIotice t h a t An and Cn a r e both s o l u t i o n of Lke equation i n [
d2
~ ~ d 5
4
-
2 a
2
, + )a n T n ] y = O .
A
IIence, Loth An ancl En can be expressed a s l i n e a r combinations of functions e with
zc
.
.
l e t u s i n d i c a t e with z1 -zl , z2 -z2 , z3 , -zg t h e s i x d i s t i n c t dctcrmination; of z [ ~ o t i c et h a t i n e q u a l i t y (29) excludes t h e occurrcnce of multiple r o o t s ] By imposing t h e boundary conditions t h e equation f o r t h e
.
eigenvalues can be found. I t i s expressed by p u t t i n g equal t o zero t h e determinant of a 6x6 matrix whose f i r s t t h r e e l i n e s a r e
G. Capriz
anci the o t h e r tlilrcc are formed
witiil
thc sane co1ur:ns i n thc
ordcr 2, 1, 4, 3, G , 5. I t can be cliccked Elat Cic determinant A i s equal t o the differencc of the squarcs of two sums S1 and S2 , where S1 is tile sum of the determinants of thc matrices of order 3 obtained by extracting the columns 1, 3, 5; 2, 3, 6; 2, 4 , 5; 1, 4 , 6 of the matrix (30) and S2 by a similar sum t h e r e the columns 1, 3, 6; 1, 4, 5; 2, 3, 5; 2, 4 , 6 a r c involved. Easy developments lead t o tile r e s u l t
from which an i n p l i c i t multivalued function
A
A
Tn = Tn (an) can be
4 computed. On one Lranch of t h i s function tklc r e l a t i o n T +_ zit n - an i s s a t i s f i e d (because then the functions e , i = 1,2,3 a r e not independent), but t h a t Lranch i s w i t l ~ o u ti n t e r e s t f o r h
conputing eigenvalues i n view of A graphical rcpresentation A .. ti12 valuc of 'I corrzspo~dingt o n. and next lowest a r e shown i n tile
-
inequality (29). of tlic two branches over which a given of an i s the lowest figure.
G. Capriz
m e numerical experiments then show t h a t t h e r e a r e r e a l p o s i t i v e eigenvalues of our problem and t h a t appropriate values of q can be found such t h a t t o an = x / q t h e r e corresponds an eigenvalue A T1 which is not an eigenvalue when a is chosen equal t o j n / q (j = 2, 3,...). To such T t h e r e corresponds then only one A A eigenvector of t h e type sought i n t h i s Section: T1 { T~ ( j = 2 ,3.. .I These r e s u l t s , based here simply on numerical evidence, can be reached without recourse t o experiments [sce 361 , througl~reference t o p r o p e r t i e s of s o l u t i o n s of a v a r i a t i o n a l problem based on t h e e q u a l i t y
-
.
G. Capriz
which follows from ( 2 4 ) and t h e boundary conditions ( 2 2 ) .
We r e c a l l now some r e s u l t s regarding the l i n e a r d i f f u s i o n problem
with t h e boundary conditions ( 2 2 ) . Ile do n o t quote here p r o p e r t i e s of s p e c i a l s o l u t i o n s corresponding t o p a r t i c u l a r i n i t i a l d a t a b u t r a t h e r s t a t e t h e general behaviour of s o l u t i o n s depending on T Remark t h a t the general s o l u t i o n of ( 3 2 ) , ( 2 2 ) i s a l i n e a r combination of functions of thc type y =c (f ) J t V=c V1 s a t i s f y Ul ) where y i s a r e a l nuculcr and )V the d i f f e r e n t i a l system
(1,s
with the usual boundary conditions.
*'P1 ,s .
.
,
.
G. Capriz
By reference t o t h i s eigenvalue problem it i s possible t o prove s e e b t P p . l ~ b - l l i J that: h
( i ) when T is smaller than the smallest eigenvalue T1 of the problem of Sect. 8, # i s necessarily negative; h
A
i s i n an appropriate i n t e r v a l T1 , T~ t6(6 > o) there are solutions of ( 2 2 ) , ( 3 2 ) exponentially increasing with time.
(ii)when
T
G. Capriz
9. The non-linear problem.
We consider here t h e non-linear problem, whose s o l u t i o n represents t h e Taylor v o r t i c e s within our approximation :
with tile boundary conditions ( 2 2 ) . We give p r e c i s e sense t o t h i s problem by s t a t i n g t h e ~ c t where we seek a n o n - t r i v i a l solution: i t i s a s u b s e t a o f a Sanach space E of v e c t o r s (ly ,V ) obtained thus. Consider t h e s e t of functions 'f ( I 5 ) cleflined i n a s t r i p S1 l a r g e r than t h e s t r i p S :06 $ 1, of c l a s s C" i n S1 , periodic i n f with period 2q; introduce i n tile norm
5
3,
5
Y m
and l e t in
-
-
5
3
be t h e closure of with reference t o t h i s norm. Then E-is t h e Banach space of v e c t o r s ( ) V I 'V) with ?y , v i n y l and t h e norm
We consider a l s o t h e s e t of a l l functions f ( 5 , y ) , C m i n S , p e r i o d i c i n S w i t h p e r i d 2q, which vanish i n a s t r i p along t h e boundary of G. The space obtained by closure of t h e s e t with
G. Capriz
yIIE,
.
1:
11 will bc indicated with reference t o t h e norm El' Then i s t h e s e t of v e c t o r s (Y),V ) of C :iitii y 6 H2 e
and V t iil ; i n f a c t vectors such t h a t y 6 V L H ~s a t i s f y , i n a generalized scnsc, t h e boundary conditions at = 0, = 1. \re w i l l look thcn f o r s o l u t i o n s of our 2roblcrn (33) i n
G2,
7
3
I t i s p o s s i b l c t o silov f i r s t of a l l t h a t t l ~ e r ca r e no
e::cept
t h e t r i v i a l one, f o r
<
n
. 7 solutioils
T1 ( f o r a proof, scc 9 , p p .
59-60). I t i s p o s s i b l e t o ailow f u r t h e r [37 - , pp. 4-51 ; s e e a l s o f o r some 1 I c o ~ n e n t sthe rlZIIrcss [46] - t h a t t h e t r a n s formation (y ,v ) 4( y ,v ) T
defined throug!l tile ilon l i n e a r pro:~lcm i n
a
Y'
,If:
~ ~ l = r a? ( ~ ~ , .v ) _ 2 5
and Ly requiring ti:at
+&y.5,
( y , V ) Be
I
(y , V'
i n E , ar.d
I
) !JC i n
2
i s a complctc f u n c t i c n a l t r a n s f o r n a t i o n of tilt s: acc C i n t o i t s c l f . Its fixed p o i n t r a r c t h c s o l u t i o n s of our probler?, I I Siriilary t h c t r a n s f o r r a t i o n d : v)+ , v- ) Bef ineci Elrourjh t h e liilcar problem
(y
.
(y
= -3 Y
Azr
13
and by requiring t h a t ( :V; he i n S, and
I
(y
I
,P
;,e i n
(k-
is
a l s o a c o ~ p l c t cl i n e a r t r a n s f o r ~ a t i o nof L i n t o i t s e l f . Tile fixeC p o i n t s a r c t h c cigcnfunctions of the proj;lcr: of s c c t . 2. -. ;ioT.r i t can l..c proved tililt tbc FrCclict i i f f c r c n t i o l of
his
tile t r m s : o r r - . ~ t i o ~ hi a t t h e p o i n t ( 0 . 0 ) of npocc 3. A l l ve Iinvc s a i d rcr-ains t r u c i f tre suLnLitutc t h c s p ~ c ci; with t h c r:ubsyacc R odd a n d t l i s cvcn A1 i s t h i s , tli':t
1
iil
of t k e v c c t ~ r s(Y,'Lr) of L r,uch t:;at
j. Thc
(as
Y is
advantarjc of considcrincj o c r problcr: i:i
rcrnilrkcG i n Scct. 3 ) t h c r c exist:
c::oiccs GT
c.;
G. Capriz
such t h a t t o t h e associated cigenvalue
A
T1 t h e r e correspons only
one eigenvector: h T1 has m u l t i p l i c i t y 1. h Then, f o r a theoren of Leray-Shauder, Tl i s a branching p o i n t f o r t h e s o l u t i o n s of t h e problem ( 3 3 ) , (22): a n o n - t r i v i a l s o l u t i o n of our problem must e x i s t i f t h e value of 'I' i s chosen w i t h i n a s u f f i c i e n t l y small i n t e r v a l (
4
h
T1, TI t d l d)O
1.
10. liw,erical study of t h e non-lincar e l l i p t i c probleri The a n a l y t i c a l developmnts of S e c t s 2, 9 assure us of t h e existence of n o n - t r i v i a l steady s t a t e s o l u t i o n s of t h e non-linear N p e r t u r b a t i o n equations ( 2 2 ) , (32) f o r T ) P1, hence of a Sranching of t h e fundanental solution. Tiley allow us a l s o t o c a l c u l a t e approxirateiy t h e value of tile Taylor nupber tiiat c l l a r a c t e r i ~ c st h e t r a n s i t i o n . I n p r a c t i c e onc vould l i k e t o know t h c a ~ p l i t u d eof t h c A p e r t u r b a t i o n a s a function of T Lcyond t h e c r i t i c a l value T1 i;nov~lcJge of t h a t arplitufic lcadr; f o r i n s t a n c c t o an evaluation of
.
t h c couples a c t i n g on t h e r o t a t i n g c:?linCers, norc p r e c i s e l y of t h e excess of those couples beyond t h e valuc t h a t would be prcclictcd f o r Couctte flow. For such an evaluation a rccourse t o n u ~ c r i c a lrethods i s c s r ; e l t i a l . One can pursue e i t h e r the r.ur:,crical i n t e g r a t i o n of t h e stcady s t a t c equations ( 2 2 ) , ( 2 3 ) tilrougli a process of d i c c r c t i z a t i o n and rucccssivc approrir:ations / 101 o r a numcricai i n t e g r a t i o n of tllc d i f f u s i o n equations ( 2 2 ) , (23) u n t i l n s t a t c i s rcaci~cdsufficientl;. near the s t c a c y s t a t c [ 2-2 ] We rjive hcrc i i r s t of a l l soi:.c d e t a i l s of t h e f i r s t process
.
O F d i s c r e t i z a t i o n i n a s p e c i a l casc considcrinq thc f i n i t c -
diffcrciicc problcn:, which Ccrivcs from ( 2 2 ) , (33) f o r tile clioicc
C. Capriz q = 1, when t h e n e t points a r e chosen t o have coordinates r=mh,
j=?
(p
1
.... .
b v i t h h = . ( n t l ) -1
,K
=
....n, a t ?=
1, 2 ,
(i, p = 1, 2, r.tl The boundary c;,nditions f o r = 1 c a l l f o r the use of f i c t i t i o u s e x t e r n a l p o i n t s , idicrcas
5
t h e conditions a t S =
5 1 inply
p r o p e r t i e s of. the operating
natrices. P r e c i s e l y , t h e f i n i t e d i f f e r e n c e problem can be v r i t t c n a s follows
y
where , a r c two vectors vitlr i n 2 + 2n corzponents, each of wllich , v over tire mesh gives t h e approximate value of t h e functions p o i n t s ordered from l e f t t o r i g h t and f r o n t o p t o Lottor?. ; , Dl a r e block-circulant ant! bloclc-syrmetric n a t r i c c s of Ul , order 2n t 2 i n subnatrices of order n :
u2
J1
=(or
-1, 0,
. . . , 0,
I,],
with 0 n u l l matrix of order n ; I i d e n t i t y matrix of order n ; A, G , C syrmetric n a t r i c e s of order n, of which t h c f i r s t i s
pentadiagonal and t h e o t h e r two t r i d i a g o n a l :
G. Capriz
Ml,
M2 a r e non-linear operators a c t i n g t h e f i r s t on t h e vector
- I
Y
and the second on t h e cornpunci vector ( If 1. As i n t h e case of Lhe d i f f e r e n t i a l problem, it i s possible t o show t h a t t h e values of homogeneous systerr,
T
, for
which t h e associated l i n e a r
has n o n - t r i v i a l s o l u t i o n s , i - e - t h e values of T which a r e r o o t s of t h e a l g e b r a i c equation of degree 2n 2 t 2n
has s o l u t i o n s , a r e p o s i t i v e (we w i l l r e f e ~ t o these values a s t h e h eigenvalues of t h e problem). I n f a c t , i f , V i s a s o l u t i o n of (35) corresponding t o t h e eigenvalue T , then
-\Y -
u22 ) a r e p o o i i i v e d e f i n i t e q u a d r a t i c Poimc x . I t follows t h a t must be p o s i t i v e ; i n t h e components of -
but xT
U1 --x , - (5T
i t follo\rs a l s o t h a t
h
f,
P
A
-
'v can be talien t o have r e a l corponents.
Again a s i n t h e case of che d i f f e r e n t i a l problen i t i s founii t h a t nor. t r i v i a l s o l u t i o n s of ti:e non- l i n e a r problem ( 3 4 ) niay e x i s t only f o r values of T cjrcatcr than t h e lowest eigenvaluc
n
T
C
of ( 3 5 ) .
Po reach a proof of a c t u a l e x i s t e n c e of a s o l u t i o n unclcr t h e contiition T
>
h
Tc
,
some preliminary r e s u l t s a r e required.
F i r s t l y we remark t h a t eigensolutions of ( 2 5 ) can Le w r i t t e n a s folloyiis
irllcrc
k. = sin 1,
[ (2i-1)
r 4
],
,,(=_n_ 2nt2
'
l$r$n
;
G. Capriz
and
-)4 , ( 1 1
+
c+
(
a r c n-vectors which s a t i s f y t h e equations 2cos 2r d l3 2cos 2 r d I
+
2cos 4 r d
)g = -
I )
-
h s i n 2rA
3 = h T sin 2 r l c
'-f
~ 1 i r r . i n a t i n g one o b t a i n s t h e equation i n
(37)
.
-'P
with
'-r
=
- ( c + 2 cosrA
I ) ( A t 2 cos 2 r d L ; + 2 cos 4 ro( I ) .
hence s o l u t i o n s of our problem (35) e x i s t provided t h a t i14? sin'(2r.O the
coinciclcs with one of,cigenvalues of t i ~ cmatrix Cr- (r = 1,
one (A +
... n) .
elementary uevclopments show C47J --- t h a t t h e matrices $1 a
2 cos 2r 4 E t 2 cos
o t h e r hand t h c rcatrices
-
-'
~ ) ( r = 1,
... n)
a r c p o s i t i v e ; on tile
( C t 2 cos 2 r d I ) ( r = 1,
... n)
are
i r r c d u c i b l e , d i a g o n a l l y dominant n a t r i c e s with pos,itive i i a g o n a l c l c n e n t s and non-positive off-diagonal c l e n e n t ; s o t n a t t h e matrices
-
+ 2 cos 2 r 4
(C
iicnce t h e C
I ) - ~ arc positive.
r a r e positive; nore precisely C n ( Cn-l
I f vc of
-f
Cr
C ~ ~ O G fCo r
, we
T
< ... < C1 . 4
a value T s o t h a t h4
(39)
?
s i n 2 2r 4 i s an cigenvalue
can c o n s t r u c t t;irough (38) t h e corresponding eigenvector
and, s u c c e s s i v e l y , througii t h e second eqn (37) and formulae (36)
t h e ciqcnvcctor
( Y-/ , y )
of our o r i g i n a l problem ( 3 5 ) .
G. Capriz
Equivalentlywe could say t h a t t h e system
admits of s o l u t i o n s . 2 Thus, we can determine n eigenva!.ues of M ( i f each i s counted with t h e appr@& m u l t i p l i c i t y ) . Actually it can be checked t h a t t o each eigenvalue s o determined t h e r e correspond two eigenvectors ( , 1 ; t h e f i r s t i s of t h e foyn ( 3 6 ) t h e second has a s i m i l a r s t r u c t u r e but t h e r 6 l e of t h e trigonometric functions i n t h e d e f i n i n g formulae i s reversed. 2 We have accounted s o f a r f o r 2n eigenvalues of Pl ; t h e remaining 2n a r e zero; i n f a c t M i s a s i n g u l a r matrix. Let us consider now t h e s p e c t r a l r a d i u s ( C r ) of C r ; a theorem of Perron-Frobenius and t h e i n e q u a l i t i e s ( 3 9 ) assure us
,
-
1
...
4 that (Cr) < 'j (Cn-l)< simple eigenvalue of C1 s o t h a t
p
C2,
. . . Cn.
'j
7
(C1) (C1)
t h a t 'j (C1) i s a i s n o t an cigenvalue of
, and
I n conclusion
is t h e minimum value of T f o r which t:.e problem (35) has solution. To t h i s value of T t h e r e corresponds a unique s o l u t i o n ( a p a r t from a constant f a c t o r ) of t h e type
whereyl
,-
a r e n(nt1)-vectors. Consider now t h e vector space Ii of t h e 4n(n+l)-vectors ( $' fl ) wit11 ) , of t h e type ( 4 1 ) ; l e t E be norned (, I f Lie choose any v e c t o r ( l/) y 1 i n D and c a l c u l a t e t h e v e c t o r s
I
- Y
-1
.
G. Capriz
.
these belong a l s o t o B Iience we can consider t h e eigenvectors of (35) and t h e n o n - - t r i v i a l s o l u t i o n s of (34) r e s p e c t i v e l y a s fixed p o i n t s of t h e following compact nappings of E i n t o i t s e l f
and
I
IJotr 2 i s thn Fr6chet d i f f e r e n t i a l of c a l c u l a t e d over t h e n u l l clement of B I f we see'- s o l u t i o n of our problems ( 3 4 ) , (35) A exclusively within I3 then we f i n d t h a t f o r T = Tc t h e r e corzesponds a simple eigcnvaluc of (35) A theorem of Leray Schauder assures u s f t!len,of t h e existence of a n o n - t r i v i a l s o l u t i o n of (34) f o r each choice of T i n an appropriate A 4 i n t e r v a l ( T ? td ) , d ) C ; i n other vords cf C - c is a branching p o i n t f o r thc s o l u t i c n s of ( 3 4 ) .
.
.
G. Capriz
11. Notes on t h e numerical experiments
We have d e a l t s o f a r with fundamental questions r e l a t e d t o t h e system (34) ; we comment now b r i e f l y on problems connected w i t h t h e planning of a c t u a l numerical experiments. Because of t h e non-linear nature of system ( 3 4 ) , i t s p r a c t i c a l s o l u t i o n c a l k f o r an i t e r a t i v e procedure of t h e Y;pe envisaged i n eqn (20) ; f o r s i m p l i c i t y we nake r e f e r e n c t h9re t o t h e scheme
although t h e a l t e r n a t i v e scheme
seems t o be f a s t e r . I t i s found t h a t , i f t h e s t e p 11 i s chosen t o be small enough, t h e v e c t o r (Y(") y ( n ' ) tends with increasing n t o tile n u l l vector when T C , whereas i t converges toiiards t h e n o n - t r i v i a l s o l u t i o n of (34) when T > Tc. A lJotice t h a t t h i s happens al.though f o r T > Tc , eqn ( 3 4 )
-%
.
A
admits always a t r i v i a l s o l u t i o n . A p r e c i s e a n a l y s i s of t h i s Lehaviour i s n o t a v a i l a b l e ; we can add hcrc only a h c u r i s t i c argument which i n d i c a t e s a bound on h f o r t h e s t a b i l i t y of t h e process (42). This bound was v e r i f i e d c l o s e l y i n p r a c t i c e ; it i s of t h e type mentioned i n Sect. 4.
G. Capriz
Let respectively
US
.
g?) ,
the e r r o r s i n y ("I Then, from (42) ve get call
A
where now y , *stand f o r the solution of (34) We accept the approximate e q u a l i t i e s
an6
.
so t h a t i t follows from system (43) t h a t
This equality implies t h a t the e r r o r decreases only i f the spectral radius f ( * of the matrix
G. Capriz
A
4
does not exceed unity. g ( W depends on h , T and a l s o on Y , ; b u t these two l a s t vectors a r e unknown t o s t a r t with : a reasonable
-
guess f o r t h e s o l u t i o n i s required i n p r a c t i c e f o r an evaluation of t h e conditions of convergence; such conditions w i l l put then r e s t r i c t i o n s on h depending on t h e value of T. However t h e c a l c u l a t i o n of t h e s p e c t r a l r a d i u s oflvllis n o t an easy matter; a s a consequence one i s forced t o r e l y on rougher estimates, such as' t h e following one.
& (k-1) Assume t h a t i n t h e v e c t o r E (k-l)=
-'
) a l l but t;$k-l) one component vanish, f o r i n s t a n c e t h e e r r o r component r e l a t i v e t o t h e value of _v over a c e r t a i n mesh p o i n t P. Then we may take, a s an approximation, t h a t only a few components of E ( k ) a r e d i f f e r e n t from zero, p r e c i s e l y those r e l a t i v e t o values of over meshpoints adjoining (
P.
I f P i s s u f f i c i e n t l y f a r from t h e boundary t h e components taken t o be non-null a r e those of order m - n, m - 1, m + 1 , m + n During t h e next i t e r a t i o n , leading t o E , there i s a "backfire" e f f e c t of t h e spread e r r o r over t h e m-th component (kt11 m A s a rough estimate of t h e condition of s t a b i l i t y it i s required t h a t
.
5
C.
Capriz
Notice i n c i d e n t a l l y t h a t , i f t h i s criterion i s adapted t o A
apply t o t h e h e a t - t r a n s f e r equation d i f f e r e n c e form untl
I
- unj
=
-3 u
3t
= d--T
l d ~ t /( A X )
a,
1
i n the f i n i t e
n [u yl-2 u j t u j l l I j
,
.
a s can it leads t o t h e s t a b i l i t y r u l e [ d ~ t / ( & x ) * ] 5 2/3 be e a s i l y checked; llerc t h e notation i s obvious. A s i s wc '.l known a more appropriate a n a l y s i s i n t h i s case suggcsts thc upp limit
-
1/2 r a t h e r 2/3 f o r t h e r a t i o [ d h t / ( ~x ) ~ ] , a t l e a s t i n t h e case of simple boundary conditions. Sirnilary, i f t h e c r i t e r i o n i s adapted t o apply t o t h e wave
&- i n t h e f i n i t c - d i f f e r e n c e = c 3t2 a x2
equation a2u
it leads t o t h e r u l e
(c A t / A x ) 6 1
form
.
Returning ncw t o our problem, we a r c - i n t e r c s t e d i n t h e s o l u t i o n of a l i n e a r system extracted from t h e system A
i n f a c t we have supposeil that'j-y be n u l l . Decause a l l components of but one a r c a l s o n u l l , we lntend t o examinc t h e approximation \?here a l l b u t f i v e components of &$I vanish. Thcse components s - t l s f y t h e following reduced system
E$-"
5
linaar wEiich can be e a s i l y . s o l v e d . I n t h e sccoiid s t e p leading t o s y s t c n i i i t h 13 unknoi~ns i s involved; we leave o u t d e t a i l s t o ricofc t;ic r e s u l t
Fror? a r1ur:crical p o i n t of v i a ; i n c q s a l i t y ( 4 4 )
I?;I'I
Lc
iiiterprctcd i~oi,a; a c o c s t r a i , l t inponeri upon tile change of)Uovcr two nlcilcsh s t e p s ; i l l f a c t ( 1 4 ) i ~ p l i c s ,i l l view of (/15),
-
An a l t e r n a t i v e , physically sigriilicant, ititerpretation
05 (45) i n possible; consider t h e noZulus
S of t h e p r o j e c t i o n i n (r, 2)-plane of t!le v e l o c i t y of t h e f l u i d and i n d i c a t e with RI! the tile Xeynolds nunbcr based on S , tile physical s i z e ( r 2 - rl) h of t h e mesh and t h e v i c c o o i t y V T:';len (45) can be w r i t t e n
.
C. Capriz
These c r i t e r i a of s t a b i l i t y , though rough, have proved t o be very u s e f u l i n t h e preparation of computer programs. For samples of r e s u l t s of numerical work we cake r e f e r e n c e , f o r instance, t o paper
[?J .
G. Capriz
1
L. F. Richardson, Weather p r e d i c t i o n by n m e r i c a l proccss.
k]
R. BsrXer, I n t g g r a t i o n des Equations du mouv2mcnt d'un f l u i d c
13)
R. Finn, S t a t i o n a r y s o l u t i o n s of t h e TJavier-Stokes equations.
C d r i d y e Univ. Press., London 1922.
visqucux incompressible. Encyclopedia of Physics, vo1.8/2, Springer (1966).
Proc. Symposia Appl. Math., _1_Z (1965) , 121-153.
D.N. de Allen, Relaxation Flethod~ i n Engineering and Science, Mc Graw-IIill, 1954. [5]
A. Thoni, C . J .
Apelt, F i e l d Computations i n Engineering and Physics. Van Costrand, 1961,
b] J. [6 d [7]
von Meumann, Collected Works, vol. 5. Perganon P r e s s , 1963. i d . , vol. 6
, ikthods i n computational physics. Academic Press. 1, S t a t i s t i c a l Physics (1962) ; 2, Quantum ilcchanics (1763) ; 3, Pundamental Methods IIydrodynanics (1364) ; 4 , ~ ~ p l i c a t i o n s i n IIyclrodynamics (13G5) ; 5, Fluclear ~ a r t x c l eKinematics (1366) ; 6, lruclcar Physics.
B. Aldcr, S.Fernbach, It. Rotenberg, ed.
ix
181 F.II. IIarlotr, The p a r t i c l e -in - c e l l nethods f o r numerical s o l u t i o n of problems i n f l u i d dynamics. Proc. Symposia Appl. Math., 15 (19631, 263-288.
b] -
[ld
J.R.
P a s t a , S. Ulam, I I e u r i s t i c numerical work i n some problems of hydrodynaniics. IIath. Tables Other Aids Conp., 13 (1953), 1-12.
A. B l a i r , IT. 1Ictropolis, J , von Ileunann, A.11. Taub, $1. Tsingou, A study of a numerical s o l u t i o n t o a two-dimensional
hydrodynamical problem. 1"ltt. Tables Other Aids Comp., (1353), 145-124.
13 -
G. Capriz
[11]
J.C.
Welch, F.H. IIarlow, J . P . Fhannon, B.J. Daly, The MAC method, a computing technique f o r solving viscous, incompressible, t r a n s i e n t fluid-flow problems involving f r e e surfaces. Los Alamos Scient. Lab., LA - 3425.
[id
F. I!.
Ilarlow, J.E. Welch, rrumerical c a l c u l a t i o n of tine-depez dent viscous inconpressible flow of f l u i d with f r e e surface. Phys. Fluids 8 (1965), 2132-2139.
-
1131
F.11. IIarlov, J.E. tlclch, Numerical study of large-amplitude free-surface motions. Phys. F l u i d s , 9 (19661, 042-351.
[14]
J.D.
[15]
1.1.R. Abbott, A numerical nethod f o r solving t h e equations of
[16]
C.C.
[17]
A. ::asai~ara, E. Isaacson, J.J. Stoker, EJumerical s t u d i e s
114
C.
1
J.C.
Bellun;~, S.W. Churchill, Computation of n a t u r a l convection by f i n i t e d i f f e r e n c e rncthods. Proc. I n t . Conference on neat Transfer, I n s t . flech. Cng., London (1361). n a t u r a l convection . n a narrow concentric c y l i n d r i c a l annulus with a horizmtal a x i s . Quat. Journ. Flech. Appl. Kath., 17 (19641, 471-431.
[h
.
L e i t h , l m e r i c a l simulation of t h e e a r t h ' s atmosphere vol. 4 . 1-29. in of f r o n t a l motion i n t h e atmosphere, T e l l u s , 2 (1965), 1.
Isaacson, Fluid dynamical c a l c u l a t i o n s i n EJumerical Solution of P a r t i a l D i f f e r e n t i a l Cquations, J.11. Cramblc, ed.,Acadcruic Press. Mew York (1366) , 35-43. Fronw,P.H. Iiarlou, IIumerical s o l u t i o n of the problem of vortex s t r e e t devclopnent. P:?ys. F l u i d s , 5 (1963) , 975-332.
G. C a p r i z
P. I?. I!arlotl, J . C . F r o m , Dynamics and h e a t t r a n s f e r i n t h e von ~ d r n ~ hwake n of a r e c t a n g u l a r c y l i n d e r . Phyc. F l u i d s , 7 ( 1 3 6 4 ) , 1147-1156. Z.J.
Daly, A n u n e r i c a l s t u d y of two f l u i d Rayleigh-Taylor i n s t a b i l i t y . The P h y s i c s of F l u i d s , 10 ( 1 9 6 7 ) , 297.
A. L. K r i l o v , E.1;.
P r o i z v o l o v a , 1Jurr.crical a n a l y s i s of t h e f l u i d f l o ~ ?between two r o t a t i n g c y l i n d e r s . P r o c c e d i n g s (CGOPIIWK PABOT) Computing C c n t r c !loscow Univ. , 2 ( 1 9 6 3 ) , 174-181.
-
G . C a p r i z , C.Ghelardoni, C.Lo%bardi, PJumerical s t u d y o f t h e
s t a b i l i t y problem f 4 r C o u e t t c flow. Phys. P l y i d s , 9 ( 1 9 6 6 ) , 1934-1936.
.
F. A . Ilarlo\?, J .E F r o m , Computer e x p e r i m e n t s i n f l u i d dynamics. S c i e n t i f i c American, 212 ( 1 9 6 5 ) , 104-110. F.H. Harlow, J . P . Shannon, J.E. :Jelch, L i q u i d waves by computer. S c i e n c e , 149 (1965) , 1092-1093. J .E. b ~ e l c h , Computer s i m u l a t i o n o f w a t e r waves,
-
Datamation 12 (1966), 41.
F.H. IIarloti, J.P. Shannon, J.E. Welch, Un c a l c u l a t e u r q u i f a i t d e s vagues. S c i e n c e s , 7 ( 1 9 6 6 ) , 14.
-
D. Greenspan, P.C. J a i n , R. Manohar, E. Ebble, A . S a b u r a i ,
Numerical s t u d i e s o f t h e Navicr-Stokes e q u a t i o n s . Math. Res. C e n t e r , Techn. Summary Rept. 402 (1964).
P.C. J a i n , Numerical s t u d y of t h e EJavier-Stokes e q u a t i o n s f o r t h e p r o d u c t i o n o f s m a l l e d d i e s from l a r g e o n e s . Math. Res. C e n t e r . Techn. Sunmary Rept. 491 ( 1 9 6 4 ) .
G. Capriz
E. De Luca, Numerical s t u d i e s of p o i n t p e r t u r b a t i o n s i n
laminar plane P o i s e u i l l e n o t i o n . Army Material Res. Agency, Tech. Rcpt. APIRA TR 63-10.
M. Capovani, G.Capriz, G.Lombardi, Studio numeric0 d e l l a
s t a b i l i t a d e l mot0 d i un f l u i d o viscoso i n un canale. Calcolo 2, Suppl. 1 (1965), 33-49.
E. Dellorno, A numerical program f o r dealing with f i n i t e - m p l i t u d e
distcrbance i n plane p a r a l l e l laminar flows. !!eccanica, 2 (1967), 95-108.
D.F. De Santo, H . 0 . K e l l c r , Numerical s t u d i e s of t r a n s i t i o n from laminar t o t u r b u l e n t flow over a f l a t p l a t e . J . Soc. Ind. Appl. !lath. , 2 (1962), 569-595. J.A.T.
L.H.
Bye, Obtaining s o l u t i o n s of t h e 1Javier-Stokes equation by r e l a x a t i o n processes. Comp. J., 8 (1965-66), 53-56. Thomas, The s t a b i l i t y of plane P o i s e u i l l e flow. Phys. Rev. 91 (1953), 780-733.
W. V e l t e , S t a b i l i t a t s v e r h a l t e n und Vcrzweigung s t a t i o n s r e r LGsungen d e r tlavier-Sto1:csschen Cleichungcn. Arch. Rat. Mch. Anal., j& (1964), 97-125. W. Velte, S t a b i l i t S t und V e r z v c i g u n ~S t a t i o n S r e r LSsungcn d e r 1Javicr-Stokesschen Glcichungcn beim Taylor Problem. Arch. Rat. Iiech. Anal., 22 (1366), 1 - 1 4 . I<. i;irchgYssner,
Die I n s t a b i l i t a t d e r StrSrnung zwischcn zwci roticrcndcn Zylindern gcqeniibcr Taylor-Wirbeln f u r belicbige Spaltbreitcn. Z.A.M.P. 12 (1961), 14-30.
C.Chelardoni, Qucstioni connesse coll'impostazione a n a l i t i c a e l a s o l u z i o r e numerica d i un problema d i idrodinarnica. Calcolo, 2 , Suppl. 1 (1965), 51-66.
G. Capriz
.
, G Lombardi , Soluzionc numerica d i un problema d i s t a b i l i t a idrodinamica. Calcolo, 2, Suppl. 1 (1965), 67-80.
bd
G .Ghelardoni
[43
G.I. Taylor, S t a b i l i t y of a viscous l i q u i d contained between
two r o t a t i n g c y l i n d e r s . P h i l , Trans. Roy. Soc (19231, 283-343.
A 223
R.J.
. (London)
Donnelly, D. F u l t z , Experiments on t h e s t a b i l i t y of s p i r a l flow between r o t a t i n g c y l i n d e r s . Proc. tlat. Acad. S c i . , 46 (1960) , 1150-1154.
D.Coles, T r a n s i t i o n i n c i r c u l a r Coucttc flow. Navy Dept Rept. , Cambridge, 1Iass.
.
S. Cfiandrasel:har,
Hydrodynamic and hydromagnetic s t a b i l i t y . Oxford Univ. Prcss (1961)
.
II. I J i t t i n g , Uber den E i n f l u s s d e r S t r g m l i n i e n - ~ r i i m u nauf ~
d i e S t a b i l i t a t laminarcr S t r h u n g e n . Arch. Rat. Ilech. Anal., 2 (1958), 243-283.
G.Prodi, Problemi d i diramazione per equazioni funzionali. A t t i V I I I Congresso U.M.1 , T r i e s t e (1967) , Under press.
.
C.Ghelardoni, Considcrazioni numeriche s u un problema d i idrodinamica. Calcolo, Under p r e s s .
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
A. DOU
ENERGY INEQUALITIES IN AN ELASTIC CYLINDER
Corso tenuto ad Ispra dal 3 - 7 luglio 1967
ENERGY INEQUALITIES IN AN ELASTIC CYLINDER by A. DOU (University of Madrid)
- Venant.
1 The nrincinle of Saint More than a centrary
ago, in 1855 and 1856, B.de Saint-Venant
stated a principle that has been applied steadily in the calculus 1 of beams, but has not yet been proved. F r o m the point of view of
[
Numerical Analysis the importance of the Principle of Saint-Venant is obvious and great. We consider an homogeneous, isotropic and perfectly elastic cylinder R
C
'
F o r simplicity we assume also
< ~ I ) c R2
Q = ~~,Y)(IxI
the
paper that there a r e no body forces
and no lateral surface forces; and also that the cylinder i s in
statical
equilibrium. T(x, y)
We a r e given the surface forces upon the bases of the cylinder we
may assume without
odd -
for
and
=
write
-t:T+)
loss.
z =
of
e
and
and
z =
generality
-
that
T-(x, y) acting respectively, and they a r e either
~ ( l ),
=\
-
-T;(')(=, y) ,
o r that they a r e even and
-
(1)(x, Y)] , x. T~(1)(x. Y) , T~
write T (2),
.
Y ~ Q
A. Dou
I:
When t h e r e i s simply T
if
T( 2) =
I
1.
T( 2) (x, y), T;?)(X, y), T~(2)(x, y ) 1 f o r z = - I , : ~ - ( ~ ) = { ~ y ) ( x ,~y(2 ~) ,I x . y -) T . (2) ~ (x.Y)} for z =
no
.x.Y(Q.
need to specify, we shall write T ( ~ ,) g = 1, 2, o r
no confusion i s possible. The specifications r e f e r to the
of the corresponding s t r e s s t e n s o r with r e s p e c t to the r(g) 33 independent variable z T h e s e corresponding s t r e s s t e n s o r will be cal-
component
.
led, respectively, odd and even ; obviously a l l t h e i r components a r e odd o r
even functions of z.
We shall consider
(34
x,Y~B.
~ ( ~ T' =5( T 1(x,Y), T ~ ( x Y ) T, ~ ( xY)} , ,
(2) which
only those s u r f a c e f o r c e s
s a t i s f y the following conditions : T~cc(G),
In (3a) and
~,Jc(G),
iZ1.2,3,
d=l,2,
throughout the paper the subindex k a f t e r t h e com-
m a means p a r t i a l derivative with r e s p e c t to x , ( x l , X2, x3) 3 (x, Y, z). The conditions (3b) e x p r e s s
that the s u r f a c e f o r c e s a r e self-equilibra-
ted a t each base, and (3c) i n s u r e s that the corresponding s t r e s s t e n s o r be contilluous in
-R .
Surface f o r c e s satisfying conditions (3) s h a l l be called p e r m i s s i b l e s u r f a c e f o r c e s (psf) , and the corresponding boundary conditions shall b e called p e r m i s s i b l e boundary conditions i s obvious by a vector s p a c e o v e r the r e a l s
(pbc)
.
. The
set
of a l l psf
A. Dou
11
Let
11s 1)
be the
(x)(( be
the
euclidean
-x p (x, y, z ) G -R .
L ~ ( Q ) n o r m of norm
tions
can
of t h e s t r e s s t e n s o r Z (g) ,
of
is s m a l l compared w i t h 1.
the b a s e s of the cylinder
=
1
~
11~ . In
z
a' >, -g-
/ >l0.t ;
,
be i:~plected altogether,
x 6 5, is
and llsmallll m e a n s that g
\If (:)I[
say
"C
p r a c t i c a l applica-
l l n e i g h b ~ r h o o dmeans, ~~ f o r instance, that the point
s u c h that
let
Then the principle of Saint-Venant c a n b e s t a t e d thus :
Except in the neighborhood
I/t (z) (1
S(x, y ) and
6 ('110) m a r
(3
IITill
i = l , 2, 3 for 1z1< 4 ( 1 5
!> 10 h . Today
,
ciple i s i t s successfuly
as
stated,
is
conditions,
true,
under
although
which
The application
the b e s t justification of t h i s prin-
steady application f o r m o r e than
y e a r s . Mathematically speaking I
of
do not
think that this
t h e r e must
principle,
it will be t r u e . the principle to t h e calculus of b e a m s
. In
and useful
with
forces
, which s a t i s f y conditions
first
of
(3 b 1 ) f o r
hundred
be some very general
indeed both, obvious F(x, y )
.
i = 1, 2;
but
ydxdy
is
g e n e r a l b e a m s a r e bouded (3a) , (3c) and the
one h a s
= M I ,
2'
Q
The principle allows to dispense with the functions F and take
R,
into account only the resultant
1' F2' F3 the bending moments
.
M and the torsion moment M T o t h e s e quantities corresponds M1' 2 3 in R a unique e l e m e n t a r y solution (x, y, z ) of Saint-Venantls type, i . e . independent
of
z,
and
therefore i s a
l i n e a r combination of
- 166 -
A, Dou
uniform traction o r compression, p u r e bending and p u r e torsion. We r e m a r k the following interesting c o r o l l a r y : The principle of Saint-Venant implies that t k only bounded solutions f o r the s t r e s s t e n s o r t ( x , y, z ) in the infinite cylinder R
a r e those WJ
of
Saint Venant's type, i. e . independent of z. In the r e m a i n d e r of t h i s s e m i n a r I s h a l l give t h r e e inequalities
that
b e a r on the principle of Saint-Venant and proved by R.A. Toupin
1 2 1 , J. J. Roseman 131 and myself, third. Finally I
14,51
and outline the proof of the
shall comment on related questions.
2 E n e r g y inequalities. The f i r s t two inequalities that one end, z =
-e
, is
z =
f
free
due to Toupin and
Roseman a s s u m e
T(x, y) and t h e o t h e r and,
, is loaded with psf
of f o r c e s . This i s achieved in o u r presentation s e t -
ting ~ ( l= )T(') = (112) T (x, y ) and + T (2)
loading the cylinder with psf T
(1) +
.
The r e s u l t of that
part
end,
Toupin a s s e r t s that
of the cylinder
U(s) , s a t i s f i e s U(s)
t h e total e l a s t i c energy of
beyond a distance
< U(O) and
.
exp
{-
s - h
0
ding on
e l a s t i c constants and on t h e s m a i l e s t Rh
away f r o m t h e loaded end, ning
part
of
f r o m the loaded
the inequality
where
of f r e e vibration of
s
).
c(h) is a c h a r a c t e r i s t i c decay length
. Substantially i t then
depen-
a h a r a c t e r i s t i c frequency
s a y s that
, if we move
the total e l a s t i c energy in t h e r e m a i -
t h e cylinder d e c r e a s e s exponentially.
The r e s u l t of Roseman a s s e r t s that, if P(s) is cylinder whose distance t o the loaded end i s s , then
a point
of
the
A. Dou where the f i r s t member and U have already been defined, K is a constant depending on elastic constants of the body and 0 and a a r e positive constants depending on the geometry of the cross-section. This result i s similar to the previous pointwise estimates of J. H. Bramble and L. E. Payne
161, but this one i s good up to
the boundary. The third inequality i s contained in the following theorem : Assume that z i s the s t r e s s tensor in R t corresponding to any psf T = = ( T I T2, T 3 ) E&,(R)
0
be the total elastic ootential energv pf 2 in
st.
< ln Then there is a constant K depending only on
(4)
IT^,
-2-
0
such that
2~1
< $ ~ ~ P , P ~ + u T J I ~ , + ~ T ~ ~+~I I~T ~ ,
"
and let
Now I shall outline the proof of this theorem given in
)
-
61. We need the
constituent equations relating the s t r e s s e s with the displacements in an homogeelastic body, the equations of Elasticity governing
neous, isotropic and
the dispalcements and the theorem of Castigliano. The s t r e s s tensor
<=
(( f ..)), i, j = I , 2, 3, analytic in R and continuous 13
corresponding to the psf T (g)= (g) (g), ~ ( g ) ), g = 1 . 2 , i s given in t e r m s (TI ,T2 of the first derivatives of the displacements u(r, y, z)=(ul,u2 us), (x,y,z)tR, by in
E,
A>
0
,
p>O
A
0
'11
' l i j Kronec ker
symbol. The displacements u
are
given by the unique solution of the
Navier equations of elasticity
satisfying the following permissible boundary conditions
(pbc)
:
A. Dou
which
express
that
corresponding to the
A tensor
no l a t e r a l s u r f a c e f o r c e s a r e present, and
psf
in
the b a s e s
of the cylinder.
-
0(x, y, z ) g ((0. .)) , i, j = 1, 2 , 3 , continuous in R i s 1.l
called a yrirtual s t r e s s
tensor
pbc (7) and (8) a s 7
same
equilibrium
R
f o r z in and
, if it s a t i s f i e s the
m o r e o v e r i s a solution
of the
equations
Let
be the positive definite quadratic f o r m of the e l a s t i c energy density due to
the t e n s o r
P
=
(( e i j ) )
. Let
t be
a s t r e s s t e n s o r in
R
1
and
0 a virtual s t r e s s
one obtains
7
. Then
tensor
the total e l a s t i c energy
&,(R)
due t o
f o r s o m e types
is
of psf,
to construct for any psf, o r a
0,
the s t r e s s tensor
s a y s that
The main idea of t h e proof that
T. Integrating W7 o v e r
, the t h e o r e m of Castiglano, o r the minimum s t r a i n
energy t h e o r e m
at least
for
virtual s t r e s s
its virtual energy o r energy integral i s bounded ,
t e n s o r such, even when
A. Dou
Now, the equilibrium by any t e n s o r
O(x, y, z)
5 1
provided that
It must sink
is
=
d
of the f o r m
11
M
to
.
(x, y )
,22
the r e a l n u m b e r s
easy
(9) a r e automatically satisfied
equations
Nu ( z )
d . .satisfy 11
conjecture and
verify, that the function
N(z)
i n c r e a s e exponentially, and in the p r e s e n t proof the functions
pz
Although
for
g = 1 and
cosh
z f o r g = 2 have been taken. ( 7 ) , (8) m u s t a l s o be satisfied,
t h e boundary conditions
there is also
abundant r o o m
try
to
t o take c a r e of e v e r y
Now we may outline the proof in t h r e e s t e p s . sufficient types of psf 1) every set of
these
2)
it will
t e n s o r in
of psf
such
can
that
condition.
F i r s t to get
meet the two following requirements:
be decomposed in
a l i n e a r combination
types ; be possible f o r each
ae.It
turns
r e a d y five types of
out psf
that,
type to construct a virtual s t r e s s for
a
cross-section
a r e sufficient, each
like
.
Q, a l -
type depending on one a r b i -
t r a r y function and possibly s o m e r e a l constants. The s t r o n g e s t condition that each type of
$1
=
( f 1 T I , Ij2T2,
f
T 3 ) , where the
A. Dou
t3 a r e
f1 , f2 ,
the
is
that
as
can be conjectured In the second
each and
T ( ~ )t h e r e for
such ,
that
from
a r b i t r a r y constants, must satisfy ,
(12 )
e a c h type of
corresponding
2 ( g ) we construct
its
.
s t e p we consider is a
each
three
virtual
stress
psf T ( ~ ) . F o r
tensor
% (g) in R(
a virtual s t r e s s
energy s a t i s f i e s f o r
tensor
8
all
o(~)
the following
inequality
where of
C
f?
Wdll
Lame
a
if
and
is any n o r m
constant of
constant
t?0 < e,
,
odd;
nent
is
the
psf
T
in
a
linear
each one of t h e m Because and
3
T ( ~ )= (
In the t h i r d and l a s t psf
only if
a=(
of
T
; and
depending on
i 3 )6 R
if ;
y
lo T ( ~ )is
isone
(x, y) is the function of the third compo-
Y1
step
g 2 T 2 ~b. 3T3) . of the proof we decompose any given at
m o s t sixteen s u m m a n d s ,
of the five types defined in the f i r s t step.
of t h e linearity of the Navier and of equilibrium equations,
of conditions (3) and (7) , the s t r e s s t e n s o r
the psf
bound
g = 1 , that i s only 3
combination of
being
a lower
T
may
be
7 . corresponding to
decomposed a l s o the s a m e l i n e a r
combination of a t -
m o s t sixteen corresponding s t r e s s t e n s o r s . Then we of
apply the second s t e p
of the proof to
each summand
the decomposition and t h e r e f o r e we get un upper bound of the total
e l a s t i c energy of
the cylinder
Re
in t e r m s of the s q u a r e s of the
n o r m s of the third component of each summand. It s q u a r e of the n o r m
of
turns
out, that the
t h e third component c a n b e always estimated
A. Dou
of the s q u a r e s of the n o r m s that
by m e a n s of a l i n e a r combination
. Therefore,
a p p e a r in the inequality (4)
t h e inequality is proved.
3. Related ,questions a) l y if
When we stated the inequality ( 1 4 ) , we said,
the psf
the constant On
are
T C of
even
, that
(14) does not depend
the c o n t r a r y , if the given
in (14) o r
K
call
psf
T
in (4) does depend on
(See the l a s t formula of I
is of
page
91
the f o r m on
a r e odd a
that, if and onT ( ~ ), then
a lower bound of in (4) , then C
lower bound
in
[5] )
the attention on the physical
e.
of
0
e.
.
interpretation
of this r e s u l t
and how intuitive it looks. b) F r o m inequality
(4)
s o r in the infinite cylinder
is bounded in
R
, then
0
Venantts type. T h i s of Saint-Venant. I tioned
i t follows that
R
03
5 is
Z is a s t r e s s ten-
, without e x t e r n a l f o r c e s , and
if
an e l e m e n t a r y soiution of Saint-
c o r o l l a r y of
recall
if
(4) may be called a weak principle
that this r e s u l t is c l o s e t o the above men-
c o r o l l a r y of the principle of Saint-Venant. It would b e
interesting to
t e r e x a m p l e of t h e principle ,
know
if t h e r e e x i s t s in 2 allowing 1 3 4 , d ( x y, z )
C
oc.1
when ) z ( + m ,
in
such
a
a counm t o diverge
R
1 3
way that a t the s a m e t i m e
Z
Y,
i=l r e m a i n s bounded. c ) The mentioned r e s u l t s of proof of
(4) , imply that in
Toupin
the cylinder
and Roseman, and a l s o the R
e
the s t r e s s e s decay
")I
A. Dou
exponentially .This rily
t r u e in
assertion
about
the neighborhood
the I1decayingN i s
of the b a s e s
not n e c e s s a -
of the cylinder, no m a t t e r
how long the cylinder may be. However, it s e e m s , that f o r any given psf T
T
aproximate
a s much a s d e s i r e d by
the corresponding the ly
b a s e s of
stress
tensor
* in.
i s possible to
such
a way, that
does take the maximum value in
t
Re
the cylinder
Y
psf T
it
, provided that
Re is
sufficient-
long. d ) Dividing the Navier
equations (6) by
grad All such elastic
- 1 < U < 112 ,
m a t e r i a l with
in
although
for t h e Poisson's r a t i o I
do
C
,
not know of any p r a c t i c a l
ct negative.
One consequence of i s that
, they take the f o r m
div
the stated r e s u l t s a r e valid
that
/U
the proved weak principle of Saint-Venant
t h e r e a r e only e l e m e n t a r y solutions of Saint-Venantls
R M
type f o r Q such
that
-1 < U < 112
.
But it has been proved by t h e a u t h o r , [7]
, that
if
6 = -1 - E , E positive and s m a l l , then t h e r e a r e , in the infinite cylinder
with
periodic
a c i r c l e for c r o s s - s e c t i o n
in
z
solutions
Venant
does
not
Navier
equations
hold
and
of (16) , s o that for Q
< -1,
r e m a i n elliptic.
without
e x t e r n a l forces,
the principle of Saint-
in s p i t e of the fact that the
[I;
de Saint-Venant, B . , M e m o i r e s de l l A c a d e m i e d e s Sciences d e s s a v a n t s & r a n g e r s , 14(1855) , 233-560 and Memoire s u r l a flexion d e s p r i s m e s , Journal de Liouville, S e r . 2, (1856) , 89-189.
Toupin, R. A. , Saint-Venantls principle, [ 21 Mech. Anal. 18(1965) , 83-96 [3)
(41
151
[ 61
A r c h . Rational
Roseman, J. J . , A pointwise e s t i m a t e for the s t r e s s in a cylinder and i t s application to Saint-Venant's principle, - Arch. Rational Mech. Anal . 2 l(1965) , 23-48 . Dou, A. , On the P r i n c i p l e of Saint-Venant, Summary Report f 472, May 1964.
MRC Thecnical
DOU, A. , Upper E s t i m a t e of the Potential Elastic E n e r g y of a Cylinder, Comm. P u r e Appl. Math. 19(1966), 83-93. Bramble, J . H. , and Payne, L. E . , *-Pointwise bounds in th_e f i r s t biharmonic boundary value problem, J. Math. and P h y s .42 (1963) , 278-286 .
--
Dou, A., Bounded solutions of the Elasticity equations in the the unbounded cylinder, Proceedings of the Inter. C o n g r e s s of Math. , Moscow, 1966. A b s t r a c t s of s h o r t co~nmunications.
[ i] in
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
TODDDUPONT
ON THE EXISTENCE OF AN ITERATIVE METHOD FOR THE S9LUTION OF ELLIPTIC DIFFERENCE EQUATIONS WITH AN IMPROVED WORK ESTIMATE
Corso tenuto ad Ispra
3-11 luglio
1967
On the Existence of an I t e r a t i v e Method f o r the Solution of E l l i p t i c Difference Equations with an Improved Work Estimate Todd Dupont Rice University, Houston
1. Introduction The i t e r a t i v e s o l u t i o n of the d i f f e r e n c e equations associated with V + ( a ( x ) ~ u )= f i s a problem which has received a g r e a t d e a l of a t t e n t i o n i n the l i t e r a t u r e .
I n t h i s paper i t w i l l be shown t h a t
f o r r a t h e r g e n e r a l domains i n the plane t h e r e e x i s t s an i t e r a t i v e method with work e s t i m a t e s which a r e b e t t e r than those now a v a i l a b l e . This i s an a b s t r a c t existence theorem, not a complete s p e c i f i c a t i o n of an improved i t e r a t i v e procedure. On a r e c t a n g l e with a ( x ) constant the Peaceman-Rachford [ 5 1 procedure gives a work e s t i m a t e of ~ ( h - ' l o g h-'1og c - l ) f o r reduction 2 of t h e L -norm of the e r r o r by a f a c t o r ;. S t i l l on a r e c t a n g l e but f o r more general equations the procedure of Gunn [31 g i v e s the same work estimate f o r the reduction of the analogue of the D i r i c h l e t i n t e g r a l of the e r r o r by a f a c t o r c . Gunn's procedure a l s o gives an estimate of O(h- 2 (log h- 1) 2log c - l ) f o r reduction of t h e uniform norm of t h e e r r o r by a f a c t o r c again on a r e c t a n g l e .
We s h a l l show
t h a t , i f the domain i s the union of r e c t a n g l e s with s i d e s p a r a l l e l t o the coordinate axes and i f a ( x ) is twice continuously d i f f e r e n t i a b l e i n the c l o s u r e of the domain, a work estimate of the form O(h 2 ( l o g h")'log c - l ) can be obtained f o r the reduction of t h e uniform norm of the e r r o r by a f a c t o r s .
I n s e c t i o n s 2 and 3 we d e f i n e t h e problem and the i t e r a t i o n and give a n a n a l y s i s of t h e work required t o produce t h e s o l u t i o n . This a n a l y s i s i s done assuming the main lemma of t h e paper, Lemma 1, which i s proved i n s e c t i o n s 4 and 5. 2.
D e f i n i t i o n of t h e Problem
We w i l l work with a domain D = R1 J R2, where each Ri is a 2 r e c t a n g l e i n R with s i d e s p a r a l l e l t o t h e coordinate a x i s . &T choice of ---
two r e c t a n g l e s i s made f o r s i m p l i c i t y , though i t i s c l e a r
t h a t what follows --
holds f o r any f i n i t e union of such r e c t a n g l e s .
Assume t h a t we have a square g r i d of mesh s i z e h covering R1 and R2 and t h a t t h e s i d e s of R1 and R2 l i e on g r i d l i n e s . t h a t 3R1 n aR2 c a D .
de a l s o assume
Let Dh denote the g r i d p o i n t s i n t h e i n t e r i o r
of D, 3Dh t h e g r i d p o i n t s on aD and
Dh
= Dh ;aDh.
Similarly, define
de a r e attempting t o solve
f o r u,,
a f u n c t i o n defined on
Bh, where Lhuh i s any one of the f i v e
point dpproximations (4.4) t o c . ( a h ) generated by (4. I ) , ( 4 . 2 ) , and (4.3) and f and g a r e given f u n c t i o n s . We s h a l l assume a t o be a 2 p o s i t i v e C f u n c t i o n i n 6. I n what follows we a r e i n t e r e s t e d only i n f i n d i n g t h e s o l u t i o n t o the a l g e b r a i c equations (although we s h a l l use t h e f a c t t h a t t h i s i s an approximation t o t h e s o l u t i o n of t h e d i f f e r e n t i a l problem).
3.
D e f i n i t i o n and ~ n a l y s i sof the I t e r a t i o n The i t e r a t i o n t h a t w i l l be examined here i s a nested one.
The
o u t e r i t e r a t i o n c o n s i s t s of a Schwarz a l t e r n a t i o n , and the inner i t e r a t i o n c o n s i s t s of a Gunn i t e r a t i o n .
The outer i t e r a t i o n i s i m -
p l i c i t and the equations involved i n i t w i l l be approximately solved by the Gunn procedure.
The o u t e r i t e r a t i o n i s defined a s follows.
Let uiyh be t h e s o l u t i o n of
where j = 1 i f i i s odd and j = 2 i f i i s even. t o R j , h of Lo.
u
i s the r e s t r i c t i o n j ,h ~ by , a Gunn ~ iteration,
As we intend t o solve f o r u
we need t o s p e c i f y an i n i t i a l guess on R
j ,h
L
which we tdke t o be
i f i 2 2 and u ~ i f , i =~ 1. i-2, h We s h a l l assume the following lemma f o r the present and proceed
with the a n a l y s i s of the i t e r a t i o n . Lemma 1. There e x i s t s q depending only on R1,R2 and a such t h a t , if
then
maxw h
r q h s q c 1 , where j '
j
.
The Domain
The Boundary Values
For example, i f D i s a s shown i n t h e f i g u r e and wh i s d e f i n e d on K l , h 9 L l , h w h ' O On R l , h , and t h e boundary v a l u e s of wn a r e a s shown, t h e n wh r q < 1 on t h a t p a r t of bR Let w
i n s i d e R1. 2,h ~ be, t h~e c a l c u l a t e d approximation t o u ~ , ~Then .
L e t t i n g uh be the s o l u t i o n of ( 2 . 1 ) and qi,h = w
~
- ,uh, ~ we
obtain
It should be noted t h a t Y ~ , ~ being , , t h e d i f f e r e n c e between the t r u e s o l u t i o n and t h e approximation t o i t , i s e q u a l t o zero on the boundary.
Let us adopt t h e f o l l o w i n g n o t a t i o n :
=
Zi
max Rj,h'Rj ' , h
lzi,hl
9
where j = 1 i f i i s odd and j = 2 i f i i s even and j # j
'
.
By the
maximum p r i n c i p l e and Lemma 1 i t follows t h a t
Theref ore,
Now i f we neglect round-off e r r o r , l e t z denote the norm (with r e s p e c t t o t h e uniform norm) of the e r r o r propagator i n the inner i t e r a t i o n , and l e t wFrUe denote the exact s o l u t i o n of ( 3 . 1 ) , we l,h obtain \.
s 3
max I twr u e~ -, ~ R i-2,h1 j, h
'
Thus, v
r s
(3.8) <
man / w ~ , T~i , h+ R j ,h
;(xi
+
Xi_2
+ vi)
.
-
h'
+
h'
-
Wi-2,hI
Therefore, (3.9)
vi
0
6
=(xi
+
.
Since
it follows t h a t
Now, (3.6) and (3.9) imply t h a t
Therefore,
This implies t h a t xi converges t o zero i f a l l t h e r o o t s of
a r e l e s s thdn one i n a b s o l u t e value.
This i s the case i f , say,
0 < : < (1-q)/5. Thus, i f z i s s u f f i c i e n t l y small (independently of h ) , t h e r e e x i s t s v such t a t 0 r v < 1 and xn < ~ v ~ ~ ~ [ x ~ f x ~ f x ~ + x ~ ~ . Hence, it r e q u i r e s O(1og c - l ) o u t e r i t e r a t i o n s t o reduce the uniform norm of t h e e r r o r by a f a c t o r c . Thus, i t remains only t o analyze t h e work required t o reduce t h e norm of t h e e r r o r by a f a c t o r
I:
a t each s t e p .
I n Gunn's paper
i t i s shown t h a t t h e work t o reduce t h e A-norm of t h e e r r o r on a
rectangle by a f a c t o r 6 i s ~ ( h - ~ ( l oh-g 1) l o g b-'),
where the a-norm
is I
2
L U A ~ U bh ) ~ being , t h e standard f i v e point approxiR mation t o t h e ~ a e l a c eoperator. It i s easy t o see t h a t t h e r e = h
e x i s t s Co and C 1' independent of h, such t h a t
t h e work a t each s t e p i s Thus, it s u f f i c e s t o take 6 = c ~ - ~ c Hence, I.
-1 2 Consequently, the estimate f o r the t o t a l work i s ~ ( h - ~ ( hl o )~ log . - I ) . Let R 1 and R2 be two-dimensional r e c t a n g l e s with boundaries aRi c o n s i s t i n g of segments p a r a l l e l t o the coordinate axes and l e t
Theorem.
D
= R1 U R2.
Let t h e r e e x i s t a sequence (h,),
0 < hm
-
0 , such t h a t
aD f a l l s on a square g r i d of mesh hm. Let aR1 ? aR2 c aD.
Let
a ( x ) E c 2 ( 6 ) , a ( x ) rao>O. Then, t h e r e e x i s t s an i t e r a t i v e procedure f o r obtaining the s o l u t i o n of (2.1) such t h a t t h e number of a r i t h m e t i c operations required t o reduce the uniform norm of the e r r o r i n the 2 approximate s o l u t i o n by a f a c t o r c i s no more than 0 ( h i 2 ( l o g hm) log
4.
F
- 1) .
Proof of Lemma 1 I n t h i s s e c t i o n we s h a l l prove a s p e c i a l case of Lemma 1 which
i s s u f f i c i e n t t o imply Lemma 1 when used with the maximum p r i n c i p l e Let R = ((x1,x2): Wxicdi).
Let 0 s c < dl and suppose dl,d2 and
c a r e a l l i n t e g r a l multiples of ho > 0. let of
nh
Eh
=
-n fl ((ph,qh):
p,q i n t e g e r s ) .
Let hm = h02". Let
anh
If 3 c
R2,
be t h e p o i n t s (x1,x2)
such t h a t (xlih,x2) o r (xl,x2ih) i s not i n
Eh.
-
Let Oh = ih'\azh.
1 Let 7 u(x) = h- 1( u ( ~ ) - u ( x - e ~ , ~ ) ) U ( X ) = h- ( ~ ( x + e ~ , ~u)( x ) ) and VXi, h 'i, h 3 f o r i = 1,2, where e l = ( 1 , 0 ) , e 2 = ( 0 ' 1 ) . Let Ahu(x) = 1 Vx 7u(x). i=l i , h 'i, h
-
L
as foltows: f o r i = 1 , 2 , d e f i n e L i , h y Li (, h~Y) L(') i,h
Thus, ),L!
and )L:
) :L
a r e t h e t h r e e most common f i v e p o i n t approxi-
mations t o ~ . ( a v u ) . Lemma 1'.
I f a i c 2 ( ~ )t n e r e e x i s t s q sucn t h a t 0 < q c 1 and such
t n a t , i f u h i s the s o l u t i o n of
on aRh r T(x1,x2): 0 :x1 < c j on hRh f o r Lh = L$),
[ ( x1 , x 2) : xl r ' c )
,
,
j = 1 , 2 , o r 3, and h = hm some m, t h e n max u h ( c , x 2 )
' 4.
The proof of Lemma 1' goes a s f o l l o w s . For s u f f i c i e n t l y s m a l l s and f o r a l l h = hm t h e r e e x i s t s q
1)
such t h a t U ~ ( C ,<~ q) For edch
2)
i
<
1 f o r y = kh and y
> 0 there exists q
i0,il
b
[d2-:,d21.
1 such t h a t f o r h s u f f i c i e n t l y
small u (c,)') < q f o r y = kh E ( ~ , d ~ - d . h 3) Take c f o r 1 ) and h < b f o r 2 ) . Note t h a t t h e r e a r e o n l y a f i n i t e number of h n t s
2
6 and t h a t we can a p p l y t h e s t r o n g maximum
p r i n c i p l e t o each of t h e s e t o g e t a q < 1 f o r each one.
Then t a k e
q t o be t h e l d r g e s t one of t h e f i n i t e s e t g i v e n by 11, 2) and t h e hnts
2
6.
I n c a r r y i n g out the proof of 1 ) and 2) we use t h e following: I f Ahvh =
Lemma 2.
i n Rh, c
-C
0 and vh = f and aRh, where
2
f E C(aR) and i s twice continuously d i f f e r e n t i a b l e on t h e closure of each edge of R, then
where K = [max f
-
min £11 m i n d . i=1,2 J
+
+ max(lDx2
[c
aR
f l , l D 2 £111 max d 1 X2 i=1,2j
.
2 (Naturally IDxq£1 i s used on s i d e s of aR which a r e p a r a l l e l t o the 1 x1 a x i s and s i m i l a r l y f o r Dx2 f . ) 2 Proof: The proof w i l l be c a r r i e d out f o r ox hvh. F i r s t extend 1' *
vh t o R i and Ri, where R ' = ( ( x1' x 2 ): -d 1 s x 1 <- 0 < x2 s d2} and R" = f ( x1 , x 2) : dl< x1 s 2dl, 0 5 x2 s d 2 ] . This extension i s made s u b j e c t t o the c o n s t r a i n t Ahvh =
.
-C
This extension i s c l e a r l y not
unique, but take any such extension and note t h a t ~ ~ ( . . ~ ~ ,= ~0 ? ~ ~ Thus, gax I P ,hvhl = rnax ) L v I . On the xl, h ' x l , h h 1 1 Rh aRh s i d e s of Rh given by x2 = 0 and x2 = d2' I v ~ l , h ' x l , vh h 1 < max IDx2 1f l -
inRh.
On the s i d e s xl
=
0 and xl = d l ' s c
Thus, I\l,h:xl,h~hl
1'; x l , h v h 1
+ max
'xl,
ID2 f X2
and vX
1'
hixl, hvh = - c - 'r2, h7'x2,hvh
1 .
Hence, t h e t o t a l change in
2 2 i s n o m o r e t h a n [ c + m a x { / D x f l , l ~f l ) ] d l . 1 X2
t h e r e e x i s t xl and x i such t h a t vx h v h ( ~ ; , ~ 2 L)
Lemma 2 holds f o r
Vx
1'
dil(max f hvh
.
-
1'
-
h ~ h ( x , l , x 2 ) ";'(man
min £1.
Foreachx f
-
2
min £1
So, t h e conclusion of
.
The proof of 1 ) goes a s follows. t o show 1 ) f o r y < s only.
It i s c l e a r l y s u f f i c i e n t
This i s done by f i r s t making t h e change I
of v a r i a b l e s vh(x,y) = a ( ~ , ~ ) ' u ~ ( x , y )This . gives nhvh = f n , wnere 3. C i s independent of h, since f h = (nha2)uh + a - ) i ( ~ i 2 ) - ~ h ) u h .
I f h [ < C. ~ 1 ~ 0 ,
where 2
g
for x < c
on dRh
for x
on aR
c
r
h
, '
i s continuous on aR, g ( c , 0 ) = 0, and f o r x < c , g ( ~ , ~ )
a(x,y)k
-
~ ( c , o ) ' , and f o r x
L
c , g ( ~ , r~ 0. )
1 ( c , ~ I) r a ( c , ~ ) ~ a x ( ~ , c d ; ' ] l,h We know by Lemma 2 t h a t I V ~ , ~ ( C 1. r, ~Ky.) F i n a l l y , by
It was shown by M i l l e r [47 t h a t Iv f o r a l l y.
Nalsh and Young [71 v
tends uniformly t o v2 a s h tends t o zero, 2,h .. where v2 i s the s o l u t i o n t o Av = 0 i n R , v = g on 3. Thus f o r h ( c , ~ 1) i s small. Hence, given 6 > 0, we can 2,h 1 take y and h s u f f i c i e n t l y small such t h a t u ( c , ~ )r max(-,cd-l) t i h 2 1 I f we i n s i s t t h a t the bound on y be smaller than t h e above bound on small and y small Iv
h, we g e t t h i s r e l a t i o n f o r a l l h.
The proof of 1 ) i s complete.
The proof of 2) i s c a r r i e d out a s follows.
The boundary d a t a
a r e increased t o a Cm f u n c t i o n g which i s s t i l l not g r e a t e r than one
.
Now i f we l e t Bh denote the
and which i s s t i l l zero f o r xl = d.
s o l u t i o n t o the new problem, then uh s iih and
i, where 6 i s
ihtends
uniformly t o
the s o l u t i o n t o t h e problem V-(avu) = 0 i n R and
u = g on bR, by the r e s u l t s of s e c t i o n 5. maximumprinciplethat
We know by the strong
max b(c,y)
t h a t f o r h s u f f i : i e n t l y smdll
Thus3q
max uh(c,y) YE[E ,d2-E1
2
q < 1.
The only t h i n g l e f t t o prove i s t h e convergence r e s u l t used above f o r uh.
This i s very s i m i l a r t o many such convergence
r e s u l t s but w i l l be included f o r completeness.
5.
Bound on the D i s c r e t i z a t i o n Error rilthough t h i s i s properly p a r t of the proof of Lemma l ' , we
s h a l l s t a t e i t s e p a r a t e l y a s it has some small amount of independent interest. Lemma
3. Let R
= {(x1,x2): O c n l c d l , O c x 2 < d2)
a > 0, and l e t g be a continuous function on bR. s o l u t i o n of ?.(a:u)
-
= 0
in R
.
Let a F
c2(R
Let u be the
such t h a t u = g on bR.
Let uh be
defined on Rh by Lhuh = 0 i n Rh and uh = g on aRh where Lh = L6) h , h, o r Lh. 6) Then max Iuh-uI
- 0 a s h - 0.
(Note no r a t e
can be given without more assumptipns; see Walsh and Young [ 7 1 .
If
g i s assumed L i p s c h i t z continuous, then an e r r o r bound of 0(h 217)
could be obtained. ) This proof i s modeled on t h a t of Pucci :61 which i s q u i t e s i m i l a r t o t h a t of Bers [ I ] . Proof of Lemma 3: 1)
Note t h a t i f Lhvh
(or min vh = min v h ) . 8h aRh
2
0 (or
5
0 ) i n Rh, then pax vh = ma* vh h
aRh
2) and B
2
I f M ~ m a x ( m a x ( I a I t l a I j . m a x a f (max(Ia l ' + l a x 2 ~ ' ) ) ) R X1 X2 R R X1
+2+
M
ho
(rnin a)-1, then t h e r e e x i s t s R
ho
such t h a t 0 < h
r
imp 1i e s Lh(e Proof:
Odl
-
e
Bx
l) = L
3x ~ : , ~ ( e')(x1,x2).=
1 ,h
(e
Odl
-
e
Bxl
) r - 1 f o r L~ = ,)!,'L
a(x ,x ) V VeBX1 + 42[ a x1 ( 5 1 , x 2) o ~ Xl,hXl,h
+ ax z (min a ) [ e R f o r h small. 3)
*
i n (Rh)
L
l,h
(a2
* \,
then mgx /vhl
Proof:
2
mag
+K
lv,l
aRh 0 i n ( R ~ )and wh*vh
Thus, 3) holds i f we take K = e R: = Rh n R 6 , "X
Let
*
h1/5 Let Rh = Rh
.
2
I
-
h < ho, and Lhvh = f
mgx [ £ I .
+ (e
( 71
.e
*
t
i
(e2,x2)yx e 3Xl, 1 l,h
(5-2
aRh
Let wh=rnag lvhl
Lh(whTvn)
-
2
0 on 3%
-
.
.
Then
--
Therefore, lvh 1
5
lwhl
1.
((x ,x ) 1 2
c
R: d i s t ( ( x l , n 2 ) , a R )
2
5).
t
Then Lh(u-uh) = 0(h1I5) i n Rh and the constant i s
independent of h. Proof:
Schauder i n t e r i o r e s t i m a t e s (see [21) imply t h a t t h e r e e x i s t s
such t h a t lux (x1,x2) 1
if I(x1,x2)
-
~3X1,
and LO) can be t r e a t e d s i m i l a r l y . l,h
\
4)
0 (xl-h)
There e x i s t s h such t h a t , i f Rh c
,
o r L0) ~ .
: :L
(xi,xi)l
I +
6
Iux2(x1,x2)
I
< Kh-lI5 and
2h and ( x 1 , x 2 ) , ( x i x i ) E
%* .
Now l e t t i n g
.
hL (x1,x2) + -a
a(xl i h,x2) = a(x1,x2) i ha
1
= ( I ) t (11) t (111)
Using u(xlih,x2) = u(xl,x2)
(I):
we see t h a t vx (11):
(111):
- u(xl-h,x2)
=
1 2 huX1 (xl ' x2 ) + ~ ux h 1 1 .*' 2 ) ( ? t
2h ux(x1,x2)
These terms a r e 0(h4I5) i n
Jc
i n Rh.
l,h
-(
Now, (a(x1,x2)
.
RE .
'
+
2,h
+ O(h2t115)
-k
i n R~
<.
,X
1 2
) = 0(h1I5)
X1t 1'x 2 )'1)2 = a(x1,x2)
.
+ 7.(apu) -
= 0(h1/5)
2(a(xl,x2)a(xl+h,x~)'l
Using t h i s r e l a t i o n and the Schauder estimates, we get
( 2 ) u = LL1)u Lh
-
i
ou(x1,x2) = (x1,x2) t 0(h1l5) i n 1,h X l , h l'xlxl
Hence Lh(1) U(X1, X 2)
chL
#
Using the above expression for u(xl*h,x2), we see that
u(xl+h,x2)
5
(4*,x2), we get *lXl
+ 0(h4I5)
a(x1,x2 + a(xl+h,x2) 2
*
= 0(h1I5) i n Rh , Next using a(xl
-f
h
+T,~2)
O(h2 ) exactly as above, we get Lh( 3 ) i
Proof:
For (x1,x2)
c
*
0 0 l e t (x1,x2) denote t h e point on
Rh\Rh
0
Iu(x1)x2)
U ( X ~ X- ~uh)(X1 , x 2)I 0
0
0
0
and u(xl,x2) ' uh(xlYx2)
0
Note t h a t (x1,x2) E aRh and a l s o
aR which i s c l o s e s t t o (xl,x2).
0
-
0
u(x1)x2)
T ~ s ,lu(xl>x2)
-
I
' Iu(x1,x2) 0
0
-
uh(x1,x2)
I
I
u ~ ( ~ ~ ) x ~ )
0 0 I . TO estimate the I + Iuh(xl)x2) 2h1I5, h < 1, f i r s t term n o t i c e t h a t I (x1,x2) - x x 1 h t h 'I5 o o 1/51 where a i s the modulus and t h e r e f o r e lu(xl,x2) - u(x1,x2)I w(2h 0 0 ~) I we of c o n t i n u i t y of u. I n o r d e r t o estimate / U ~ ( X ~ )- Xuh(x1)x2) i
Iu(x1,x2)
-
0
0
u(x1,x2)
L
again make t h e change of v a r i a b l e vh(x1,x2) = ~ ( x ~ , xh (X ~ 1), X ~2 )u. 3-
Then, a s b e f o r e , Ahvh = f h , where f h = (nha2)uh Now, e s t i m a t i n g crudely, we g e t I f b [ h'
+
-
- 'l,h
L ~ ~ ) .) u ~
(a-')(,Li)-
C , C independent of h.
Next,
t V 2 , h 9 where
I V ~ , ~ ( X ~- ,v2,kx1,x2) X ~ )0 0
ch1I5.
i
-
I I.
By Lemma 2 the second term i s bounded by
-
Thus, i t remains only t o e s t i m a t e I Y ~ , ~ ( X ~ , vX~ ~, ~) ( x ~ , x1 . ; ) converges uniformly t o the s o l u t i o n l,h 5 and v = a 2 g on 3R which i s continuous i n R . dence,
This i s done by noting t h a t v v1 of Av = 0 i n
RO
we get 1vl,h(x1)x2)
-
0
0
'l,h (x 1, x 2 )I
+
0. This proves 5 ) .
Combining 31, 4 ) and 5) we g e t max lu-uhl Rh This completes the proof of Lemma 3 . 6)
-
0 as h
- 0.
6.
Remarks 1)
It seems q u i t e probable t h a t t h e arguments used here can
be extended t o n dimensions and somewhat more general equations 1 2 with a work estimate of O(h-"(log h- ) log c - I ) . I n the proof of Lemma 1 we were only a b l e t o prove the
2)
existence of the q and not estimate i t .
It i s , of course, t h i s
f a c t which makes t h i s procedure only an a b s t r a c t existence theorem.
It i s s u f f i c i e n t , by arguments i n M i l l e r ' s paper, t o f i n d q f o r c = 1 and i n t h i s case we can g e t a good estimate f o r q depending
ply
only on the constant of e l l i p t i c i t y f o r what seems t o be the worse possible case. f o r xl
That i s , of course, the case where a(xl,x2) = max a
*
c , a ( x x ) = min a f o r x 2 c . I n t h i s case 1' 2 q = (max a + min a ) 1max a . However, I have been unable t o show
-
t h a t t h i s i s t h e worst case and it seems t o be a q u i t e i n t e r e s t i n g problem.
3)
I g r a t e f u l l y acknowledge the encouragement and a s s i s t a n c e
of J i m Douglas, J r .
References
1. L. Bers, On mildly nonlinear p a r t i a l d i f f e r e n c e equations of e l l i p t i c type, J . Res. Nat. Bur. S t . , u ( 1 9 5 3 ) , 229-236. Friedman, P a r t i a l D i f f e r e n t i a l Equations of Parabolic Type, P r e n t i c e Hall, New York, 1964. (Chapter 3, Section 8 ) .
2.
A.
3.
J. E , Gunn, The s o l u t i o n of e l l i p t i c d i f f e r e n c e equations by semi-explicit techniques, J . Soc. Indust. ~ p p l .Math., Numerical x n a l y s i s , Ser . B, 2(1964), 24-45.
4.
K , M i l l e r , Numerical analogues t o t h e Schwarz a l t e r n a t i n g
5.
D . W. Peaceman and H. H. Rachford, The numerical s o l u t i o n of parabolic and e l l i p t i c d i f f e r e n t i a l equations, Journal of the Society of I n d u s t r i a l and Applied Mathematics, 3(1955), 28-41.
procedure, Numerische Mathematik
,
l(1965 1, 91-103.
-
6.
C . Pucci, Some Topics i n Parabolic and E l l i p t i c Equations, University of Maryland Lecture S e r i e s , No. 36, 1958.
7.
J . L . Walsh and D. Young, On the degree of convergence of
s o l u t i o n s t o d i f f e r e n c e equations t o the s o l u t i o n of t h e D i r i c h l e t problem, J . of Math. and Phys., 2 ( 1 9 5 4 ) , 80-93.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
J. R. CANNON and
J I M DOUGLAS
T H E APPROXIMATION O F HARMONIC AND PARABOLIC FUNCTIONS ON HALF-SPACES FROM INTERIOR DATA
C o r s o t e n u t o a d Ispra d a l 3-11 L u g l i o 1967
The Approximation of Harmonic and Parabolic Functions on Half-Spaces from I n t e r i o r Data J. R. Cannon (Minneapolis) and J i m Douglas, J r . , (Chicago)
1. I n t r o d u c t i o n .
The o b j e c t of t h i s paper i s t o d i s c u s s the
numerical approximation of functions e i t h e r harmonic o r parabolic i n a half -space given t h e approximate values of the functions on some subset of t h e open half-space. obvious.
Two t h i n g s a r e immediately
One i s t h a t , without some knowledge of the g l o b a l be-
havior of the f u n c t i o n s , nothing can be s a i d about the functions i n t h e whole half-space.
The second i s t h a t , i f t h e r e e x i s t s one
f u n c t i o n extending t h e d a t a t o the half-space, i n general t h e r e a r e i n £ i n i t e l y many.
Throughout t h e paper t h e n o t a t i o n i s based
on approximating any one of t h e s o l u t i o n s ; i t i s c l e a r t h a t t h e e r r o r e s t i m a t e s apply e q u a l l y w e l l t o every s o l u t i o n . The g l o b a l c o n s t r a i n t imposed on harmonic functions i s a convenient g e n e r a l i z a t i o n of boundedness (with a known bound) on the h a l f - s p a c e ; i n t h e p a r a b o l i c case the s o l u t i o n s a r e assumed nonThese d i f f e r e n t assumptions lead t o minor d i f f e r e n c e s
negative.
i n t h e arguments and r e s u l t s ; however, teach assumption could be employed i n e i t h e r c a s e .
Two cases a r e t r e a t e d f o r t h e measurement
F i r s t , i t i s assumed t h a t t h e f u n c t i o n i s measured with
of t h e d a t a .
a known accuracy on an e n t i r e hyperplane p a r a l l e l t o t h e boundary of t h e h a l f - s p a c e .
Later t h e d a t a a r e measured with a prescribed
accuracy on a r e c t a n g l e on t h e same hyperplane. -
-
-
-
- -
-
--
This r e s e a r c h was supported i n p a r t by t h e National Science Foundation and t h e Air Force O f f i c e of S c i e n t i f i c Research.
When the d a t a a r e given on a l l of t h e hyperplane, i t i s proved i n both the harmonic and p a r a b o l i c cases t h a t an approximate a n a l y t i c continuation can be found i n a p r a c t i c a l manner and t h a t the accuracy of the approximation on any s p e c i f i e d rectangle i n the open half-space depends i n a Holder-fashion on the accuracy of the measurement and the amount of computation involved i n the d i s c r e t e procedure.
When the data a r e given on a compact subset
of the hyperplane p a r a l l e l t o the boundary of the half-space, the s i t u a t i o n i s somewhat d i f f e r e n t .
I n the harmonic case the dependence
of the approximate s o l u t i o n on the d a t a i s very weak; i n f a c t , the a p r i o r i estimate of the e r r o r derived h e r e i n i s of such a nature a s t o preclude p r a c t i c a l c a l c u l a t i o n .
We do not know whether the
estimate can be improved; however, we do show t h a t , i f t h e d a t a surface i s r o t a t e d ever so t r i v i a l l y , no a p r i o r i e s t i m a t e i s feasible.
Indeed, even uniqueness f a i l s .
In the parabolic case
f o r d a t a on a compact subset of the hyperplane, our a p r i o r i e s t i mate of t h e e r r o r i s again weaker than t h a t f o r d a t a on a l l the hyperplane, but the e s t i m a t e does i n d i c a t e t h a t t h e computing problem i s practical.
No example i s known t o us t o i n d i c a t e t h a t t r i v i a l
r o t a t i o n of t h e d a t a s u r f a c e should lead t o f a i l u r e ; t n e autnors disagree a s t o the l i k e l i h o o d of such a f a i l u r e f o r parabolic functions Harmonic f u n c t i o n s i n two v a r i a b l e s a r e t r e a t e d a t considerable length i n s e c t i o n 2 and t h e r e s u l t s f o r two v a r i a b l e s a r e extended r a t h e r e a s i l y t o s e v e r a l v a r i a b l e s i n s e c t i o n 3.
Solutions of t h e
heat equation a r e t r e a t e d analogously i n s e c t i o n 4. A number of the r e s u l t s i n t h i s paper have been presented
[ 2 , 3 , 4 1 i n s e v e r a l e a r l i e r l e c t u r e s by one of the a u t h o r s ; however,
no proofs have been given and the theorems here s t a t e d a r e stronger than those given e a r l i e r .
I n p a r t i c u l a r , the r e s u l t s f o r d a t a
given on a compact s e t a r e much more p r e c i s e here. numerical experiments were reported i n [I].
Some r e l a t e d
2.
Harmonic Functions i n Two Variables.
Let
u be a harmonic
function i n t h e half-plane (y>O] having t h e r e p r e s e n t a t i o n
where (2.2)
P(x,y)= n
-1
2 2 -1 yCxty1
and u i s a signed measure such t h a t
where
p =
ut
-
u- i s t h e decomposition of u i n t o t h e d i f f e r e n c e of
nonnegative measures.
The c o n s t r a i n t .(2.3) i s t h e global
r e s t r i c t i o n imposed upon u t o f o r c e continuous dependence of u on t h e d a t a t o be s p e c i f i e d l a t e r . Assume f i r s t t h a t u i s known approximately on t h e l i n e
where g(x) i s continuous.
L a t e r t h i s condition w i l l b e modified.
It i s c l e a r t h a t
t h u s , only t h e values of u on the s t r i p (O
u on t h e s t r i p {6syO. Note t h a t , i f msx+Xsmtl, mtl kt1 P(x-r,Y)dp(~)t 1 P(x-~,y)du(~) xtX k=mtl
I
I
by (2.3)
< ~ ~ - ' Y M [ x I - ' ,~ 2 2 ,
.
Thus,
where C denotes a constant depending on Y. *1, + 2 ,
. . . , where
h = n-l
,n
an i n t e g e r .
Let xi =
F~ =
ih, i=O,
I f m and q a r e integers
such t h a t msxi<m+l and qsxi+X
X
P
mtl
i FY
)
i
q-1 k+l ) = ,[ P(xi- F ,Y)du(?)+ L ,f P(ximF, Y ) ~ w ( ? ) k w l k Xi
(2 .a)
xi+x
+ .r
Then, from (2.3) it follows t h a t mtl
P(xi-F , Y ) d v ( ~ .)
4
I ,r P ( x ~ - F , Y ) ~ w (-F )i X
P(x.-F. , Y ) ~ ( (, FF ~~ + ~1 ) ) 1 J x. se . < e l
i
1
3
(2.9)
s
a P (xi-E ,Y) I. max ks~sktl
- 200 Since Px(x ,y)=
- 2n"xy(x
2+y 2)- 2
,
L
Hence, it follows from (2.7)- (2.11) t h a t
since (2.13)
max IT3 P (xi-? ,Y) xi?;~5mtl
I+
m
C
k-1
mar I-3P kscrk+l "
xi-^ ,Y) ( LC.
L e t A denote t h e c l a s s of a l l sequences [(ai ,bi)) = a such t h a t
C (a.+b )sM, rise .
n = 0 , + l ,k2,
....
J
For a € A s e t
The sequence (ai] r e p r e s e n t s t h e d i s c r e t i z a t i o n of t h e unknown measure
u+ and (bi) t h a t of p ; n o t e t h a t , a s a consequence
of (2.7)
, t h e Poisson i n t e g r a l can be truncated w i t h a
predictable
e f f e c t on t h e s o l u t i o n .
data on ( y = Y) a s follows.
Now, f i t u(x,y;a) t o t h e
- 201 Set (2 :16)
e(a) =
sup l u ( x i , ~ ; u ) - g(xi) I ,a € A. -w
For a E A t h i s supremum
i s f i n i t e , s i n c e both u(x,Y;a) and
g(x) a r e bounded a s a r e s u l t of (2.3) and i t s d i s c r e t i z a t i o n (2.14).
Let
since
i ( P ( r F j , c j + l ) ) , i ( [ , F~ ~ ~+ ~ ) ) E) )A , (2.4) and (2.12)
+
imply t h a t
What i s wanted i s a s o l u t i o n of t h e minimizdtion problem given by t h e equation e(a) = [ f o r some a E A ; b u t , s i n c e t h i s i s an infinite-dimensional l i n e a r programming problem, i t i s not a t a l l clear that a solution exists.
However, (2.12) implies t h e existence
of a ' € A such t h a t
although i t i s not obvious how one would o b t a i n such an a ' . For t h e moment assume t h a t an a ' has been found and s e t
The function v i s then t h e approximation t o u i n the s t r i p {O<~
Note t h a t s i n c e terms appear and disappear from
v a s x moves from nor harmonic. L e t
-at0
-,consequently, v i s n e i t h e r continuous
z(x,y) = c P(x-4 ,y)(a.-b.) , { ( a . b . ) ) = a ' ; -a,
(2.21)
z i s both continuous and harmonic i n {y>07. Moreover, i t follows from (2.6) t h a t (2.22)
Iz(xi,Y)-v(xi,Y)
11
CM[XI-l, i = 0 1 , *2,.
..
From ( 2 . 4 ) , (2.19), and (2.22),
The g l o b a l c o n s t r a i n t s (2.3) and (2.14) imply t h a t ux and zx a r e bounded on {y = Y] by a m u l t i p l e of M depending only on Y . Hence,
The bounding of z-u i n t h e s t r i p { O < y < ~is ) facilitated by t h e following lemma. Lemma 1: Let w(x,y) be harmonic f o r y>O and l e t w have the r e p r e s e n t a t i o n (2.1).
I w(x,Y) I <
E
Let
1 1 ~ ([n,n+l))IM, 1
and rp0, t h e r e e x i s t
8
0
n = 0,*1, k2,.
.. .
If
>O and a constant C such t h a t
f o r 0< c < cO. Proof:
It i s easy t o s e e t h a t i t i s s u f f i c i e n t t o bound w(0,y).
Let O
- 204 Then, g i s a harmonic function t h a t i s nonnegative on t h e segments { ( x , ~ ) :
(2.34)
-
ASXs8,y
a
6 1 and E(x,y): x = * 0 ,
bsysY+b), and
g(x ,Y+b) 2- l o g
A harmonic minorant f o r g i s given by t h e harmonic function vanishing
on t h e t h r e e lower s i d e s of t h e r e c t a n g l e and equal t o y c o s 1 ~ x / 2 ~ on t h e upper s i d e . (2.35)
Thus,
g(x , Y ) >
y ~s i n~h 9 ~ E,
-
and (2.36)
i f (iy) 1
a
9
si n h g
[11+4n
s i n h n(y- 6) 128
-1 -1
K
6
8)r
~ i n hnY/20
1 2 2 2 Since sioh qxlsinhqy = xy-l[1+6- q (x -y )+. . .I,
provided t h a t
Choose 9 s o t h a t t h e second i n e q u a l i t y holds. 6 c min(q,kYq).
F i n a l l y cnoose co so t n a t Beo
Then choose
M.
Tnen,
K s c0nst.M dnd i t follows t h a t (2.39)
Iw(O,y) I s l f ( i y )
Is
1-.pl
$7
CM
, T ~ Y ~ Y ,
and t n e lemma nas been proved.
The r e s t r i c t i o n t o small c i s n o t important, but i t is o n l y i n that case t h a t the r e s u l t is useful.
The t r i v i a l example
- 205 u = e - Y s i n x shows t h a t t h e lemma i s c l o s e t o b e s t possible. The a p p l i c a t i o n of Lemma 1 t o t h e f u n c t i o n z-u i s immediate, s i n c e
It f o l l o w s from (2.24) and (2.40) t h a t
R e c a l l t h a t z d i f f e r s from v ( x , y )
=
u ( x , y ; a l ) , t h e approximate
harmonic c o n t i n u a t i o n , by n o t more t h a n c M [ ~ 1 - ' a l o n ~t h e l i n e {y=Y).
It i s e a s i l y s e e n t h a t
Thus, t h e f o l l o w i n g theorem h a s been proved. Theorem 1. L e t a ' E A be a n y s o l u t i o n of (2.29) and d e f i n e ~ ( x , ~ ; a by ' ) (2.15).
I f q > 0 and i f c
, xel,
and h a r e
s u f f i c i e n t l y s m a l l , t h e r e e x i s t s a c o n s t a n t C such t h a t
provided t h a t u has t h e r e p r e s e n t a t i o n (2 . l ) and ( 2 . 3 ) and (2.4) hold. Note t h a t t h e e r r o r e s t i m a t e i s i n terms of t h e g l o b a l constraint M , t h e measurement e r r o r L : , and t h e parameters h and X
of t h e approximation.
Obviously, t h e d a t a need have been known
o n l y a t t h e p o i n t s (xi,Y), b u t t h i s i s a t r i v i a l r e d u c t i o n .
As was remarked on e a r l i e r , t h e infinite-dimensional l i n e a r programming problem (2.17) imposes s e r i o u s p r a c t i c a l l i m i t a t i o n s . I f an approximation t o t h e s o l u t i o n u i s d e s i r e d on t h e r e c t a n g l e R = { / x 1 <XI, v
values of ai and bi assigned a t p o i n t s (xi,O) a t a g r e a t d i s t a n c e from R should have a n e g l i g i b l e e f f e c t on t h e approximation i n R and, consequently, could be s e t equal t o zero.
programming problem would r e s u l t . t h e proof seems n o n t r i v i a l . be presented.
A f i n i t e linear
The i n t u i t i o n i s c o r r e c t , but
Two arguments, both complicated, w i l l
The f i r s t method w i l l g i v e a b e t t e r e r r o r e s t i m a t e ,
but t h e harmonic conjugate
w i l l be introduced, l i m i t i n g t h e
argument t o two-dimensional problems.
The second argument a l s o
i s based s t r o n g l y on complex a n a l y s i s , but the a n a l y t i c function a r i s e s from extending an independent v a r i a b l e t o t h e complex domain.
This method of a t t a c k can be a p p l i e d t o harmonic functions
i n s e v e r a l v a r i a b l e s and t o s o l u t i o n s o f t h e h e a t equation. pays f o r t h e g e n e r a l i t y by obtaining weaker
One
error estimates.
During t h e argument t o be given below ' q u a n t i t i e s X2 ,X3 ,X4, and X 5 , i n a d d i t i o n t o t h e X1 of t h e d e f i n i t i o n of R , w i l l be introduced. 'Xi ,i = 2 , .
..,5,
w i l l tend t o i n f i n i t y i n obtaining
t h e e s t i m a t e s , and t h e following r e l a t i o n s w i l l hold: X1< X5< X4< X3
= X4
+ X2.
More p r e c i s e requirements f o r t h e s e terms w i l l
appear l a t e r . Retain t h e r e p r e s e n t a t i o n (2.1) (2.4).
,
(2.3) and t h e measurement
The e s t i m a t e (2.12) c a r r i e s over i n t h e form
where X2 > 0.
Let X3 = X2
+ X4,
r e p r e s e n t t h e sequences a =
where X4 > X1.
(ai ,bi)
Let A now
1, where
Set (2.45)
and l e t (2.46)
e(a) =
max lu(xi ,Y;u)-g(xi) Ixi l cx4
1,a
E A.
t h e determmation of an a ' E A such t h a t e ( a l ) = 5 i s a standard f i n i t e - dimensional l i n e a r programming problem.
It follows
again t h a t
s i n c e t h e e r r o r i s t r e a t e d only f o r ( x i 1 not s e t t o zero u n t i l [ x i [ > X4
+ X2.
X4 and the weights a r e
Again s e t v(x,y) = ~ ( x , ~ ; a ' )
f o r some s o l u t i o n a ' of t h e mimimization problem and s e t
(2.49)
z(x,y) =
c P(x-F y ) ( a . - b . ) , jy J J I F j la3
[(a. b.)] = a'. J' J
As b e f o r e , i f w = u-z,
C l e a r l y , w i s harmonic i n (y>O) and W(X,Y) = . ~ P ( XY- ~Y ) ~ v ( F ) , (2.51)
I v 1 ([n ,n+l)) 5
Since Iwx(x ,Y) I
ICM,
2M.
(2.50) holds f o r a l l x E [-x4,x4j with a
different C. The remainder of t h e a n a l y s i s of w i s s i m i l a r t o t h a t given i n Lemma 1 , but t h e l i m i t a t i o n of t h e bound i n (2.50) t o { l x l < ~ leads ~) to certain additional features.
First, it i s clear
that (2.52)
IW(XYY) 1s CM, I x k
Now l e t X1< X5< X4.
x4.
Then, i f 6 > O ,
Similarly, (2.54)
-
1 J w ~ ( x , Y + ~46) I ~yl
+-2CM
X4 X5
=
- y3.
Ixlsx5
.
C l e a r l y , Iwx(x,y) 1s C(6)M, yr6>0.
The estimation of w i s reduced
t o e x a c t l y t h e same problem a s was t r e a t e d i n Lemma 1, except t h a t i t is not s u f f i c i e n t t o look a t w (0 ,y)
.
The r e c t a n g l e used
t o estimate t h e harmonic function g introduced i n (2.33) should be centered i n x a t an a r b i t r a r y x
* conjugate w should. vanish
0
E [-X4
,X41, t h e harmonic
a t (xo ,Y+6), and 13 should b e replaced
The constant K of (2.32) should be replaced by
by X5-X1.
Choose 6 s rnin(n,%Yq) and X5 such t h a t s inhn (y- 6) /2 (X5-X1)
(2.56)
s innrrY/2 (X5-X1)
Then, choose
Xi
and
Xi
1 Y
-
q , y26.
s u f f i c i e n t l y l a r g e and h ' s u f f i c i e n t l y
small t h a t
For X 2 s
Xi,
(2.58)
Xi,
and hsh' , i t follows t h a t 1-f + q I y -, Iw(x :Y) 1s CM '3 Y2T, lxIs
X4r
xl.
I t i s easy t o s e e t h a t
Theorem 2 .
Let a ' E A be a s o l u t i o n of t h e l i n e a r programming
problem (2.47) and d e f i n e u ( x , ~ ; a ' ) by (2.45). Then t h e r e e x i s t X5 > Xl and Xi > 0 ,
if X2
2
xi, x4 2 x;,
Xi
Let . q > 0.
> X 5 and h ' > 0 such
and h r h'. Moreover, v3 can be bounded a s
i n (2.59). In p a r t i c u l a r , t h e e r r o r i n t h e approximation depends i n a
Holder-continuous fashion on t h e measurement e r r o r s and t h e amount of c a l c u l a t i o n performed, a s indicated by h , X 2 , X4, and X5' A second deviation of an e r r o r estimate can be based
on t h e extension of t h e v a r i a b l e y t o t h e complex domain,
*
(2.60)
a = y + iy.
This a n a l y s i s begins w i t h t h e r e l a t i o n s (2.50) (2.52).
,
(2.51)
, and
Note f i r s t t h a t
Consider t h e i n t e g r a l
Since
+
~ ( x - F ) ~ += u( x~ - ~F )~~
and
1 (x-F) 2 +
(y2+y* 2)2
y2 I2 2 y4, t h e denominator
+ 2(x-E)2 (y2- y *2 )
of P(x- F ,a) i s bounded
away from z e r o f o r any a such t h a t Re u > 0. converges f o r any x and any o such t h a t Re
D
Thus, t h e i n t e g r a l > 0 , and w(x ,o) i s ,
f o r each x , a holomorphic function of a i n t h e half-plane ( ~ ue > 01.
- 211 -
r;'
Moreover, f o r y > 0
,
Hold x f i x e d i n t h e i n t e r v a l [-XI ,Xll and consider t h e r e c t a n g l e Q = (Y r y
5
Y1,
0 < y
?;
< Y1
J(
Then, i t follows from
1 i n t h e a-plane.
(2.61) and (2.63) t h a t = 0, u
(2.64)
E
2Q
E ?Q
,C
=
*
C(Y,Y1,Y1 ) .
I t then follows from (2.31)- (2.36) t h a t
where
* *
*
(2.66)
a(y) =
s inh n (Y1-y ) / (Y1-Y) sinhfl;/(yl-y)
1 Now, consider t h e r e c t a n g l e Q ' = bay9(YtYl)
*
*
*
*
O s y
* , where ,/ y* I rY2)
and 0 < Y2 < Y1 and x remains f i x e d . The same r e s u l t holds f o r y < 0. Then, 1-a a 1 ~4 , Y = 7 (Y+Y1) , a E "',a = a (Y2), (CM) (2.67) Iw(x,o) 1s 1 C(6,1(YtY1), Y2)M, a i Q' .
0 < 5;
q
*
*
Hence, (2.68)
(w(x,y) 1 1
1 [ ( c M ) ~ - ~ Y ,~lxIsx1 I ~ ( ~ ,) 6syq(YtYl)
,
where
*
(2.69)
P(Y)
s i n h n (y- 8) /2y2 i
s i n h ~ ( + ( Y + Y ~6))12~: -
Since R = { ( x1s X1, q r y r Y) , there e x i s t a constant C and a positive number 8 such t h a t
The remainder of the argument leading t o a theorem of the same type a s Theorem 2 i s the same a s before, although i t i s c l e a r t h a t the exponen R
i s not so l a r g e a s before.
However, the advantage of t h i s approach
w i l l be seen immediately below, a s well a s l a t e r i n several other
app 1ica t ions. So f a r it has been assumed t h a t the data g(x) has been known f o r a l l x.
Clearly, i t i s not p r a c t i c a l t o measure g f o r d l 1 x ;
consequently, it i s both p r a c t i c a l l y useful and mathematically i n t e r e s t i n g t o l i m i t the data t o the following:
the choice of the subscript being made t o coincide i n the argument below with the previous X4.
Again l e t us approximate u on the
rectangle R , but i t i s not assumed t h a t X1 < X4.
I f the approximation
i s made a s before, r e l a t i o n s (2.43)- (2.52) remain v a l i d , although X4 i s fixed now and not a parameter t o be chosen. The estimation of the e r r o r w w i l l involve f i r s t the extension
of the x-variable t o the complex domain and then the y-variable. Let
.g
= x+ix
* and
consider the i n t e g r a l
which can e a s i l y be seen t o converge and t o be holomorphic
x for
* Ix I
< y.
In fact, i f
* Ix I
s pY, O < p
in
< 1, then
and
Thus, t h e holomorphic function w (y ,Y) s a t i s f i e s t h e following bounds :
By t h e argument leading t o (2.36)
,
I W ( ~ * , Y1s) (C(P)M)
1-a(x*) a(x*) . Yl '
By symmetry t h e above r e s u l t holds with y replticed by 1 y f o r 1 y / r
P
Y.
Thus, w (.c ,Y) s a t i s f i e s t h e fallowing c o n s t r a i n t s on t h e boundary of the half-strip
i / Imr
r
yo
2
01:
Let x be fixed so t h a t x > X4, and l e t xl > x.
Then, by the
same drgument a s above,
(2.78) @(x,xl) = sinh
n(xl-x) 2yo
/ sinh rrxl 2yo
,
The r a t i o e(x,xl) i s an increasing f u n c t i o n of xl and has the l i m i t
a s x 1 tends t o i n f i n i t y .
Hence.
The argument leading from (2.50) t o Theorem 2 can be repeated
*
with the symbol X4 replclcing the parameter X4 i n the e a r l i e r argument.
Let X
?t
be chosen so t h a t (2.56) i s v a l i d and Let l l < X 5 < X 4 .
5 It follows t h a t the q u a n t i t i e s vl,v2, dnd v
3
defined i n (2.501, (2.531,
and (2.54), r e s p e c t i v e l y , can be replaced a s follows:
The r e l a t i o n (2.58) becomes
Note t h a t t h e estimate (2.83) contains i n p a r t i c u l a r two parameters, >t
X4 and X5, t h a t were introduced i n t o the a n a l y s i s f o r convenience
but which a r e not involved i n the c a l c u l a t i o n .
Obviously, these
parameters should be s e l e c t e d so a s t o minimize the bound.
The
minimum i s c l o s e l y approximated by equating the two a d d i t i v e terms i n the bracket:
As
P
+ xi1 + h
tends t o zero,
* nX4 2yo
log l o g ( € 1
+ xi1 + h)
2yo -7 log
a(y0)
* - X5)
l0g(X4
nx4
Consequently,
The constant C depends on M,p, 6,yo, and 7 . The i n e q u a l i t y (2.87)
implies the following r e s u l t . Theorem 3. Let a ' E A be a solution of the minimization problem (2.44)
- (2.47),
where X4 is defined by (2.71).
defined by (2.45).
Let u(x,g;a? be
Then, there e x i s t positive constants C,Y, and 6
depending on tne rectangle R,X4, and M such that
provided t h a t u i s a harmonic function i n the upper half-plane having the representation (2.1)
- (2.3).
The e r r o r estimate above is not adequate f o r practical usage, although it does e s t a b l i s h a quite weak continuous dependence relationship.
It i s perhaps of some theoretical i n t e r e s t t o consider
the asymptotic form of the estimate a s X4 tends t o zero.
As X4
tends t o zero,
and
becomes small quite rapidly.
It should a l s o be noted t h a t
the method of analysis used t o derive (2.87) could be extended to give a unique continuation tneorem f o r harmonic functions i n a half -plane. One consequence of Theorem 3 is a u~~iquenees theorem for harmonic functions.
Let
be a l i n e segment of length 2X4 centered a t (0,Y)
and making an angle 9
with
the x-axis.
Theorem 3 implies that a
narmonic function t h a t is bounded. on l;y>O] and is constant on (i.e., Q=O) is identically constant. This i s f a l s e f o r 9 # 0. example, l e t La be tne l i n e determined by section with the x-axis.
Let
ra and
r0 For
l e t xa be i t s i n t e r -
i s a l e v e l curve of u This i n d i c a t e s t h a t t h e weak continuity 8 8' r e s u l t i n g from Theorem 3 i s t o be expected. It a l s o implies t h a t L
no a p r i o r i estimate can be obtained from data given on Tg
.
3.
W
i
c Functions i n More than Two v a r i a b l e s .
The r e s u l t s
of the l a s t s e c t i o n can r a t h e r e a s i l y be generalized t o harmonic Let x = (xl, , . , ,xn) E R n 1 Let ~ ( x , ~ ) = c ~ ~ be( the ~ nPoisson ~ ~ tkernel ~ ~ f)o r~ ~
functions i n more than two v a r i a b l e s . and y E R1.
the half -space {y>O]. Let u be a harmonic function i n the h a l f space {y>O] having the r e p r e s e n t a t i o n
Denote t h e cube { sicx i<sit l ) , s i an i n t e g e r , by S. (3.2)
I ~ I ( s )r M ,
Also, assume t h a t , f o r some Y r 0
Assume t h a t
all S .
,
Clearly, (3.1)-(3.3) a r e t h e d i r e c t analogues of (2.1)-(2.4). Again l e t
be the r e c t a n g l e on which i t i s d e s i r e d t o approximate u.
The
approximation w i l l be accomplished i n p r e c i s e l y the same fashion a s before and the method of proof w i l l be a straight-forward gene r a l i z a t i o n of the e a r l i e r proof based on extending y t o the complex variable a = y
+ i y* .
Let I = ( i l , . . . , i n ) be a multi-index of i n t e g e r s , x i = i h , and 1 j . Then i t i s xI = (X , . . . ,x: ) . s e t zI = (x! < xJ < il n j J easy t o see t h a t
Let A denote the sequences a = { ( a I , b I ) ) such t h a t aI,bI
2
0
, all I ,
a I = bI = 0
i f m a x l ~ iI >
,
j
zI
c (aI+bI) ~ S
s
M
,
all S
x,,
=
x2. + x 4 ,
.
For a E A, s e t (3.7)
u(x,y;a) =
C
P(x-51,~)(aI-bI)
I'iJ - x j ICX,
,Y
> 0
,
j
and l e t (3.8)
e(a) =
max lu(xI,y;a) [ x i I <x, j
-
g(xI)j
.
Then,
and again a s o l u t i o n a ' E A of the minimization problem can be obtained by l i n e a r programming.
Set v ( x , y ) = u ( x , y ; a t ) f o r some
s o l u t i o n a ' and s e t
Then, v i s the approximation t o t h e harmoaic function u.
As before,
-
220
-
v i s not harmonic, but z i s and d i f f e r s from v on R by
-
i t i s s u f f i c i e n t t o estimate w (x,y) = u(x,y)
cMXil.
Thus,
z ( x , y ) on R.
The harmonic function w s a t i s f i e s the following r e l a t i o n s :
Iv(S) I s 2M
(3.11)
, all
S
,
I W ( X , Y )I~ 2c t CM(X;'+~) IW(X,Y) I g CM
r
yl
,
I X ~ I r x4
,
all x
,
.
Repeating the argument beginning with (2.60) with o a s before and Y
2
y sY1, we find t h a t
The i n t e g r a l
converges and i s holomorphic f o r Re a > 0, and I w(x, a ) 1
* *
Hold x fixed and s e t Q = {YsyaYl,'Osy
6 i y < Y1.
*
CM i f
The
r e l a t i o n s (2.64) and (2.65) again hold with a ( y ) s t i l l given by (2.66).
Define Q ' a s follows (2.66).
The remainder of the argument
i s unaltered and
where
f~ >
0.
Cleariy, X4 can be chosen so t h a t maxlw) s C(M,R)(C+X;1t h ) B R
.
In any event, the e s t i m a t e s i n the two v a r i a b l e case c a r r y over e s s e n t i a l l y unaltered t o the higher dimensional case.
The argument given i n the two v a r i a b l e case f o r the d a t a known only on an i n t e r v a l can be c a r r i e d over, but t h e dependence on the data and parameters remains weak.
4.
Parabolic Functions.
The approximation of s o l u t i o n s of the
heat equation from i n t e r i o r d a t a can be accomplished i n a fashion very s i m i l a r t o the method applied t o harmonic f u n c t i o n s .
It i s
s u f f i c i e n t ro consider t h e case of a s i n g l e space v a r i a b l e ; t h e extension t o s e v e r a l space v a r i a b l e s i s a s immediate a s i n the harmonic c a s e . The most n a t u r a l g l o b a l c o n s t r a i n t t o apply t o a s o l u t i o n u of the heat equation i s t h a t i t be nonnegative, s i n c e temperatures a r e n a t u r a l l y bounded on one s i d e along with a l l the o t h e r common physical q u a n t i t i e s s a t i s f y i n g parabolic equations.
In addition,
l e t us assume t h a t u i s bounded on each half-space Tt r to > 0) by M(to), where M(T) = 1 and M(t) i s otherwise unknown:
Assume a l s o t h a t
It follows from a theorem of Widder [51 t h a t
where
$ -xL/4t
K ( x , t ) .= ( 4 n t ) - e
(4.4) and u =
+.
Now,
1 2 u(nG,'f)
ntl 2
n
~(nG-t,~)dp(~)
Hence,
Thus, the constant M assumed i n the harmonic problem has been c a l c u l a t e d under the n a t u r a l assumptions above.
( I f the harmonic
function had been assumed nonnegative, the same c a l c u l a t i o n would have been v a l i d . ) It i s easy t o see t h a t (4.7)
0 s u(x,T)
Also, (4.8)
IJ
-
-x;/~T ,
x-X2
K(x-5,T)dW(r) 5 CMe
xi+x2 xi-X2
~(x~-:,~)d~(e) r Ix.-5.
-
1
I<x 3
K ( x ~ -. FT ~ ) u ( [ F ~C,j + l ) ) I 2
s CMh.
Let A denote the s e t of sequences a = f a . ) such t h a t 3 a 2 0 , all j j (4.10) aj = O , and s e t
1x.l 3
>x3=x2tx4,
Note t h a t t h e c o n s t r a i n t (4.6) does n o t appear i n a d i s c r e t i z e d form i n t h e d e f i n i t i o n of t h e admissible sequences A; only the nonnegativity i s r e q u i r e d .
Let
It follows from (4.9) t h a t
(4.13) Let a
'
= inf e ( a ) < E t CM(h + e
2 -X2/4T
1.
= (a.'] be a s o l u t i o n of t h e optimization problem (4.13) and
J let v(x,t) = u(x,t;al).
I n d i s t i n c t i o n from the previous harmonic
c a s e s , v ( x , t ) i s not t h e d e s i r e d approximation t o u.
Again, l e t us
attempt t o approximate u on t h e r e c t a n g l e R = (1x1 r XI,
6 r t
5
Ti,
noting t h a t i t i s again t r i v i a l t o e s t i m a t e u f o r t > T.
It follows from (4.13) t h a t "
Consequently, i f X2
: !
1,
1I s 1h; then, Choose x 1. s o t h a t [xi-n- '2 '2 (4.16)
2 -X2/4T 1 a' r ~(#,~)-~[1+2ctC~(h+e 13 ns! . < n t l j J
2
1 1 ; (4.16) i s t h e d i s c r e t e analogue of i f [ n , n + l ) c ( 1x1 s X4+Z-7h) (4.6) and i s a consequence of t h e procedure f o r 1x1
Ix44-:h.
This
is the reason t h a t some v e r s i o n of (4.6) was not imposed i n t o
the d e f i n i t i o n of A . Since X2 grows r a t h e r slowly t o enable the 2 CM exp(-X2/4T) term t o become small, it i s not too r e s t r i c t i v e t o require that
Set (4.18)
"1
, otherwise ,
=
and l e t (4.19) The function V i s the approximate s o l u t i o n of the continuation problem.
It follows t h a t
-
v(x,t)
1
(4.20)
(v(x,T)
s CMe
and w(x,t) = z ( x , t )
-
(4.22)
0 s v ( [ n , n t l ) ) r 3M
2 -X2/4T
,
1x1
g
X5
.
u ( x , t ) , then
/ w ( x , ~ ) Ir y1 = 2c Iw(x,T)
I
< CM
+
,
xi1 and
h s u f f i c i e n t l y small, -x;/~T CM(e t h ) , 1x1 r X 5 '
, all x
.
The e s t i m a t i o n of w on R w i l l follow the harmonic argument based on extending y t o the complex domain.
F i r s t , note t h a t
i f T < t s T1, T1
-
T < ~ ( x ~ - x and ~ ) ~1x1,
Let o = t
X1.
I
+ it* .
Then, -2 *2 IK(x,o)I s ~ l + i t * / t ~ ' ~ ( x , t ( l + tt )),
(4.24) and (4.25)
w(x,o) =
/ K(x-!,a)dv(~)
,
Re
0
> 0
It follows from
i s holomorphic i n o i n Re a > 0 f o r each fixed x. (4.22) and (4.24) t h a t
* *
Let x be f i x e d , 1x1 r XI, and l e t Q = (Tit
*
*
I W ( X , ~ ( T + T ~ ))I+ I I~ (CM) 1-a(t*) Y2 a(t )
*
a(t ) =
,
* * )/(TI-T)
s i n h n(T1-t
s i n h n ~ i (TI-T) /
9
*I *
* *.
1 by the proof of Lemma 1. Let Q ' = ( + ~ S ~ ~ ( T + T 1 t ~ )sT2), , T2
*
Then,
1-o ( T ~ a) (T;) Y2
(4.28)
Iw(x,a)l s
c
,,
ocaQ1
,
1 a€a~'n(ty(TtT~))
.
Thus, by another a p p l i c a t i o n of the proof of Lemma 1,
*
B(t) =
In particular
*
s i n h n(T+T1-6)/4T2
,
f o r some a > 0 Theorem 4.
1 sinh ~ ~ ( t - ~ 6 ) / 2 T ~
.
Let a
'
E A be a s o l u t i o n of t h e minimization problem
(4. l o ) , (4.12), (4.13), and l e t u(x, t ; a ') be defined by (4.11).
Then,
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 , c and C depending on the rectangle R and M such t h a t
P,
The approximation of a parabolic function u from d a t a on a f i n i t e interval,
can be c a r r i e d out i n a f a s h i o n q u i t e s i m i l a r t o t h a t f o r a harmonic f u n c t i o n ; however, t h e dependence of the s o l u t i o n on a r e c t a n g l e R = {brtsT, l x l r ~ on ~ ) t h e d a t a i s somewhat stronger i n the p a r a b o l i
case.
Again l e t
x
= x
+ i x* and
set
Since IK(x, t ) in
x
1
< ~ ( xt ), e x *214t, it follows t h a t w ( ~t, ) i s e n t i r e
f o r each t > 0 and t h a t l ~ ( ~ , t )s I CM i f 0 c 4 r t , lx
*I
i
*.
X1
The r e l a t i o n analogous t o (2.76) i s
*
*
*
Iw(ix , T ) [ r (CM) l - a ( x I y1 a(x ) (4.33)
*
a(x ) =
* *
,
s i n h n(X1-x )/2X4 s inh nxf/2x4
9
since (4.22) implies t h a t w i s bounded by yl on (t=T, 1x1s x 4 ) . k;
0 < X2
* < X.
1'
If
i t again follows t h a t
*
Since X1 i s not r e s t r i c t e d , then the bound (4.34) can be put i n the form
f o r any
7
The argument leading £rod (4.23) t o (4.30) can be
> 0.
repeated with
a = a(R) > 0.
yl
replaced by the e s t i m a t e (4.35).
Hence,
If the q u a n t i t y i n the brackets i s approximately
minimized by equating t h e two terms, i t can be seen t h a t
Hence,
where c and C a r e p o s i t i v e .
Note t h a t t h i s implies, i n p a r t i c u l a r ,
that maxlwl R
5
-1 -M C(logyl
,
any M ,
i f y1 i s s u f f i c i e n t l y small. Theorem 5.
If a ' E A i s a s o l u t i o n of t h e minimization problem
(4.10), (4.12),(4.13) f o r f i x e d X
> 0 and i f u ( x , t ; a l ) i s defined 4 by (4.11), t h e r e e x i s t p o s i t i v e constants c and C depending on
R, M, and X4 such t h a t
While the e r r o r bound derived i n Theorem 5 i s not q u i t e a s strong a s Holder c o n t i n u i t y , i t i s nevertheless s u f f i c i e n t l y strong, t o be of p r a c t i c a l use. It i s c l e a r t h a t the r e s u l t s above extend t o the case of s e v e r a l space v a r i a b l e s .
References
1. Cannon, J. R . , Some numerical r e s u l t s f o r the s o l u t i o n of the heat equation backwards i n time, Numerical Solutions of Nonl i n e a r D i f f e r e n t i a l E uations, e d i t t e d by D . Greenspan, John w a n d Sons, Inc. , ew York, 1966.
+
2.
Douglas, J r . , J r . , Approximate harmonic continuation, A t t i d e l Conve no su & e uazioni _alle d e r i v a t e p a r z i a l e , Nervi, 2 h b ~ a i bo z l o n l Crernonese , Roma .
3.
, Approximate continuation of harmonic and parabolic i u n c t i o n s , Numerical Solution of P a r t i a l D i f f e r e n t i a l I n c . , New E u a t i o n s , e d i t t e d by J. H. ~ r a m a z m i Press, c -6.
4.
, The approximate s o l u t i o n of an unstable physical problem s u b j e c t t o c o n s t r a i n t s , Functional Analysis and 0 t i m i z a t i o n e d i t t e d by E. R. C a i a n i e l l o , ~ c a d e m l cP r e s s , ~ . h i 9 6 6 .
5.
Widder, D. V . , P o s i t i v e temperatures on an i n f i n i t e rod, Transactions of the American Mathematical Society 2 (1944) 85-95.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M.E. )
B. E. HUBBARD
"ERROR ESTIMATES IN THE FIXED MEMBRANE PROBLEM
C o r s o tenuto ad I s p r a dal 3-11 Luglio 1 9 6 7
ERROR ESTIMATES IN THE FIXED MEMBRANE PROBLEM
B. E. HUBBARD 1) E s t i m a t e s of the discretization e r r o r in the fixed membrane problem appear to be considerably m o r e difficult to obtain than for the Diri-
6
chlet problem. F o r example, the detailed e s t i m a t e s by Ssulev s t r a t e t h i s point
. I shall
will illu-
give h e r e a particular d i s c r e t e analogue which i s
a natural one to considcr and yet one whose e r r o r analysis i s easily r e l a ted to that f o r the Dirichlet problem.At the s a m e t i m e we shall considerably rc,ciucc the usual regularity assumptions in o u r e r r o r analyses. The r e s u l t s presented h e r e a r e d r a u n , f o r the m o s t p a r t , f r o m a p a p e r
2
with J . H . Bramble. Let
R
be a bounded domain in
E
n
. We shall approximate
the
problem
with eigenvalues O <
A 1 < A2 < ...
and eigenfunct~ons u l , u 2 , .
. . by
tilose
of the d i s c r e t e problem
where
Ah
The o p e r a t o r s For
where
a Rh ,
R = R i U R and h I1
I. = I
direction and
a r e those defined by Bramble
2 is the usual O(h ) approximation a t polnts
.
1
x q ; .
x E R I 1 we define
0
. 110,
0
1
.. 0
. pj <
1,
is the vector of 1en;:th
h
to admit tile possibility that
l ) ~ l l i v e r s l t y of Maryland, supported in p a r t b,; Mathematik, E. T. I-I. Zurich.
i n the x
I
x + L Y . ~ I,. 1 1
tlie P'orscl~ungsinstitut f u r
- 234 x
B. E. Hubbard
- P . h . a r e points where the mesh l i n e s cut
aR
1 1
. If
v(x) i s sufficiently
smooth then we note that
A
s o that, in general, the o p e r a t o r
i s only a n
h
defined by
O(1) approximation to the Laplace o p e r a t o r f o r
For
V, W
functions defined a t m e s h points of
E
n
X ERh
.
, having com-
pact support we define (V, WIh = hn
1 V(x)W(x) X
E:
where the s u m s a r e taken o v e r grid p o l ~ t t sin It i s
easily s e e n
that
p1
b(x)
I1
11-2 x E R'
-
min V(x) = 0 @
where
.
s a t i s f i e s the variational principle
Dh(v, V )+
p1 =
n
b(x)
2
V (x)
II
(V, VIh
Rh
i s defined by
T h e r e f o r e we s e e that the m l t r i x of o u r d i s c r e t e problem i s s y m m e t r i c , positive definite and hence,
p
,U
exist
and
We a s s u m e the usual n o r m a l i z a t i o ~ l s
0
< p 2 ( .. . .
B. E. Hubbard
I uk (x)2 dx
( U k Uk)h
=
.
1
=
R
Multiplying the equationfor
u
k
by
U. and suniming we J
Likewise, multiplication of the equatio.1 for yields the second equation
-
h k
U.) J
h
=
-(uk.
A
h
U. J
by
see
that
uk and summing
U.) = p j ( u k UjIh ~h
.
while t h e f i r s t i s the d i s c r e t e analogue of Green's second identity applied to
u
k'
U. vanishing outside of J
If we w e r e t o s e t
j =k
-
Rh , Adding t h e s e equations yields
t h i s equation
relates
.
(Pk-d k )
to the
(Auk A huk) If t h i s local e r r o r w e r e uniformly local e r r o r , 2 O(h ) then , upon proving that f o r h ho(k)
<
2 we would have established that ( p - A k) = O(h ) under t h e usual k 4 assumptions that u E C (R) Since the l o c a l e r r o r at points of
.
R'
is only O(1) however, we take the following additional s t e p s . We h d h on R , which, introduce t h e Green's function f o r the o p e r a t o r h f o r p a r a m e t r i c values y E Rh , s a t i s f i e s t h e d i s c r e t e equation
B. E. Hubbard
Many properties of
G (x,y) a r e discussed by Bramble h example, the discrete representation formula applied to
after using the equation for
U. and J
the symmetry of
. For
1
U. J
G
h
is
over the s e t
Rh Thus (
(3) where
a)
But this
k
-
A Uk A
hUk.
UjIh =
p j(Q)h. U .)
Jh
is defined by
is the mapping
discretization
error,
u - A u k - + e , of the local e r r o r to the h k e(x) , in the Dirichlet problem. Just a s in the
A
Dirichlet problem we obtain the estimate 4
u E C (R) from the local estimate k
and from the inequalities
Q)
2
= O(h ) when.
B. E. Hubbard M , , is a function of the derivations of J l e s s than o r equal to j The number
.
u
k
of order
To ?rove (2) and thus complete the classical e r r o r estimate for the eigenvalues we will make two assumptions a )
pj
(b)
Ak
dj
as
h-0.
j=1,2,
...
i s simple.
Both a r e made for convenience since,
(a) convergence theorems
exist for much more general domains than those which shall be considered here and (b) the corresponding e r r o r estimates can be obtained for multiple eigenvalues
2
, but with somewhat greater effort.
It follows from (1) and
If
j#
k
( 3 ) that for given
our assumption that
pj+dj
as
k
h+O
implies
that
and consequently
Since the matrix of our problem i s symmetric, and positive definite the eigenvectors
{u]
span the space of
have from Parsevalls identity
Then (6) implies that
our discrete problem and we
E.B. Hubbard
The inequality
$
(uk, uklh+
(2) follows immediately in t h e classical c a s e s i n c e 2 uhdx = 1 'This fact can be shown m o r e directly by writing
.
R
where
s o that
8k + O
as
h
+
0
. In fact,
if we choose the sign of
uk
s o that (u U ) k' k h
> 0 , then 8 k c a n be defined by
Setting j =
in (5) l e a d s to the f i r s t inequality of the following
k
theorem Theorem 1 :
If
-
Uk
where lut
k
4-
is a simple eigenvalue then
m aRh x luk(x)
-
Uk(x)
1 , and
K
k
i s a bounded
quantity. Proof :
T o prove the second inequality we u s e (7) and (8) in the equation
- 239 E.B. Hubbard
T o prove the t h i r d inequality we substitute
u (x) - U (x) into the dik k
s c r e t e Green's formula,
A s w a s mentioned in the l e c t u r e s by Bramble,
G (x, y) h a s the following h
majorant
where
p ( (x
=
(x l 2 + a h
and
y n ,a, do
a r e geometrical constants.
Thus if n = 2, 3 we s e e that the r e s u l t of apllying Schwarzl inequality to the second and third t e r m s of the above identity yields
- 240 E. B. Hubbard
from which the final inequality in the theorem follows. If n
>
4
the
result can still be obtained by an iteration process that begins by noting that
where
j
i s large enough s o that Schwarzl inequality will give bounded
quantities. Having done this we have an inequality of the form
where Q i s a bounded quantity, computable in t e r m s of known quantities. By repeatedly substituting this inequality into itself we will obtain t e r m s
to which we can apply Schwarzt inequality. Thus the theorem i s proved. Ah analogous theorem holds if
Ak
i s a multiple eigenvalue.
We shall now give some specific e r r o r estimates a s corollaries of this theorem
. The first of these gives an e r r o r estimate of the clas-
sical type. Corollary 1 :
If
A
<
c and
4 u Z C (R) then , for
ilk
simple,
Proof : Just a s in the Dirichlet problem it follows from
I
(4) and esti2 = O(h ) It i s
mates for the discrete Green's function that (a) k h 2 easily seen that. E (u ) = O(h ) which completes the proof. h k
.
E. B. Hubbard In the remaining two c o r o l l a r i e s we s h a l l r e l a t e the e r r o r estimat e s directly to the smoothness of the boundary. Corollary -
2 :
If
3 R € H1(2, A, y ) , i. e. if the functions which give the
equation of the boundary in local coordinates have two derivatives and the second derivatives satisfy a Holder condition with exponent and
y >
0
A then
where
K i s a computable constant.
P r o o f : It i s known, Gunter 3 that u k H(2, ~ At, 7 I) , the c l a s s of 2 u € C (R) whose second derivatives satisfy a Holder condition with exponent
/' E
( 0 , ) , and constant
and the mean value t h e o r e m
where
d(y) = m i n x-y X E R
we s e e that ( @ gKh k clusion follows.
I
Corollary 3 : L e t i s composed of
2
R p
for
At
u
k
. In view
. Once again
. Using this information it i s e a s i l y established that
of the inequality
2 E (u ) = O(h ) s o that the conh k
be a bounded domain
analytic a r c s . L e t
n - ,i
ai
in = 1,
E 2 whose boundary
...,p , the
interior
E. B. Hubbard
angles at the corners,
Proof :
x. , 1
be such that
The work of S. Lehman
if we require that
" < 2Q -
<
27d.
Then
i
5
implies that
x. a r e greater than a distance 1
lines. We can investigate the behaviour of
k
ch
from the mesh
(x) by means of the
following inequalities
where 0 as
p+
<
a .
1
and p sufficiently small. We note that K( p) ' A direct calculation then yields for x E Sp (xi)
/3
.
1
, w
E. B. Hubbard
This estimate is consistent with those obtained by P. Laasonen [4]
for
the Dirichlet problem and in certain respects gives improved estimates. Clearly,
[lak11
will have the form indicated in the theorem
[
k
.
E (u ) < K ( E ) h 2 i u - r h2] so that h Ic the corollary follows. We see from (9) that luk(x) Uk(x) satisfies Likewise we can show that
a point dependent estimate of the type (10)
.
-
I
-
244
-
REFERENCES -----------1
J. Brabmle :
2
J. Bramble and
3
(C. I. M.E.
Lectures, this volume)
B. Hubbard :
Effects of boundary regularity on the discretization e r r o r in the fixed membrane eigenvalue problem. (To appear)
Gunter
Die Potentialtheorie und ihre Anwendung auf Grundlagen der Mathematischen Physik B. G. B. G. Teubner, Leipzig (1957)
:
.
4
P.Laasonen:
On the degree of convergence of discrete approximations for the solutions of the Dirichlet problem. Ann. Acad. S i. Fenn. Series A. 246, 1-19 (1957)
.
5
S. Lehman
:
Developments at an analytic corner of solutions of partial differential equations, J. Math. Mech .8, 727-760 (1959)
.
6
V.K. Saulev :
On the solution of the problem of eigenvalues by the method of finite differences, Translation in English in A. M. S. Translations (2) Vol. 8, 277-287
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. f
K. JORGENS
CALCULATION OF THE SPECTRUM OF A SCHRODINGER OPERATOR
Corso tenuto ad Ispra dal 3-11 Luglio 1967
- 247 CALCULATION O F THE SPECTRUM OF A SCHRODINGER OPERATOR by
Let
T denote a selfadjoint linear operator in a complex Hilbert
space H , U (T) the spectrum of T,
u d(T) the
discrete spectrum of
T (that is the set of all isolated eigenvalues of T of finite multiplicity)
- ad(T)
Q ~ ( T=) U (T)
tor
T is said to be semibounded (below) , if
wer bound; in this case we denote by lower bound of U(T) and
T
the essential spectrum of
and
v
(T
(T)
(T) and
has a finite lo-
(T) the greatest
a ( T ) respectively, s o that
e holds for every semibounded operator and only if V (T) belongs to
u d(T)
T, and
. The opera-
v (T)
V (T) ,< p (T)
(T) holds if
.
Semibounded operators play an important role in the quantum theor y of systems with a finite number of degrees of freedom. The operator T of total energy of a N-particle s'ystem in the Schrddinger representation is a selfadjoint semibounded differential operator of second order in the Hilbert space
L ~ ( R (with ~ )
m = 3N)
. An eigenvalue .a < IIL (T)
is the total energy corresponding to a stable state of the system; in particular
V (T) (if this number is smaller than p (T) ) i s the enrgy
of the ground state. The number
p (T) is the smallest possible energy
corresponding to an unstable state of the system and ,U (T) - u (T)is the minimal energy required to lift the system from its ground state to an unstable state. Considerable effort in numerical analysis has been devoted to the problem of computing V (T) and other eigenvalues of
T
below
p (T).
There a r e the well-known variational procedures (Rayleigh-Ritz o r Galerkin) which give upper bounds to these numbers, and more recently the methods of Weinstein, Aronszajn and Bazley (see [ 2 )
and the litera
ture quoted there ) for obtaining lower bounds, such that it i s now possible to compute these eigenvalues up to a given e r r o r with a reasonable
K. Jorgens
amount of work. On the other hand little has been done to find numerical approximations for ,U (T) as
. We
v (T)
although
this number is clearly a s important
discuss here, for Schr6dinger operators T , two me-
thods which make it possible to compute ,U (T) in many cases : The perturbation method. Let
T
be representable a s a sum
T = T +A, 0
where T
0
i s selfadjoint and semibounded and
is the domain D(A) of {un) C D(To) such that
A contains llun -k
{ A U ; ~ , )converges
is T -compact (that 0
D(T ) and every sequence
1<
1
with
A
0
C contains a subsequence
strongly in H)
. Then it follows that
J
and in particular e (T) = ue(To) ,
p (T)
=,U(T )
o
. The
in finding a representation of this sort such that ,U (T ) at least is computable by known methods
0
.
J
method consists is
known o r
N The method of splitting. Let m = with
where
m.
x.J R ' T. J
for
x
R
m
j=1 mj ,
. Let
is a Schrijdinger operator
in
m L2(R j )
acting on functions
-
of the variable x . and p . the operator of multiplication by a real funcJ ~k tion p . ) Now consider the operators ~k J k
(x., .
acting on functions in
L ~ ( R ~ for - ~ i ~= 1.2,. )
.., N . Then under certain
hypotheses we have the formula
p(T)
= min U (Si)
(i)
+ ,U (Ti)
)
K. Jorgens
which reduces the problem to the (supposedly easier) computation of the numbers ,u (T.) and to the well-solved problem of computing the numbers
($1
1
.
Schrtldinger operators.
We consider operators of the form
satisfying the following conditions : a . bnd J
p
2 (aj) L2, ~
'
are real , m
(4)
.(Rm) , L2, loc Then the operator
T
1 j=1
R ~ ,)
in the sense of
3 ja J.
i s defined and
0
O C(
= 0
symmetric on
(R"')
C:
and
can also be written a s
We now use a
uliqueness theorem which
is
special case of a theorem of Ikebe and Kato Theorem 0
(Ikebe-Kato). Let the coefficients
(4) and 1 m a; E C (R )
a
slight extension of a
14 1 . a . and J
p
satisfy
and
J
for some a > 0 , C > 0 and for all x E Ftm , Then T (defined in 0 m C: (R ) ) has a unique selfadjoint extension (also denoted by T )with 0
domain
Furthermore
T
The perturbation method.
We consider perturbations of
m T =
(7)
.
i s semibounded below
0
c i a . + a . t b.) J J J j=l
where the coefficients
a . and J and the perturbed coefficients
p of
2
T
0
+
are a s
in theorem satisfy
.
Theorem 1.
Let
assume that
a . and J
(i)
the derivatives
(ii)
the functions
m
p
a J.ak
B
of the form
0
p+q
a . t b . and p+q J J p respectively) Then T
substituted for a . and J C: (Rm) and can be written in metric
T
as
T
0
(4) (when
i s defined and sym-
* T0 t A
with
satisfy the assumptions of theorem 0 and
a r e uniformly bounded
-
(bj)\nd
satisfy condition
(6)
j=1
(when substituted for Then
T
p ) with some a >O and
C>0
.
has a unique selfadjoint extension with domain
D(T) = D(To)
which i s semibounded. If in addition the function
tends to zero
for ( x
+
co
, then
This theorem and some similar
A
i s T 0-compact.
ones a r e contained in [5
]
. The most
remarkable feature of these results i s that the coefficients of the unperturbed operator tors
T
0
T
0
need not be constant o r even bounded. F o r opera-
with constant coefficients Birman [3
1
and Balslev
[I lhave
K. Jorgens
obtained very general perturbation theorems using the theory of quadratic forms in Hilbert space
.
The proof of theorem
f o r functions
r e s t s on an identity of the form
1
u r H ; ~ ~ ( R ~, ) where
a r e integral operators with kernels
I x-y I
zero for
I X - ~ (-< p with T =
Example 1.
K p (x, y) and
0
a constant
-A
c 1 x-y
C independent
(that i s
T =
C
(i
j=1
Hp (x, y) identically
a J. + bJ. ) 2 + q
therefore p (T) = 0
x, y
of
and
P
.
a . = p = 0 ). It i s well known that J
o
m
for e v e r y p ~ ( O . 1
1 2-m
= U (T ) = ( 0 , m) in this case. By theorem
e
H
> p and satisfying the inequalities
I K p ( ~ Y)1 ( 5
for
K p and
u (T
0
the operator
1
has the essential spectrunl
u (T) = (0, co ) and e
.
1 and a (x) = - - x2 , a2(x) = Example 2. Let m = 3, x = (x x 1 1' 2"3) 2 1 T with these coeffi=5 x l , a (x) = 0 and p(x) = 0 The operator 3 0 cients describes the motion of a charged particle under the influence of
p
.
p
a homogeneous magnetic field of strength spectrum we get
u
of
T
0
the
x -direction. The
3 can be found by a separation of variables (see
( T ) = oe(T0) =(lpl, 0
/? in
)
. By theorem
1 any perturbation
151 ); A
satisfying the assumptions of the theorem does not change the essential spectrum.
In particular we can take
b. = 0 for J
j = 1 , 2 , 3 and
)=
q(x) =
- y ~ x I -1
; the p e r t u r b e d o p e r a t o r
T= T
0
f
q
now c o r r e s p o n d s t o
the so-called Zeeman-effect in quantum t h e o r y . According to t h e o r e m 1 we have
(IPI,
(Te(T) =
case where
y> 0 )
s p e c t r u m of
T
0)
; f u r t h e r m o r e it i s possible to show (in the
by a Ritz-type variational argument, that the d i s c r e t e c o n s i s t s of infinitely mahy eigenvalues below
p ( T ) = I p l ( s e e 15))
.
The method of splitting. We s t a r t with the following example : Example 3. The S c h m d i n g e r o p e r a t o r f o r a s y s t e m of N
p a r t i c l e s of
equal c h a r g e and m a s s in a c e n t r a l Coulomb field. H e r e we have m=3N and
-
- ..,n ) N
x = (xl, x2,.
where
,;
J
E R ' ~ i s the position
of t h e j-th particle.
The o p e r a t o r i s
with
positive constants
(with
T = 0
-A
p
and
) only gives that
p
. Theorem T
1 applied to t h i s o p e r a t o r
h a s a unique selfadjoint and s e m i -
bounded extension. Condition (iii) is not satisfied, and in fact we shall show that
p
(T) can b e computed with the help of formula (3) and i s
found to be s m a l l e r than
p ( T0 )
=0
for
N
-> 2
.
...,
T h e o r e m 2. L e t T. f o r j = 1 , 2 , N be o p e r a t o r s of the f o r m J m j ( 7 ) in L2(R ) with coefficients satisfying the assumptions of t h e o r e m 1. N Denote by G . the v a r i a b l e s in T . and put m = m , and J J i=l J m,+m k x = (xl, x 2 , , ,xN) L e t p . be non-negative functions defined on R J ~k
-
-
..- .
satisfying (6) with and such that
m = m.+m J k the functions
for s o m e positive constants
and
C,
1 ijI-ca
I
and as ; k r n for fired ;.. Then the k J m-m. operators (1) and (2) in c ~ ( R and~ C) ~ ( R 1 ) respectively have unique selfatend to zero a s
for fixed
; k
0
0
ae (T.)=I,U(T,), ca) 1 1
djoint and semibounded extensions and formula (3) i s valid. If for the index i giving the minimum in (3), then we also have
A theorem
0
e
(T) =(p(T),oo).
of this type has for the first time beer. proved by
1 for an operator somewhat more general than (8) . Theo2 above i s contained in [63 . Related results can be found in the
M.G. ZIslin [8
rem
papers of
C.van Winter [7] ; here the interaction t e r m s
a r e not 'jk assumed to be non-negative and consequently formula (3) has to be replaced by a more complicated one. Finally Zislin and Sigalov
[9]
have
refined the theorem for the operator (8) by taking into account its symmet r y with respect to permutation of the variables similar to (3) for the restriction of
zJ. . They
prove a formula
T to the invariant subspace of
L ~ ( R ~corresponding ) to an irreducible representation of the symmetric
.
group S A summary of this work and of related results has been giN ven by Sigalov in his recent article [lo1
.
Theorem 2 clearly applies to example 3. Denote the operator in que1 -1 . 3 stion by T~ ; then T = T = A (x in L (R ) and 2 1 j 1 . is the well-known operator for a hyue(T ) = ( 0, WJ) by theorem 1. T 1 drogen-like atom; its lowest eigenvalue i s v ( T ) = The opera4 N tors S. corresponding to T a r e all identical (except for the numbering 1 N- 1 of variables) and equal to T Therefore by theorem 2 we have N N- 1 N N- 1 N- 1 ue(T ) = ( V ( T ), o o ) . It followsthat ,U ( T ) = V ( T )
-
-p 1
-lp2 .
.
-I -'p2 4 N
v (T
for N > 2 . I f / ? ) N y Z i s l i n N
< p (T )
)
,..., N .
M=l,2
[8]has
; in this case we have
p (T
shown Mi-1
)
p
that M ( T ) for
REFERENCES -----------Balslev, E. in LP(R ) n Birman, M. (russianj
: The singular spectrum of elliptic differential operators
. Math. Scand. 19 (1966),
193-210.
the spectrum of singular boundary value problems . S.Math.OnSbornik 97 (1961) , 125-174. :
Bazeley, N. W. : Lower bounds for eigenvalues. J.Math. and Mech. 10 (1961), 289-308 -
.
Ikebe, T. and T. Kato: Uniqueness of the self-adjoint extension of singul a r elliptic differential operators. Arch. Rat.Mech.Analysis 9 (1962), 77-92. Jorgens, K. : Zur Spektraltheorie d e r Schrodinger-Operatoren. Math. Zeitschr. 96 (1967) , 355-372, Jorgens , K. : Uber das wesentliche Spektrum elliptischer Differentialoperatoren vom Schrodinger-Typ, Report, Inst. f.Angew. Math. Univ. Heidelberg (1965)
.
van Winter, C. : Theory of finite systems of particles; I the Green function, I1 scattering theory. Math.-Phys. Publications of the Danish Royal Academy of Sciences, Volume 2 (1965). Zislin, M. G. : On the spectrum of many-particle SchrCfdinger operators (russian) Trudy Mosk. Mat. Obsc. 9 (1960), 81-120
.
-
.
Zislin, M. G. and A.G. Sigalov : On the spectrum of the energy operator of atoms on subspaces corresponding to irreducible representations of the permutation group (russian) Izvestija Akad. Nauk SSSR 29 (1965), 835-860
.
.
Sigalov, A.G. : On an important mathematical problem in the theory of atomic spectra. (russian) Uspechi Mat. Nauk 22 (1967), 3-20
.
.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO
(C. I. M. E. )
A. LASOTA
CONTINGENT EQUATIONS AND BOUNDARY VALUE PROBLEMS
C o r s o tenuto ad I s p r a dal 3-11 Luglio 1967
CONTINTENT EQUATIONS AND BOUNDARY VALUE PROBLEMS by A. LASOTA (Krakow) It i s known that the theory of differential equations with multivalued right-hand sides (sometimes also called contingent equations) is closely related to the optimal control theory. Our aim i s to show that the technique of the contingent equations is useful in the theory of continuous and discrete boundary value problems too. In section 1 we s t a r t wtih a new topological method which permits to establish the existence of solutions provided a criterion of uniqueness i s fulfilled. Section tains a result of A.PliS 3
and
concerning contingert nquations [5]
2 con-
. Ir
section
4 existence theorems for continuous and discrete boundary value
problems a r e given. The main result, an approximation theorem, is stated in section 5
.
1. Fixed point theorem
. Let
E
a Banach space and let
n(E)
denote the family of all nonempty subsets of E. F o r a set A in n(E) , a mapping
H:A-2
i s closed in
n(E) i s called upper semicontinuous if its graph
A xE
bounded subset B
i s compact in
. The
map
of A,
H
E. The map
H
is
to be compact
called completely
h:
A3E
i s equivalent
to
compactness of
H
ty of H
the complete continuity of h.
means
IS
if, for any
continuous if it
compact.
F o r a single valued mapping H:x+ h(x)
said
the closure of the set
is upper semicontinuous and
of the mapping
is
equivalent to that of
the upper semicontinuity the continuity of h, the
h and the complete continui-
The following theorem i s due to A.Lasota
and Z.Opia1
[4]
.
A. Lasota
.Theorem 1. Let and let
U
be a
neighborhood of
0
in
the space E
H: U - - , n(E) be a completely continuous mapping such that
Then for any mapping
h: E-+E,
the condition
h(x) - h(y) H ( ~ - Y )
(1.3)
for x-y
eU
implies that the equation
has exactly one solution. The special case
of theorem
1 was presented at the second con-
ference on differential equations (Equadiff 11) in Prague 1966. The proof of theorem 1 is based on
the old idea of Hadamard to
use the monodromy principle. If h A
is
a
mapping of the form h(x) = Ax
is a linear completely continuous operator
1 yiews the essential ces to set
part of the first
H(x) = {AX}
and
+
b where
b c E and
(A: E - t E ) , theorem
theorem of Fredholm. It suffi-
to observe that condition (1.1) means
the uniqueness of solution of the linear homogeneous equation x =Ax. Note that in the Fredholm theorem the map
A may be non-contrac-
tive in general.
2=
2 . Plis lemma. Let [a, b] be a compact interval of the real n line R and let cf(R ) denote the family of all nonempty, convex, clon sed subsets of the n-dimensional real space R The space of all n continuous mappings of into. R with the usual norm
.
2
A. Lasota
(
1. 1 stands
P (p, A) point
p
f o r the Euclidean n o r m in
(p 6 Rn , A to
Rn) will b e denoted
cn.
By
n(Rn)) we denote the Euclidean distance of the
set A .
the
9
L e m m a 1. Let
G be a mapping of into cf (Rn) and l e t n v k { c C of absolutely continuous functions satisfy
a sequence of
t h e following conditions (2.1)
lirn k + m
vk(t) = ~ ( t )
lim
(2.3)
for t
(v;( (t) , G(t) ) = 0
ii -+a
Then the function
v
is
3
e
almost everywhere (a. e. )
a. e.
absolutely continuous
and
3. Continuous boundary p r o b l e m s Now p a s s F:
to
JXRL cf(Rn) , a
the differential equations. Consider a map map
f:
JXRL Rn and
(i) F(t, x) i s homogeneous and completely continuous , { ( t , x, Y ) :
i.e.
in x ( F ( t , the
Y € F ( t , x)
(ii) f(t, x) i s continuous in
Rn s u c h that :
9 x) = AF(t, x)
for real
a)
set
, 1x
1 = 1j
(t, x) and s a t i s f i e s the condition
f ( t , 4 - f ( t , y ) 6 F(t, x-y)
(3.1) is
Given the mapping gent equation
cn+
R~~ ;
i s compact in
(iii) L
L:
l i n e a r and continuous F, f, L
and a point
(t, x, y ) e &RZn
. re R n , we
;
consider t h e contia-
A. Lasota
with the homogeneous
linear
condition
and the o r d i n a r y differential
equation
with the l i n e a r condition
An absolutely continuous (in tion
Caratheodory
sense)'
of equation
(3.2)
almost e v e r y a h e r e
From
the t h e o r e m
T h e o r e m 2. and
if
(3.3)
then
there
exists
( 3 . 4 ) satisfying
Note
that
usual
if
it s a t i s f i e s
condi-
3.
F , f, L
one
0
s a t i s f y conditions
0
133 ) 0
1 ,2 ,3
solution of equation
(3.2) satisfying
and
of
only one solution
the equa-
(3.5).
the t h e o r e m
the
(3.2)
be called a solution
1 we can d e r i v e the following ( s e e [2),
i s the unique
tion
(3.2) in
on
If the functions x = 0
n x t C will
function
2
is
not
true
(non Caratheodory)
if t o mean
solution of
sense.
In
o r d e r t o prove t h e o r e m 2 c o n s i d e r the mapping H of n E = c n~ R~ into cf(R ) s u c h that for e v e r y point (x, p) i t s image H(x,p)
is a s e t
for
u(s)
for
e v e r y point
of a l l p a i r s
F(s,x))
the formulae
and (x, p)
( y , q ) given by the formulae
the mapping its
image
h
of E
h(x, p) i s
a
into pair
itself (y, q)
s u c h that given by
.
A. L a s o t a
It i s e a s y
to
H, h satisfy conditions (1. 1) , (1.2),
s e e that the maps
(1.3) and that
the existence
of
problem ( 3 . 4 ) , ( 3 . 5 ) is
solutions of
equivalent to the existence of solutions of the functional equation (1.4)
H is
Map to
show
Then
evidently compact. Thus
H
that
to
end
the proof it i s sufficient
is upper semicontinuous. Suppose that
we have
(3.6)
yk(t)
uk(s) d s
=
+p
k
,
k
k
u ( t ) g F ( t , x (t)) ,
a
and consequently k / (Y (t))E F(t. Xk(t))
(3.8) From
(3.8)
and
the upper semicontinuity of F lim
k -tco
Thus from the l e m m a
In and
From
addition,
(3.7)
p ((Yk ( t ) ) ' 1 it
, F ( t , x(t)) = 0
follows
upor1 passing
we
obtain
.
immediately that
the l i m i t
in (3.6) (for t = 0)
we have
(3.9), (3. 10) and (3. 11) it
a r e done.
to
.
follows that
(y, q) H(x, q) and we
4.Discrete boundary problems. In order to obtain a discrete analogue of a finite sequence
theorem 2 consider
.
a = t < t <.. . < t = b o 1 n Then we can replace the contingent equation (3.2) by the multivalued difference equation
A Xi
(4.1)
€ F(ti, xui))
I
and the differential equation (3.4) by the difference equation Axi i = 0, ..., n-1 . TA = f(ti , x(ti)) (4.2)
.
and A t , = t . - t . 1 1+1 1 solutions of equations (4.1) and (4.2) a r e continuous in
where A x . = xitl-xi 1
We assume that
0and linear in each Theoreme 3. If (iii) and
if
x=0
interval
.
[ti , titl]
the mappings
F, f, L
satisfy conditions (i) , (ii) ,
is the unique solution
of equation (4. 1) satisfying
(3.3) , then there exists one and only one solution of (4.2) satisfying (3 .5) More exstensive theorems and some applicaticns in the theory of discrete boundary value problems a r e due to The proof
of theorem
F?H. Szafraniec [6
3.
3 is based on the same idea a s the proof of theo-
r e m 2. But in this case the lemma 1 is not
5. Approximation theorem.
needed.
The development of the theory of differential
inequalities allows to obtain precise uniqueness theorems for a general class of differential boundary value problems. It is interesting that on the same way we can establish the existence of solutions and the convergence of finite difference approximations. Namely setting %= (to,
f ( ' L ) = man Theorem 4. x = 0 is
titl
- ti : i = 0,. . . . n-1 1
If the mappings
the unique solution
of
F, f, L
the
. . . ,t n) and
we have the following
satisfy (i) , (ii) , (iii) and if
contingent
boundary vqlue problem
A. Lasota
( 3 . 2 ) , ( 3 . 3 ) then : 1) there exists a unique solution
x
J(z)
for
2) 2
of problem lim
3)
S(%) +o
and
S (@
lo i s
2
0.
an
it is
immediate1 consequence of theorem 2 .
sufficient
( 4 . 1 ) , ( 3 . 3 ) has
to
show that for sufficiently
only the trivial solution
x =0
3 . To this end suppose that for each integer k
a sequence
and a corresponding
Since xk
0
problem
to use theorem
there exists
of problem ( 3 . 4 ) , ( 3 . 5 ) ,
sufficiently small there exists a unique solution
bxr-xO\\=
order to prove
small
0
(3.5) , (4.2) ,
The statement In
x
is
nontrivial solution
linear
in
each
xk of problem (4. 1) , ( 3 . 3 ) , i.e.
k k interval (t., t . ) we can write ( 5 . 1) in 1 1+1
the form
and
since problem ( 5 . 1 ) , ( 5 . 2 ) i s homogeneous we
can assume that
(5.4)
From this and ( 5 . 3 ) it nuous
on
J
.
obtain
the functions
xk a r e equiconti-
Upon passing to suitable subsequence, if necessary,
we may assume that x e c n . Thus
follows that
the sequence
passing to the limit
(see lemma 1 )
{ xk 1 convergences in ( 5 . 2 ) , ( 5 . 3 ) and
to a function ( 5 . 4 ) we
A. Lasota
which is
impossible.
1
k. To prove statement 3' consider a sequence ) + such that k ) - P O and the corresponding sequence {xk] of so1utio.n~ of
(
. We
( 4 . 2 ) , (3.5)
and on
other
have
hand (xO(t)) = f(t. xO(t)) ,
From
this
(uk(t)) where
and
e F(tik
= r .
k k , u (t. )) t
ek(t)
for
k
tE(ti>
Lu = 0
1
k k o u = x - x and
that
sequence we k v = uk 1 uk
may
The functions
k
1) u I\+0 assume
.
Then upon passing to k that )lu I\ . + c (C E (0, t o 3
the sultable sub.
1 . Setting
we obtaln
vk a r e evidently
equicoi~tinuous and we may a l s o a s s u -
i vk { converges
m e that the sequence
lo
a function
v
C"
.
The
1 yields
v' (t) and obviously and
0
(3. 1) it follows that
Now suppose
lemma
Lx
Lv = 0 ,
we a r e do~:e.
(I
F ( t , v(t))
v 11=1 which
k .
i s impossible. Thus J J u (\-,0
A. Lasota
In o r d e r to illustrate theorem
4 by an,
equation
( 3 . 4 ) the following aperiodic
We s t a r t
with
Lemma 2 .
then
x = 0
is
Note
From
y ( t ) i s nonnegative and
the unique solution
2
the estimation i s due to
(5.7)
of the inequality (5. 6) satisfying
theorem
4
( 5 . 7 ) i s the best possible of this type.
S. Kasprzyk
and
[I].
.J. Myjak
and lemma 2 it follows immediately
Theorem 5 . If the function
where
such that
value condition
that
lemma
boundary value condition
lemma concerning differential inequality
If the functions
the boundary
The
a
example, consider for the
f(t,x) i s continuous and
the continuous nonnegative function
9
satisfies
if
inequality
then l o there 2'
exists a
unique
solution
of ( 3 . 4 ) , ( 5 . 5 ) ;
for sufficiently small $(%) there exists a unique solution x
of ( 4 . 2 ) , ( 5 . 5 ) ;
A. Lasota REFERENCES
1
S.Kasprzyk, J. Myjak, On the existence and uniqueness of solutions of F'loquet boundary value problem , Zeszyty Naukowe U. J., P r a c e Matematyczne (to appear)
2
A.Lasota , Une generalisation du premier theorem de Fredholm et s e s application naires,
3
.
&
la
theorie des equations
Ann. Polon. Math.
A. Lasota, Z. Opial,
On
differentielles ordi-
18(1966) , 65-77.
the existence of solutions
of linear pro-
blems for ordinary differential equation^ , Bull. Acad. Polon. Sci., Ser. s c i . math. , astronom. et phys. , 14(1966) , 371-376. 4
A. Lasota, Z.Opia1, of nonlinear
On the existence and uniqueness of solutions
functional
equations, ibidem
Mesurable orientor
A. Plis,
6
F. H. Szafraniec, Existence theorems for discrete boundary problem,
.
ibidem,
.
5
Ann. Polon. Math. (to appear)
fields,
(to appear)
13(1965) , 565-569.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
REDUCTION A DES PROBLEMES DU TYPE CAUCHY-KOWALESKA
torso t e n u t o a d I s p r a d a l 3 - 1 1 L u g l i o
1967
REDUCTION A DES PROBLEMES DU TYPE CAUCHY-KOWALESKA Par J. L. Lions
Universite de P a r i s Introduction. I1 intervient assez souvent en pratique des systemes
differen-
tiels . qui ne sont pas d~1. lype de c a u c h y - ~ o w i l e s k a: en hydrodynamique l e s dquations de Navier Stokes , l e s equations de l a magneto-hydrodynaen meteorologie (Marchouk 1 2 ) ) etc..
mique,
On donne ici deux procedes ,
.. .
de portke semble-t-il a s s e z gdne-
rale,
pour approcher de tels probl6mes par des problemes du type
Cauchy
-
Kowaleska
thodes numeriaues
qui sont eux memes II~tandard~~.
Le p r e m i e r procede,
dit de penalisation , est une
procede introduit p a r Courant
l a methode Temam
en calcul des variations
,
es
s u r lfexemple des equations de Navier-Stokes, suivant
[1],[27
.
Une idee vojsine
avait deja ete introduite pour l e s
lineaires dans Bramble-Payne [l
poses
[l]
variante dtun
maintenant frequent en contrble optimal ; nous donnons au N, 1
dl;sage
Lattes
ensuite approches par des me-
- Lions
[I]
] . Une variante
a ete
equations utilisee dans
pour llapproximation numerique de problemes ma1
. Le deuxihme procede, dit des derivees artificielles, e s t expose s u r
deux exemples : au Stokes et
au n. 3
n. 2
s u r ltexemple des equations
Rachford
rll ) .
Navier
s u r un probl6me lineaire quelque peu "singulier" -
mais qui intervient dans l e s applications problgme relevant
de
(cf. Garipo-r
de techniques analogues
dans
[I]
un
un
Douglas- Peceman-
J. L. Lions
1. Equations
-
1.1
de Navier-Stokes-; Methode de penalisation.
Position du probleme (1)
Soit R assez
un ouvert borne reguliere
R~ , n = 2 ou
de
3
de frontiere
.
On c h e r c h e une fonction u
{ ul, ..., u
=
Q = Rx
definies dans l e cylindre
e t une fonction p
1 o, Tr telles
que
n
3u i=l (1.2)
div
(1.4) ob
t
u =0
f f l , ..., f n ( e t
de ( 1 . 2 ) (1.1)
f
dansQ,
(9>0)
u
dans R o
sont d o n n e e s d a n s
Naturellement il faut p r e c i s e r h de ( 1 . 1 ) - (1.4) ; mais
p
dans Q ,
u(x, 0) = uo (x) f =
grad
1
QetR
.
quel s e n s on c h e r c h e une solution
toutes facons c l a i r que, B cause 3 P ) l e problhme mixte (qui ne contient p a s de derivee
. . . (1.4)
,
il e s t de
n t e s t p a s du type de Cauchy -Kowaleska.
Cela conduit
d e s difficultes numeriques : l o r s q u e l l a n p a s s e p a r ex. aux differences finies, l e s conditions ( 1 . 2 ) introduisant d e s ncontraintesltcompliquant s e r i e u s e m e n t l a m i s e en Notre objet
oeuvre d e s calculs.
e s t donc ici d l a p p r o c h e r l e s y s t e m e (1.1). . .(1.4)
p a r un s y s t 6 m e de Cauchy-Kowaleska. Cela
peut s e f a i r e g r o s s o
(i) pait "suppression" d e
mod0 d e deux
facons ( 2 )
la "contrainten ( 1 . 2 )
..
;
(ii) p a r introduction dlup " t e r m e artificiell1E 3~ A '
3 t
'
' ) ~ f . a u s s i l e s conferences de C. Capriz, c e Volume. 2) A n o t r e connaissance precedes.
. I1 y a
t r e s probablement beaucoup d l a u t r e s
.I. L. Lions
fait llobjet de ce N
La llmethode" (i)
la methode (ii) 1)
fait llobjet du n. 2 suivant.
1.2. Penalisation. L1idee generale
est
de supprimer (1.2) et d'ajouter dans (1.1)
un t e r m e de l a forme
-&
I1petitl1- ce
> 0
(1.2) mais
5
-
(div u )
qui conduit
l a place
Pcecisons cela
grad
de
5 une fonction u
(1.2)
Introduisons
,
une condition
?i
dans ce
but l e s outils suivants :
1 n V = j v l v;(H0(R:) ,
div
v = 0 } ,
H= I f 1 fb(~~(R )",)
div
f = O
V et H sont des espaces de Hilbent produits
]
(2) ;
R
sur
(3), pour l e s
scalaires dx i, j = l
dont
ne verifiant plus
R
9
J
l e s normes correspondantes sont notees
2)~A(~)
11 u (1 , I f 1 ,
est defini dans l e s konferences de Raviart, ce vol.
"Toutes l e s
fonctions sont 5 valeurs reelles
J. L. Lions
Pour u, v, w & V, on pose
Le probleme (1.1) Con
. . .(1.4) peut
suivante ( 2 ) : trouver
alors s e f o ~ m u l e r
2
u t L (0, T;Y))'(
de la fa-
tel que
Remarque 1.1. La condition "div
u = 0"
V et la pression p n'a l e produit
est contenue dans
"disparufl que parce que l'on a p r i s
scalaire avec v
On montre
l'appartenance A
de
ceci (Leray
divergence
nulle (et nu1 au bord).
El] , Ladyrenskaya
[I]
, Lions [I],
Lions-Prodi
a) il existe une solution u de (1.7) (1.8), verifiant en outre
b) si n= 2, (5 (1.8) On
il
y a unicite
de
va maintenant approcher
des fonctions Aprks
u
ne
u dans
u
verifiant plus
ce qu'on a
dit
2 L ( 0, ;V) verifiant (1.7)
(dans un sens convenable) par (1. 2)
au debut de cette section, il est nature1
On defiliit ainsi une forme tri-lineaire continue s u r ' v . Suivant l e travail classique de Leray [I] 3) Espace defini dans Raviart, ce vol. 2 4, On montre, Lions r d , q u e s i u t L (0, f ; v) et satisfait alors (1.8) a un sens.
-
5 , Dans
le
cas
'In = 3.
on ignore
s'il
y
a
unicite ou
(1.7)
non.
'J. L. Lions
1 W = (Ho ( 9 ) )"
(1.10)
K = ( ~ ~ ()" 9 )
(1.11)
l e s produits scalaires
dans
V et
chercher
H,
puis
Mais il
y
de
a
K
W et u
etant
"les m^emestt que dans
verifiant
18 une difficult@ :
on verifie
sans peine que s u r
V , o n a
ce qui conduit
aux
inegalites de
dtune solution globale par
contre
2, l e probleme
le modifier de
l e facon
forme trilineaire
V ut
n,
(1.5)
b (u,u,u)
1) Naturellement 2)
Mais cela
T quelconque) de u6W
(1.13) nlest pas vrai pour
probablement
Mais
(i. e pour
ltenergie pour (1.7) et
qui
(1.12) e s t
suivant
(1.7) (1.8)
de sorte que
. On
ma1 pose
(cf. Temam [I]
Cvidemment
coincide
sur
& llexistence
1) ,.
tres
doit done
[2] ) : on introduit
V
avec b(u, v, w).
(H: (62) )n = W on a :
=2I
(u2)dx +: 1, u.1 3 3x. J
1, J
(1. 13) a lieu
1
div u S u 2 ) d x = 0 J j
"8 cause de laphysique"
ntest pas demontre
.
- 274 -
J . L. Lions
et on peut maintenant introduire le probleme penalise suivant trouver
&'(o,T
(t(tl .V)+
;W) verifiant
( ( ~ ( ( t )v)) +b 9
(U
1 (t) . v) +-(div
(t),
UL( '
).di~~)=
1.3. Resultats On montre alors (Temam [I]
p] )
:
Theoreme 1.1. Le probleme (1.16) admet une solution ; il y a unicite -- s i n = 2,. En outre Theoreme -(i)
pl n
(1.17) (ii) (1.18)
. Lorsque 1
1.2.
+
0
on a -
= 3 , on peut extraire
u +u
dans
I'
de facon que
2 L (0,T ;W) faible et
2 L (o,T;K)
-
fort 1)
si n = ' 2 , o n l a : : u
-+ u
I
dans
2 L ( o , ~ ; w ) faible et
L
i
(0,
T;K)fort .
En outre
Remarque
1.2
Faisant
v =u
t
dans
(1. 16) on
en
deduit aussitot
" ~ x t r a c t i o n inutile s i l e probleme de Navier Stokes admet une solution unique. par R (avec extraction "Quotient de llespace des distributions s u r
Q
de suite s i n=3) .On peut preciser ce resultat.
J. L. Lions
1.4 Applications. Du point de vue numerique l'approximation s e fait maintenant en deux temps : 1.4.1
. On
1.4.2 finies
choisit
au
moyen
de
On integre numeriquement
(1.19) ;
(1. 16) par
l e s differences
(ou Galerkin) ; il s1agit maintenant dlun probleme non lineaire
mais ou
les
methodes standard slappliguent. Cf. T e m a p
c]
Remarque 1 . 3 R. Temam la methode cfr.
egalement
pas
Equations
Temarn
[2]
2 (1.16)
fractionnaires (variante des directions alternees):
, Yanenko [I] , Temam [3],
. de
Navier-Stokes ; methode des derivees artificielles.
2.1 Idee generale de L1idee est
la hethode
grosso mod0
O P a?mais
adopte dans
Douglas (ce volume), Marchouk [I]
Lieutaud [I]
2
des
a
cela conduit
+
div
de
.
"remplacer" (1,2) p a r
u = O
des difficult& avu l e s integrales dlenergie
. Pour
I1recuperers l e s integrales dlenergie on modifie llequation (1.1) comme suit : on cherche. 1)
avec
l ) ~ e notations s sont celles
du
n. 1
.
J. L. Lions
(6)O)
(2.3)
a qtL + div u E L Q
= O
(2.4)
\ Cr ( x , t ) = ~s i
x e ,
(2.5)
~F]o.T[,
0
2 q0 choisi quelconque dans L (52)
.
On peut montrer (par le m&me genre de methodes que dans Lions [I] , Lions-Prodi [l])les
. Le
Theoreme 2.1.
probleme- ((2.1)
-I1 y a unicite
si
(2.6)
ut
+
. . . (2.5)
admet une solution
n= 2 on oeut extraire une sous suite
Lorsque encore notee
resultats suivants :
u L , q b , telle que 2 u dans L (0,
r,W) faible
et
2 L (0, T, K) fort
En outre grad
(qt-
lu'
1
2)+
grad
p
(Q)
dans
.
2.3. Remarques
et applications
Si l'on compare
l e s methodes de penalisation (N. 1) et
des derivees
artificielles on peut faire l e s observations suivantes : (i) l a methode de penalisation
copduit
k une mise en oeu-
vre numerique plus simple (notamment puisqu'll y a une equation de moins) ; (ii) l e choix de
t
est plus simple
dans la methode de penali-
sation ; (iii) p a r contre la methode des d e r i d e s artificielles est de por-
J . L. Lions
tee peut et1.e
plus generale comme l e montre l'exemple donne au N.
a p r e s , oh une methode de penalisation ne semble pas commode.
ci
. Equations avec
3
conditions d'Cvolution au bord : hethode des derive-
e s artificielles.
Soit
R C R~ ouvert borne de frontiere u=u
On cherche une fonction
A u = 0
(3.1) A?=
(3.2) oh
pour
-13 xi
(
1J
(
(x, t ) solution
xeR
3 xj
r reguliere .
,
t >
de
o
X
avec (3.3) (oh
3u
3u i t a i j 3 7 ; cos J
--=
av
Donc l e s d e r i d e s en de
(3.1)
le probleme
' ) ~ t done A 2' Pour un
ne contienl
peu
(n, xi) ,
n = normale 5
: nlapparaissent que s u r
r
eit6rieure 5 R)
2
et 2 cause
n'est pas du type de Cauchy - Kowaleski.
pas
de derivation en t-
simplifier on prend des fonetions
P valeurs reelles
.
,I. L. Lions
3 . 2 Solution "fontionnelle" du probleme -- (3. 1)(3.3) ( 3 . 4 ) On utilise
ici,
[I]
de Lions-Magenes Soit
s a n s l e s r a p p e l e r , quelques notations et r e s u l t a t s
gbHli2
.
[2]
(r); soit
A w = 0
w dans
l a solution
de
R
(3.5)
wl Alors
r
g
=
est
F
defini --
et
g H -112 (,-)
dtoh un o p e r a t e u r
avec
Cet
operateur
B
est
Le probleme (3. 1)
coercitif
(3.3) (3.4)
s i w designe l a t r a c e de u s u r w
solution
On lution
equivaut ,
alors
au suivant il
faut
trouver
de
peut
a l o r s appliquer
de (3. 10) (3. 1 1 ) Du point
lton peut
au s e n s
de .ue
.
les
(cf. p a r
numerique,
metilodes standard --- pour la r e s o ex. Lions-Magelles
r21
, chap 3)
.
h o r m i s d e s c a s p a r t i c u l i e r ~ ob
expliciter l t o p 6-r a t eu r integro differentiel singulier ----
R ( et
oh
.J. L. Lions
l'on peut a l o r s appliquer l'integration
l e s methodes de c h e r r u a u l t
[g . chap. 51,
de (3. 10) ne s e m b l e pas t r e s facile.
C1oil l'inter'et (eventuel) de l a methode d e s d e r i v e e s artificielles.
-
3.3. Methode d e s d e r i v e e s artificielles. On c h e r c h e
solution
1
t
& "~7 '+As
(3.12)
de 0
=
, C > O ,
oh = 0
prolonpment
de
On peut a l o r s m o n t r e r Theoreme --
3. 1. ---L e probleme
3.2. -Tlleor.&me .... --
Lorsque u
3.4
t
, u
u
3 R(ou suppose que u
0
C H (r),d>o) . Id
0
l e s r e s u l t a t s suivants : (3.12)
--+
0
dans
(3. 13) (3. 14) -admet une solution --
on
a
L 2 ( 0 , ~ H1(fl) ; )
.
Applications. On peut nlaintenant d i s c r e t i s e r (3. 12) (3. 13) (3.14)
desormais
I1 slagit
d'un problPme standard. Cette inethode a e t e donnee
r21.
Lions 3.5
.
-R--e m a r q u e s Si
A
e s t s e l f adjoint,
ou
peut c o n s i d e r e r l e probleme
dans
J. L. Lions
Au
Alors
le
=
dans
0
oh
On a
>OJ
,
mCthode d e s d e r L v e e s artificielles conduit
m e suivant : on cherche
(3'21)
O X [t
ui
u
t
solution d e
= prolongement
d e s r e s u l t a t s analogues
Voir probleme
de
3
de
u 1,
ceux
de
,
c e type dans Garipov [l]
3.2
.
.
au proble-
BIBLIOGRAPHIE --------------Bramble-Payne
Y. Cherruault R. Courant
[I]
[ll
cf. Payne, dans Numerical Solution.. P r e s s , 1967.
.
Bramble ed. Acad.
111 Approximation d l o p e r a t e u r s l i n e a i r e s e t applications. Variational methods for the solution of problems of equilibrium and vibrations- Bull. A. M. S. 49 (1943), p. 1-23.
J. Douglas et P e a c e m a n - Rachford [I]
SIAM, 1955, p. 42-65 e t 28-42.
R.M. Garipov [I] On the l i n e a r theory of gravity waves. Archive Rat. Mech. Anal. 24, 1967, p. 352-362. O.A. Ladyzenskaya [I]
Equations de l'hydrodynamique..
. Moscow
1962
R. L a t t e s e t J. L. Lions [l] Methode de quasi r e v e r s i b i l i t e e t Applications. Dunod 1967. J. L e r a y [l] J . Math. P u r e s e t Appl. 12(1933), p. -82 ; 13(1934), p. 33-418; Acta Math. 63(1934), p. 193-248.
J. Lieutaud [I] These. P a r i s 1968. J . L . Lions [I] Equations differentielles [2] Cours CEA-EDF
- Paris.
1965
J. L. Lions e t E . Magenes [I] Probli?mes aux l i m i t e s non homogPnes (11) Annales Inst. F o u r i e r , 11(1961), p. 137- 178. 121 ProblPmes aux l i m i t e s non homogenes e t applications-vol. 1 e t 2 Dunod. 1968. J . L . Lions e t G. P r o d i G:I.
[I]c.R. Acad.
Sc. P a r i s , 248(1959), p. 3519-3521
.
Marchuk [I] Methodes numeriques.. Novosibirsk 1965 121 P r o b l & m e s de Metheorologie . Leningrad 1967. (en r u s s e ) Traduction F r a n c a i s e , Armand Colin, P a r i s , 1968.
R. Teman
[I] C.R. Acad. Sc. P a r i s , t. 263, p. 241-244. [2] Article s u r Navier S okes B P a r a i t r e [3]
T h e s e , P a r i s , 1967.
N. N. Yanenho [iJ Metilode deb pas Ir*ctioll~~dii-es. . . . (sir r u s s e ) Novosibirsk. 1966 - Traduction F r a n c a i s e , Armand Colin, P a r i s , 1968.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
J . L . LIONS
"PROBLEMES AUX LIMITES NON HOMOGENES A DONNEES IRREGULIERES ; UNE METHODE DfAPPROXIMATION
I'
Corso tenuto a d I s p r a dal 3-11 Luglio 1967
PROBLEMES AUX LIMITES NON HOMOGENES A DONNEES IRREGULIERES ; UNE METHODE D'APPROXIMATION Par J . L. Lions Universite de P a r i s Introduction. eme Pour l e s operateurs elliptiques du 2 ordre, une methode d'approximation pou l e s probl6mes aux limites homog6nes ?I donnes irregulieres (le 2eme membre etant p a r ex. une rnasse de Dirac, la solutiun c o r r e spondante etant alors le noyau de Green) est donnee darls
Bramble [I]
et pour l e s donnees frontigres irregulieres (la solution correspondante &ant alors p a r exemple le noyau de Poisson) - une autre methode est donnee dans Jamet
1
Wous introduisons ici une methode differente valable pour l e s operat e u r s elleptiques d'ordre quelconque
et des donnees au bord arbitraire-
ment irreguligres et valable aussi pour l e s equations d'evolution. Mais supposant
(A l a difference des travaux cites de Bramble et Jamet) que et l e s coefficients dp, y I1tr&s reguli5rs". Les details te-
le frontigre
chiniques sont
dans
de Lions-Magenes [21
le cas general fort longs et utilisent l e s resultats ; nous expliquons ici
l a mCthode s u r des exem-
ples simples.
5 1.
Cas elliptique
1. Example 1
1.1- Position du probleme Soit libre
R un ouvert borne
. Dans
R
de
de frontiere
ou s e donne un operateur
A de
7 a s s e e regula forme
J. L. Lions
a , a , . c*(R) , avec 0
1J
On considere
oh
l e probleme de Dirichlet
l1on suppose que
On
montre
(cf. Lions-Magenes
bleme
(1. 3) (1.4) -admet, s o u s l e s 2 unique qui appartient & L (Q) . Notre
objet
(dans L'(R) )
e s t de
El]
,
[z]
chap. 2) que l e -pro-
conditions (1.5) (1.6) , une solution
t r o u v e r une approximation de c e problbme
.
1 . 2 . IdCe generale de l a methode.
On va changer l a nature du en
probleme (1. 3) (1.4) (du type l t ~ i r i c h l e t " )
l l a ~ o r o c h a n tp a r d e s o r o b l e m e s de Neumann. Soit
b ij3
l ' o p e r a t e u r de derivee conormale a s s o c i e
a
On r ~ m p l a c e a l o r s (1.4) p a r 6, 2 t u = g a3 On e s t ainsi trouver
(pour
conduit
au probleme -
> 0 donne) u €
solution
sur
r
,
A. >0
.
(du type Neumann) suivant de
I ) Tout c e qui suit s t e t e n d aux o p e r a t e u r s elliptiques conque (on n'ulilise janiais de principe du maximum).
dlordre q ~ e l -
J. L. Lions
E a g On
verifie
t
.
u6 =
fadement
que
sur
ce probleme admet une solution , m i -
Clue dans H 1(R) l ) . Reste ensuite 1.3. Resultats
E
faire de
.
0
convergence
Theoreme 1.1. Lorsque --
-4
p.
f
lorsque
0, u
E
4
.
2 L (R) .
dans
-,u
0
Principe de la demonstration.
On deduit
de 18 ltexistence et ltunicite de
2) Introduisons
"E.
A% v = u (1.10)
t 2 . t y~ L= Puisque
ul 6 ~ ' ( 0 ) ,
solution
dans
dans 0
€
dans
1 H (R)
H'(R) de
a;
F, v
6
E
3 H (0) (cf. p a r
ex. Lions-
, chap. 2 t s i lton prend dans (1.9) v = vE [2] l t o n integre p a r parties , on obtient 2)
Magenes
')l?ormellement,
.
7
sur on
u
(1.1) (1.8) est un probleme voisin et
e t que
regularis6 de
.T.
L. Lions
3) Mais en reprenant l a demonstration de la regularit@ au bord, on montre que
I1 1I
(1.12) A1Ors
, 1 (uE
\Ia%+v~ 1 HI/^(^)
, C, independant
IuI
6 C2
(
de
&
,
et
Mais alors (1.11) donne 2
6
lutl
lfl la l
IUtItC1
c211gllH-~
(f1
d'oh
4) D'apres
(1.13)
on peut
extraire une suite encore notee
u C
telle que (1.14)
u --, w
t
Mais =
f
et
alors
r
(1.8) donne
12 J ,
(Lions-Magenes
.
= g
wlr
,
chap. 2 ) , puisque
H - ' / ~ ( P)
u F 4 w J p dans
,
2 L (R) faible
dans
AuE =
UL+?-Wdans a
Comme
Aw = f
av
,
on
a
w
H -3/2(r) =
u et
donc 0
(1.15)
u~
5) Alors et
= 0
u
dans
~ ' ( 0 ) faible.
+
dans
v t t v
luh -u12
=lqf
H2(R) faible , oh 2 +Iu( - 2Re (uL ,u)
1 =(g,~ 1 -
€
vt
-
- 3- v + --3 v i)u* 3u*
(uL-ul
2
-+
3v -(g.--
v')~
dans )
r
t(f, H-lI2
f v )
V
E
) t
\uI2 -
(r) faible
- 1
2
= 0
.
A'V
=
u,
2Re ( u L , u ) : et
donc
1.4. Applications. On peut effectuer une approximation dlabora de
Aubin
puis
on approche
111 ,
(1.7)
on ctloisit
(1.8) par les metholies generales
. Methode.
On considere maintenant encore
(1.3)
(1.4) aver
f 6 L'(R)
mais
fois
La solution (Lions
:
.
Cea el]
2.1. Position du probleme
avec cette
"en deux temps"
u
- Magenes
Cette fois
correspondante
[I]
2 )
.
est
cette fois dans
H"(Q)
l a brCgularisationll (1.7) (1.8) ntest plus suffisante et
il faut une double regularisation, On considere cive
choisie
scalaire
sur
~ ~ ( une 0 )forme
Ifla plus simple possiblev ,
sur
alors 2)
u
t
On peut alors montrer Theoreme 2.1 (2.3)
..
par
ex. b(u, v)
= produit
.
~"$2)
On definit 2 dans H (R) de
b(u, u) sesqullineaire coer-
. Lorsque u
€
(oh
=itl,
t2] ) . comme
la
solution -
le €40
-, u
, on dans --
a H-'(R).
Cela sgffira pour comprendre comment resoudre le cas general : H-' (7) , s > 0 quelconque,
&
Comparer
J. P. Aubin
. J. L. Lions [l] (ce volume) .
- 290 La demonstration consiste A choisir v = v
C
J. L. Lions
dans (2.2) :
solution de
(t l ~ ' t ~ f ) v E= ( - ~ t l ) - ' Lu 08 (-At11 phisme
(-A + l ) de
- 1 e s t Itinverse de ltiaomor-
1 H (0) s u r H-I (R) , et avec des conditions aux limites 0
convenables. 2.2. Application. On a
(theoreme de Sobolev fractionnaire; cf
Donc
:
Si n 6 3 on peut prendre dans (2.1) g = le solution
u correspondante
est le
noyau
de
.J. Peetre 111 )
x ) ,x E r; S (x - 0
Poisson
.
0
Cf. pour cela une autre methode (utilisant l e principe du maximum) dans
P . Jamet
fi]
.
2.3 Remarque P a r l e genre de mation
A
mt'hode donnee ici,
on peut construire ltapproxi-
(Iten deux t e m p s r t ) des problemes genCraux non variationnels --
I1 devait
.
B . recouvrant A J e t r e possible de trouver une approximation directe de
elliptique
d t o r d r e 2m,
les
ces problemes , mais cela n r e s t pas fait B notre connaissance.
4 2. Cas parabolique. 1.
Nous nous bornons
Dans le cylindre
A
un cas particulier t r e s simple.
Q=R x
] O,T [ ,
on considere l e probleme
-
291
J. L. Lions
oh
A
et
avec
est
Alors
donne
comme
la solution
(cf. Lions-Magenes
[2],
On t t a p ~ r o c h e tce ' la
solution
37
2
tL
f
,
u est
en particulier
chap. 4)
.
(Q)
dans
. avec
Z
L (Q)
probl&me de la facon suivante : on designe par
de
'3 Ut
(1.5)
91
au
+ A u
f ,
=
t
On peut alors montrer le Theoreme 1.1. Lorsque u!
(1.8)
t
-
0, on
&
u
dans
a 2 L (Q)
On peut , apres avoir I1regularisell l e (1.5) (1.6)
(1.7)
. problhme initial
. utiliser l e s methodes de Raviart
2 Remarque. P a r double regularisation (analogue au N. 2 , d r e le c a s
oh
g g H-'
'I Les coefficients de suit s1etend aux
(r), s >
A
0
quelconque
.
4 1) on pourra atteo-
.
peuvent dependre de t
operateurs
PI
en
d l o r d r e quelconque.
et
tout
ce qui
BIBLIOGRAPHIE --------------J . P . Aubin [I] Approximation . . . Bull. J. P. Aubin - J. L. Lions [I] Ce volume. J.H. Bramble [l] J. Cea
S. M. F. MBmoire N. 12, 1967.
Ce Volume.
[I] Approximation
P. Jamet
variationnelle des problgmes aux limites. Annales Inst. Fourier 14 (1964), p. 345-444.
[I] A paraitre
J . L. Lions-E. Magenes Ll] Probl6mes aux limites non homog6nes (11). Annales Inst. Fourier, 11(1961), p. 137-178.
-
[2] Problgmes aux limites non homog6nes et applications. Vol. 1 e t 2, P a r i s , Dunod 1968. J. Peetre
[I]Espaces d'interpolation
P. A. Raviart
et theorgme de Sobolev. Annales Inst. Fourier, 16(1966), p. 279-317.
[I]Ce Volume.
CENTRO INTERNAZIONALE MATEMATIC0 ESTIVO (C. I. M. E. )
J. P. AUBIN et J. L. LIONS
REMARQUES SUR LIAPPROXIMATION REGULARICEE DE PROBLEMES AUX LIMITES
C o r s o t e n u t o a d I s p r a d a l 3-11 Luglio 1 9 6 7
REMARQUES SUR LIAPPROXIMATION REGULARISCE DE PROBLEMES AUX LIMITES Par J.P.Aubin et J . L . Lions
Introduction ----------On donne ici un procede assez general permettant d'approcher dans la topologie l a plus fine possible ( I ) l a solution de certains problemes aux limites variationnels lineaires de nature elliptique, par des solutions de problemes aux differences finies (ou dtautres problemes approches)
.
La methode proposee utilise essentiellement : (i) une Vegularisation" des operateurs differentiels : (ii) des operateurs de prolongement et restriction ( p r ) h' h
.
Une technique du type (i) a dejA ete utilisee (mais non assortie de discretisation) dans J. L. LIONS -G. STAMPACCHIA [I] (cf. (3.14) de ce dernier travail)
. La technique (ii) est utilisee dans J. P. AUBIN [l]
[z] .
Ltusage simultane de ces deux techniques conduit aux resultats de cette note; comme cons4quence on obtient une estimation Sde l l e r r e u r dans les espaces tlusuels" de fonctions differentiables. Le meme genre de methodes peut slappliquer aux problkmes non lineaires ou unilateraux dans l e s cas ( helas r a r e s ) oh l1on conduit un theor e m e de regularite. 1 . .................... P o s i t i o n du p r o b l e m e I . 1. Soit
soit
V
$
un espace de Hilbert s u r
(
,
de norme
(I I( , et
V1 son anti-dual, ( f ,v) design2r.t !o forme shsquilineaire mettant en
anti-dualite
Vt et
.
I. e. la topologie raisonnable de ltespace oh est l a solution du problgme que lton approche, sans perte de regulariete. Pour dlautres resultats dans ce sens, cf. l e s conterences de BRAMBLE (ce Volume)
.
- 296 Soit A~ f((V
:vl) I )
J. P. Aubin et J. L. Lions
, tel que
2 Re ( A v , v ) 34\\v\l , d > 0 , v v € v
(1.1)
L1adjoint A* de on sait que
A
et
A$
A
verifie alors Re
sont
.
(Rv , v ) >oflvV
des isomorphismes de
V
bl v h V et
surv'.
On suppose maintenant connu un "theoreme de regulariten (cf. section 1.2 c i apres pour des exemples) : on suppose connus deux espaces de Hilbert
W et F avec W
,FCV1
C V
l'injection de chaque espace dans la suivant etant continue (mais W -resp. F
-
ntetant pas necessairement dense d a n s v - r e s p y ' ) , de sorte que
la -
a (mais A ntest pas
&s
solution u
V
de
fl
Au = f
" appartient
w. I
necessairement un isomorphisme de W s u r F , sf.
aussi No 5). 1.2. Example
.
Si R e s t un ouvert borne de
fk
, de frontiere
reguliere , on
prend
espace de Sobolev d t o r d r e m muni
On considere ltoperateur
de
la structure hilbertienne
A defini par
De facon generale, $ (X, Y) designe ltespace des applications lineaires continues de X dans Y .
- 297 J. P. Aubin e t J. L. Lions
oh l e s que
a
sont des fonctions donnees dans Pq (1. 1) a lieu . L1espace V1 n'est pas un
R
espace de
et
l'on suppose
R.
distributions s u r
k l'on prend F = H (R) , a l o r s l a solution u dans V de
Si
Au= f
verifie
(le4)
I
A u. = f
des distributions dans R, A = I ( - 1 ) IpI Dp (apqDq),
au sens
et m conditions aux limites "du type NeurnannM s u r r (1) on a a l o r s
et sous des hypotheses de regularite convenables u6 et s i donc l1on prend
W =H
kt2m
(R) (R) alors (1.2) a lieu
1.3. Le probleme e s t maintenant l e suivant : s i f V1 , on connait l a solution donne
de
dans -
est donne
une methode systematique d'approximation l1
Au
F , peut-on approcher
un procede assez
dans -
(cf. J. P. Aubin [l] ) ; s i maintenant
=f1I
,
u
dans - W
?
f
dans
V de est
Nous allons donner
.
general ayant cette propriete
2 ................................. Schemas variationnels regularises.
Soit h
un parametre de
analogue 3 Aubin (i) (2.1)
tels
[l] , pour chaque
un espace
V
h
(Rn destine h
on
tendre v e r s 0. De facon
suppose connus
de dimension finie
(ii) un operateur
p 'injeotif de V dans W h h(iii) un operateur r (lineaire ou non ) de V dans h que ,
((1 111
(') Pour details,
designant. l a norme dans
cf.par
ex. Lions-Magenes
V h
W :
PI ,
Chap. 2
.
- 298 J. P. Aubin et J. L. Lions
(2.2)
d(h)+o
(ii)
(iii)
s i h d o,
p h r h v 4 v dans V (resp W ) Y V a V (resp. W) lorsque h - r o
P l a ~ o n snous dans
l a situation de
1.2. ph et rh satisfaisant
On peut alors construire cf. J. P.Aubin [2] 2
(2.1) (2.2)
(2.3)
.
et avec
$(h)=O
kt2m (si W = H ( R ) ) (1)
(Ihlmtk)
2.3. Schema variationnel regularise
.
Remarque preliminaire. Une fois en possession des operateurs p h"hJ ttnaturellel' de l1Cquation
(ou encore, en posant
(A u,v)
= a(u, v)
l'approximation
:
est :
1
Uhevh Mais il nly a pas de raison pour que, sans hypothese supplementair e , on ait
alors ph u
h
+ U
dans W lorsque
f
est
dans
F.
La methode proposCe consiste B regulariser l e schema
(2.4)
.
La forme b(n, v) On choisit
'2'
b(u, v) forme sesquilineaire continue s u r
W, telle
I). hn) designe alors l a maille du reseau s u r lequel on d i s c r e i s e le ~ r o b l e m eaux limites
2)0n aura interet t~prendre b(u, v) l e plus simple possible produit scalaire l e plus simple possible dans W.
, p a r exemple
- 299 -
V v C W , J3 > 0 .
Re b(v, v ) > t l\lvl\l On definit alors
u
J. P. Aubin et J . L. Lions
(
h
Vh comme
la
( I ) solution de
~ ( hb(ph ) uh, ~ ~ v ~ ) + a ( ~ =~(I.uphvh) ~ . . ~Jvh ~ r c~vh )
(2. 6,
oh f ( h ) > 0
E( h ) 3 0
et
lorsque
.
(2)
h3 0
Ctest l e schema variationnel regularise qui a la propriete recherchee comme le montre le theoreme suivant, qui s e r a etabli au N. Th@or+me 2.1. que
ph
satisfait
3 :
On suppose que A a l e s proprietes (1.1) (1.2) , (2.1) (2.2), et que b
B
satisfait A ( 2 . 5 )
7
la solution du schema
. Si
u
u 3 u dans V lorque h-)o h h (ii) si f e E' et i E (h) est choisi de s o r t e que - s(2.7)
+J&4 0
alors
p h
Estimation de
h
est
-
(2.6) on a :
(i) si f 6 Vt alors
u
h
C )
p
(3).,
7
lorsque
u
h3 0
dans -W
,
lorsque h a 0
,
lterreur (4) on peut obtenir l1esti-
la theorie de ltinterpol;tion
En utilisant
mation de l t e r r e u r suivante ; de facon generale, Hilbert intermediaire
(J. L. LIONS [l]
soit
[w, V]
) de parametre 6
.
l'kspace de 0 <8< 1 (5),
Alors
("11 es! immediat
n
de
verifier
faudra p r e c i s e r ce point
-
que
(2.6) admet une solution unique.
cf. (2.7) c i dessous
.
( 3 ) Cette propriete est verifiee (comme on le rnontre sans peine)pour le schema (2.4) (4)0n utilise ici seulement la theorie hilbertienne de. ltinterpolation, comme introduite dans Lions [I] . Un expose detail16 en est donne dans Lions-Magenes [I), Chap. 1.
J . P. Aubin et J. L. Lions
Sous l e s hypoth6ses du Theor6me 2. I . , o n :
suivant.
Le theorkme e s t egalement demontre au N. 2.1. -Remarque. ---
La theorie de l'interpolation est un outil
commode pour ltobten-
tlc \'estimation de l ' e r r e u r . Pour un autre exemple dvapplication (utili-
tiorl
sant cette fois l'interpolation d'espaces de Banach, comme dans Lions-Peetre
il))
P S ~ donne
dans Peetre-Thomee
3 . ....................................... 1)emonstration d e s T h e o r e m e s 2. 1. et 2 . 2 3.1. Le c a s (i) du Theoreme 2.1.
v = u dans (2.6) et on prend l e s parties reelles des deux h h
On fait membres. I1 vient
On peut alors supposer, p a r extraction dvune sous suite encore notee
h , que Ph Uh
-
Mais comme ( f , p (3.2)
k h
) = a( u
, phvh ), on deduit de (2.6)
R h ) b(phuh; phvh) + a(phuh-u, PhVh ) = 0 Si Iron prend
fort
dans V faible.
"+
on peut p a s s e r
dans (3.2)
v
h
= r v , et puisque p r v
h
la limite :
--
1) L e s c designent des constantes diverses.
h h
4
v dans V
- 301 J. P. Aubin e t J. L. Lions
donc
u
+
= u
. Enfin,
prenant
v = h
u
dans
h
I(h) b ( ~ h u hphuh) , + a ( ~ h u h - uPhuh ' -u)' 7
dans W)
tlloh ( i ) (et en o u t r e \[(h) p h u h - + O
( 3 . 2 ) on obtient
C(Ph uh -u, U)3 0
.
3 . 2 -------L e c a s ( i i ) du T h e o r e m e 2.1. Evidemment l e s estimations du c a s
3. 1 sont valables.
P ~ . e n o n sdans ( 3 . 2 ) v = u - r u e t posons h h h
Sh
= PhUh
- Ph rh U.
I1 vient
d'ob
d
t(h)b(phuh.~ ~ u ~ ) - + a ( $ ~ t(h)b(phuh. >&)= ~ ~ r ~ u ) + a (hur h- u,p h et comme
u W
on peut u t i l i s e r (2.2)
< -
, 1
2
[ (h) \\\ph~hJI1
1
(i)
. Donc
!lShll 2 'c ( f ( h ) 'l ( h )2)
On peut done d l a p r @ s(3.5) supposer p a r e x t r a c t ~ o nque ge dans W faible e t
d'aprhs ( 3 . 1 ) v e r s u,
donc
phUh conver-
- 302 (3.7)
dans W faible (1)
"j u
phUh
Mais on deduit de (3.3)
d'oh b(
J . P. Aubin e t J. L.Lions
que :
1 6 h. $h *. [a ( g; Shl5 Ib( p h r h ~ih) J (h)
( (h)
+
6.
et grace A (2.7) , b ( d , J )d 0 d'oh l e theoreme, c a r h h Ph Uh
On deduit de
(cas oh
- u = Jh+(phrb u-u) .
(3.6) st
de (2.7) que
Ilu - p u 11 I C h h 9 = 1) e t p a r ailleurs, dans l e c a s oh
On interpole
entre ces deux
9 = 0 : (\(u- p u Ill < c h h majorations: on en deduit l e resultat.
.
4. Exemple 4.1. On s e place dans l e cadre de 1 . 2 e t 2.2. On considere donc un probleme de Neumann
avec
Si
A operateur elliptique d'ordre 2m F = H 40) , a l o r s W = H
kt2m
correspond un operateur differentiel (soit
.
( 0 ) de s o r t e qu'A b( u , v )
B) d'ordre
2(kt2m).
introduits .dam J. P. Aubin h et ClJ , conduisant A (2.3) , on doit a l b r s choislr dans
Si on utilise l e s operateurs p et
4
qhap I1 , 2 et
63.
(2.6)
(4.21 avec
(1) Cela, sous lthypoth&se , ~ ( h ) / pour l a convergence forte
< c. L1hypoth&se plus
-
restrictive (2.7) intervient
J. P. Aubin et J. L. Lions
Le schema variationnel regularise (2.6) e s t alors un schema aux differences finies, qui peut e t r e explicit6 6 4 4
1. Si 0
< 0 j 1 , ltespace [w,V]
Mais si
est
-
5
r e , (S. L. Sobolev [I]
.
on a,
[I;]
, chap
11
ltespace
r C (R) designe Itespace des fonctions
differentiables dans
(4.5)
. (cf. J. P.Aubin,
r fois continament
dtapr&s le theoreme de Sobolev fractionnai-
J. P e e t r e [I] )
(R) C
cr(5)
k+2m(l-0) >
-2 + r.
~i
n
n Soit k donne tel que kt2m > - + r Alors s i l1on choisit 0 de 2 n 1 facon que 0 < (k - - - r) + 1 , on deduit du Theoreme 2.2 et de 2 (4.5) que 0 ((phuh-u 5 c p verifiant (4.3)
.
I ~ T.
(1
c
.
(Q)
P a r consequent
5
- -------------Transposition Supposons maintenant que
II
i) W e s t dense dans ii)
V
F e s t dense dana Vt
.
J. P. Aubin et J. L. Lions
et que
k
(5.2) A et A est un isomorphisme Alors le transpose 'A*
de W s u r
F 1)
est un isomorphisme de F 1 s u r W' qui
prolonge A donc :
A
(5.3)
est un isomorphisme de
F' s u r W1
.
Soit f un Clement de W1. Considerons l a solution u dans F' de llCquat ion (5.4)
Au = f
Sans les hypotheses du theoreme 2.1, l e schema variationnel regularise (2.6) a encore un sens et l'on s e pose le probleme de savoir s i p u h h converge vers u dans l'espace F'
.
5-2. Exemple. 011
s e place dans le cadre de 1.2. D1apres J. L. LionszE. Magenes
[l] , chap. 2 , on peut choisir des espaces W et F tels gue llequation
(5.4) conduise teur
des problemes aux limites non homoghes pour lloperaIPI P - ) D ( apq Dq ) $I
.
A =
Sous les hypotheses (5. I ) , (5-2) et celles du theoremes 2-1, s i fE
W 1, 3lors p u converge fortement vers h h Demonstration Soit
de
Vh
pi
(W1. Vh)
le transpose
0
dans
F1
.
de p
et Ah lloperateur h s u r lui meme defini par la forme sesquilineaire :
I
" ~ f e s ~ a c eW contient donc maintenant l e s conditions aux limites ,
J. P. Aubin e t J. L. Lions
oh (fh , vh ) Vh
et Vi
est l a forme sesquilineaire s u r V mettant en antidualite h
.
La solution
u de h
(2.6) est alors
-1 t hi = A h Ph
(5-6)
f
et l e theoreme 2.1 affirme que s i f
converge v e r s 0 dans W
appartient
-1
u - p h uh = Thf
(5.7)
egale B
=(A
-ph%lp\
Les operateurs aP(F, w ) .
)f
.
On montre de m&mequten remplapant
converge v e r s 0 dans
h F, a l o r s
W pour tout f T* h
de
A par A?
F
forment done un ensemble equicontinue de -1
-1
T = T I = (A -p A p t ) h h. h h h forment egalement un ensemble equicontinu de d ( w t ; F t ) P a r transposition, l e s operateurs
Pour que l e s operateurs (
W F ) , il suffit a l o r s que
element
f
11
.
T converge6 simplement v e r s o dans h Th f converge vers o pour tout
/IFl
dlun ensemble dense dans Wt. Prenons a l o r s Vt
: Vt est
dense dans Wt et, dtapr&s l a premigre partie du theoreme 2-1 , on sait que
\I ~ h (IFf
Q I ( T ~f
11
= J U -phuhJJ:end
On en deduit donc que
tend v e r s o avec h.
si
vers 0 f 6 Vt
.
BIBLIOGRAPHIE J.P. Aubin 611 Approximation des espaces de distributions et des operateurs diffdrentiels Bull. s. M. F. Memoire N. 12, 1967
.
[2] Evaluation des e r r e u r s de troncature des approximations des espaces de Sobolev. A paraitre dans Journal of Math. Analysis and Applications. Bramble
[I]
Conferences de ce Volume et Bibliographie de ce travail.
[I) Espaces intermediaires entre espaces hilbertiens ;applications. Bull. Math. R.P.R. t.2 , (1958) , P. 419-492
J. L. Lions
b]
J . L. Lions-E. Magenes cations Vol. 1, 1968
.
Problemes aux limites non homog&nes et appli-
( 1 1 Sur une classe dlespaces dfinterpolation. J . L. Lions-J. Peetre Institut des Hautes Etudes Scientifiques N. 19 - 1964
.
J. L. Lions-G. stamPacchia[1] Variational Inequalities Math XX (1967), p. 493-519.
.
-
Comm. Pure Applied
[I] Espaces dfinterpolation et theoreme de Soboleff. Annales J. Peetre Institut Fourier, 16, 1 (1966), p. 279-317
.
J. Peetre-V. Thomee
value problems.
.
[I] On the rate of convergence for discrete initial A paraitre.
S. L. Sobolev [l] Applications de lfanalyse Fonetionnelle en Physique Mathdmatique - Leningrad. 1950 ,
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
W. V. PETRYSHYN
ON THE APPROXIMATION-SOLVABILITY OF NONLINEAR FUNCTIONAL EQUATIONS IN NORMED LINEAR SPACES
C o r s o tenuto a d Ispra. dal 3-11 Luglio 1967
ITERATIVE CONSTRUCTION OF FIXED POINTS OF CONTRACTIVE TYPE MAPPINGS IN BANACH SPACES by W. V. PETRYSHYN (University- Chicago) Introductory remarks. It i s known that the problem of finding solutions of nonlinear functional equations (e. g. , nonlinear integral equations, boundar y value problems for nonlinear ordinary and partial differential equations, etc.) can be formulated in t e r m s of finding fixed points of a given nonlinear mapping defined on some subset of an infinite dimensional functional space. F o r compact mappings a general existence theory of fixed points based upon topological arguments has been constructed over a number of decades Lassociated with the names of Brouwer, Poincare, Lefschetz, Schauder, Leray, Tichonoff, Rothe, Krasnoselsky, Altman, and others). More recently, there has begum a systematic study of fixed points (their existence and actual construction) of various classes of noncompact mappings (BrodskiMilman
- , De M a r r
Kachurovsky [19,20], [8,9]
Bg
p 8 1 , Browder
p a , Sb, 5 , 6, 4 1 , Kirk
, Edelstein E l , 12,131 , Belluce-Kirk
De P r i m a D 0-3 , Lions and Stampacchia D7] -
, Kaniel p 2 ]
, Shinbrot
Opial i30) , Lees-Schultz [26], redo 1 7 1 , Petryshyn-Tucker [37
PO] , Petryshyn
23
,
rl] - - , Gohde
, Browder-Petryshyn
[31,32,33,
35, 3 6 1 ,
de Figueiredo L15, 161, - Browder-de Figuei-
1 , and others) .
The purpose of this part of my talk is to survey, unify and extend a number of recent results concerning the iterative construction- of fixed points of noncompact contractive and strictly pseudocontractive mappings acting in a Banach o r a Hilbert space. We first devote a section to the definitions of various concepts used in this paper and to the historical development
of the iterative method in the construction of fixed points of contrac-
tive mappings; then we devote a section to the complete proofs of the most recent results which, a s will be shown below, include all those men%-
W. V. Petryshyn
ned in t h e s u r v e y section. In the l a s t section we discuss t h e convergence of fixed points of s t r i c t l y contractive mappings
kA studied in [6],
where
< 1. It is worth noting that o u r proofs, which use the a r g u m e n t s analogous t o those in (Is , 8 , 9, 3 0-1 a r e k
is
a r e a l number such that
0
s u r p r i s i n g l y simple. 1.1. Historical r e m a r k s and s t a t e m e n t s of r e s u l t s and definitions. Let A
X
b e a Banach space,
X and
C a closed convex s u b s e t of
a (possibly) nonlinear mapping of
C
X which i s contractive
into
( o r nonexpansive), i. e.,
1 Ax-Ay I( 5 IJx-y I(
(1.1)
for a l l
x and y in
C.
The study of the iterative construction of fixed points of contractive mappings i s an extension of the c l a s s i c a l theory of nthe method of successive approximationsn f o r s t r i c t l y contractive mappings developed by Picard, Banach, Cacciopoli and o t h e r s , where A is s t r i c t l y contractive on
1 x-y \(for
(1.2) It i s known 24
a l l x, y C and s o m e kc1
that if a s t r i c t l y contractive o p e r a t o r
then the P i c a r d sequence { x x
ntl
= Ax
ntl
n
C if
A
maps
C into C,
) given by
n+ 1 = A x (x given in C; n = 0, 1,2, 0 0
...)
converges t o the unique fixed point of A in C. of C into C, the P i c a r d s e -
In the c a s e of contractive mappings A
quence need not converge, t h e r e need not b e any fixed points, nor need the fixed point b e unique if it does exist. T o obtain s o m e positive r e s u l t s on the P i c a r d sequence f o r contractive mappings, we need to impose further r e s t r i c t i o n s on the Banach s p a c e s X , the s u b s e t s
A
.
C and o r the operator
F o r the s a k e of completeness and c l a r i t y we r e c a l l the following de-
finitions: X i s uniformly convex if f o r any such that
1 X-y 1 1 r
for
IIx \I< 1
and
IIy II-<
6
> 0 there exists
a 6 ( 6 )>O
1 implies that l(xty tyl<2(1-6
(t));
W.V. Petryshyn X
is
x, y e H such that
l(x
tely continuous if
P
if
x
n
)( txt(1-t)y
sti-ictly convex if
2
x implies
1 <1
for all
I= 1 y (1 = 1; a mapping
P of
i s continuous and compact; P x n Px, where
t
nLn
e
(0, 1) and all
X into
X i s comple-
P i s weakly continuous and fl?n
denote the
weak and strong convergence, in X, respectively; P is strongly continuous if
x
3 x implies P x +Px. Using Schauderls Existence Theorem 1 [38], n n Krasnoselsky [251proved the following positive result for completely conti-
nuous contractive mappings. Proposition 1.1. vex subset of
X
If
X is uniformly convex,
and A a completely continuous contractive mapping of
{xntl} determined by
C into C, then the sequence
-
x n t 1 = 1/2Ax n t 1/2xn
(1.4)
converges to a fixed point of In
39
C a closed bounded con-
A in f2
(x0 given in C; n = 0,1,2,
...)
.
Schaefer extended Proposition 1.1 in two directions by proving
the following. Proposition 1.2. If X, C and A satisfy the conditions of Proposition 1.1 and
A
i s a fixed number such that 0 <
a < 1, then
the sequence
{xnt 1)
determined bv
-
converges to a fixed point of Proposition 1.3.
A
- C.
x
n t1
X
i s a real Hilbert space
H and A
ded convex subset of into C then
If -
A in C. a weakly continuous contractive map of C
determined by (1.5) converges weakly to a fixed point of
in
Let us
H, C a closed boun-
add that the proofs in
assertion that when
[25,39
1were essentially based on the
X i s uniformly convex and
-
A i s a contractive map
C, then the mapping AX=A A + (1-A )I is asymptotically reguntl lar, i. e., (AX x x ) 0 as n -+a for any given x in C 0 X O 0
of
C into
-
- 312 (see Browder-Petryshyn
W. V. Petryshyn
14 Edelstein extended the validity of
[8] ). In
Proposition 1.2 to strictly convex Banach spaces stein used
Mazurls Theorem
X. In his proof Edel-
(see [29] ) since in case
X i s strictly con-
AX i s asymptotically regular without first proving the strong convergence of the Picard sequence { A: vex we cannot conclude that
XI.
As was already noted in Introductory remarks, the study of the existence and of the iterative construction of fixed points has been recently carried further for non-compact and contractive type mappings. Namely, suppos'e
'
r > 0 in a real o r complex r Hilbert space H, S the boundary of B and A a contractive mapping r r of B into H which satisfies on S the Leray-Schauder condition r r C = B (0) denotes the closed ball of radius
Ax - p x { 0
(1.6)
for all
x
Sr and a n y p > l .
in
Applying the recent theory of monotone operators Browder P r i m a [ l o ] ) proved that
A
existence theorem the writer micompact mappings whenever
{ un\
has
a fixed point in
(see also De
.
Using the above r 1 3 5 1 has derived the following result for de-
A ( a mapping
P
i s a bounded sequence
B
[ 3a]
i s said to be demicompact if
{ un
and
- Pun 1 i s strongly con-
vergent then there exists a strongly convergent subsequence { u n1. \ L
-A
Proposition 1.4. Let B
r
into
H which satisfies
E ( 0 , l ) the sequence
i'n+l\
converges to a fixed point of Proposition
1.4
be a demicompact contractive mapping of (1.6)
. Then for
any
x o Br ~ and any
determined by the retraction-iteration method
-
A in Br.
extends in two directions the iterative construction
of fixed points of contractive mappings. First, we replace the assumption of compactness of of assuming that
A by demicompactness; second, for such mappings, instead
A
maps
B
r
into
B
r
we assume only that
A
satisfies the Leray-Schauder condition
on S
In our view the latter result r' for demicompact operators is the main advancement of the theory. Note that if
A is compact, then it is clearly demicompact but the converse is
not true. For example, it is easy to see that strictly contractive and
T is
S is compact, is demicompact but it is certainly
into B then r = 1 for each r r n (1.7) reduces to the method (1.5). In this case, Proposition 1.4.
not compact. Clearly, if n and
A = . T+ S, where
A maps
B
i s valid for an arbitrary closed bounded convex subset
C without any chan-
ge in its proof not only in Hilbert spaces but also in uniformly convex Banach spaces. Thus, in this case, Propositions 1.1. and 1.2 Proposition 1.4. Let us add that
(1.6) i s implied by the following conditions: for all x
Remark 1.1. the retraction
in
S
r
It was shown by de Figueiredo-Karlovitz F8 ]that if C = B (0) i s contractive for a Banach spa-
R C of X onto
X of dimension
ce
follow from our
> 2 , then X
1 i s a Hilbert space. This fact indicates
that the retraction-iteration method (1.7) i s essentially restricted to Hilbert spaces. Further progress in the theory of the iterative construction of fixed points was made in Browder-Petryshyn
[8]where
concerning the Picard sequence (A"x\
was established. We mention here
only two results
from [8 ]which a r e directly relevant to our discussion.
We first reaall that a mapping
closed' *I
a number of results
if for any sequence
P
of
tun/ in
X
into
X with
-
X is said to be strongly u - - \ u and Pu -\v n n
we
have Pu = v. (*)some authors refer to the above property a s ndemiclosedness. U We prefer the tirill ?strongly closedn(employed alsi, ill since it i s more in harmony with the concept of wstrong'continuityuused, especially, by Soviet mathematicians On the other hand, the therm ndemiclosednessn of P was used in [41 , 24 , 361 to desci-ibe the situationu u +un and Pu --Iv implies that Pu = v" n n which seems to be more in harmony with the concept of ndemocontinuityn(i.
pg)
U n + ~ implies Pun-Pu thematical community. J
)introduced by Browder and accepted by the ma-
- 314 -
A""
Proposition 1.5. n x-A x + 0 a s
-
into X -
such that
Let -
A
W. V. Petrhyshyn
be an asympotically regular. (i. e. -
n , w for every
x in X) Contractive mapping of
I-A i s strongly closed. Suppose that the s e t
A is not empty. Then for any
fixed points of
any weakly convergent subsequence
{ inx0 /-l i e s in
F(A)
a single point
y,
. In particular,
X
F(A) of
-
xn € X the weak limit of
1 A"(') x0 \ of the Picard sequence i s reflexive and F consists of -if X -
n then - A x0 -y We r e m a r k that Propostion. 1.5 i s valid, in particular, when
.
X
is
a Hilbert space o r a Banach space having a weakly continuous duality mapping ( * ) J. In this case it has been shown in
[ 36 ] (see also
15.91 ) using
A
i2 contracti-
the theory of monotone and J-monotone operators that if ve then
(I-A) i s strongly closed. Hence in this case we drop in Proposi-
tion 1.5 the assumption that (I-A) i s strongly closed (see Theorem 4 in
Remark 1.2. X instead of
Proposition 1.5 was stated for mappings
C into
A
of
X into
C in order to employ the theory of monotone'and
J-monotone~operators in the proof of strong closedness of
(I-A). This i s
not a restriction in case of a Hilbert space since a contractive mapping defined on
H. But
can be extended to a contractive mapping defined on all of
C
it is a restriction in case of a Banach space since, in general,
such an extension theorem i s not available.
-A
Proposition 1.6. Let mapping of
X into
- X with
further that
A satisfies
(m) (I-A) Then for each
be an asymptotically regular contractive a nonempty s e t
F(A) of fixed points. Suppose
the followinn condition.
maps bounded closed sets into closed sets. x in X the Picard sequence 0
convergence to a fixed point
(*)L% X be the dual space of X and (f, x) the value of the linear functional f in X? a s the element x of X. Let y be a continuous strictly increasing real values function defined on R+ = {t:t > 01 with p (0) = 0 A mapping J of X ) i s called a duaiity mapping with a gauge function (r if into (see (Jx, x) = JX ( x 1 and 11 J x r ( 11 x (1 ) for every x in X.
.
1
I(=
W. V. Petrhyshyn
in F(A ). It was shown in A'
= A I t (1-A
[8,39] that for any fixed
) A (or
A E(0,l)
the mapping
A + (1-A )I) i s a contractive asymptotical-
AX =
F ( A i ) = F(A) (or F(Ah ) = F(A)) provided that A
ly regular mapping with
i s a contractive mapping of
a uniformly convex Banach space
with a nonempty s e t F(A) of fixed points. Since, furthermore,
- Ata =(I-A)(I-A) ( o r I-AX =A(I-A)) we s e e that
we have
I
condition
( a )if
and
only if
X into X forJ€(O, 1)
A satisfies
A\ (or AA ) does also. Using the above re-
m a r k s we have the following corrollary of Proposition 1.6. Proposition 1.7. Let
A be a contractive mapping of a uniformly con-
-
vex Banach space X into X with a nonempty s e t F(A) of fixed points. Suppose that
A
satisfies condition (0). Then the sequence
(xn.1)
determi-
ned by (1.5) o r by
converges to a fixed point of A. It will be shown in the next section that Proposition 1.7 remains valid when
A
i s assumed only to be a contractive mapping of a closed bounded
convex subset C of a uniformly convex Banach space c a s e we drop the assumption that points since in
X into
A has a nonempty set
C. In this
F(A) of fixed
this case the existence of fixed points of A in C follows
from the recent existence theory proved independently by Browder [4], Kirk 1231 and Gohde 1191 tness dibon
. Since,
a s i s not hard to see, the demicompac-
(and, in particular, compactness) of
(a)(*), it follows
A implies the validity of con-
the modified Proposition 1.7 i s an extension of Pro-
position 1.4 for the case when A map
C into
C.
(*)1t was stated in [ 8 1 that condition ( a ) i s equivalent to the demicompactness. This i s c l e a r 6 not the case since the mapping A=I satisfies condition ( t r ) but certainly is not demicompact. The authors of 181 caught this inaccuracy too late for the correction to be inserted in the paper 8
- 316 -
W.V. Petrhyshyn
When in Proposition 1.5 the space X is a Banach space with a weakly continuous duality mapping (and, in particular, a Hilbert space) Opial /30] strengthened the result of Proposition 1.5 (or rather of Theorem 4 in
8 ) by showing that the entire sequence
convergent
Anx0) is necessarily weakly
(see also Theorems 7 and 8 in [9) )
following was provetl in [30] Proposition ----
.
. More
specifically, the
1.8. -Let C be a closed set in a uniformly convex Banach
space X having a weakly continuous duality mapping and let -
A be a con-
tractive mapping of C into C with at least one fixed point in
C
--
x in C and any A( ( 0 , l ) the Picard sequence 0 convergent to a fixed point of A in C.
. Then for
any -
-
We recall that fixed point of A
if
C is bounded, then the existence of at least one
in C follows from the results in
@,
19,233. Let us add
that Opial did not use the theory of monotone o r J-monotone operators to prove the strong closedness of Schaeferfs [39]
(I-A). His arguments were similar to
. We recall that for weakly continuous contractions in real
Hilbert spaces the weak convergence of [A; Schaefer 1391
-
xO}was first proved by
. The extension of this result to general contractions was
thus carried out in two stages, the proof of Proposition 1.5 by BrowderPetryshyn [8],
and the proof of Proposition 1.8 by Opial
PO] .
Remark 1.3. It was already noted in [9] that Gohdels assertion in 1 1 4 that the weak convergence result in E9]
follows already from the
existence theorem i s inaccurate since the proof by Schaefer uses two facts, the existence of !it least one fixed point and the fact that if a subsequence of {A;
xo) converges -ueakly to y, then y is a fixed point of A. This
latter fact follows a s in the proof of Theorem 7 i n p ] and Opial has utilized this argument together with a simplified form of'schaefer's proof. Let us add that under the stronger assumption that the strong limit set of the iterates i s nonempty, convergence results for contractions have been given by Edelstein E l , 12,131 and Gohde
LO]. However,
there seems
W. V. Petryshyn
to be no explicit way of extracting convergent subsequence and therefore the methods a r e not really constructive.
In [9]
Browder and the writer studied various classes of nonlinear
operators (contractive, strictly pseudocontractive, psedncontractive) mapping
C of a real Hilbert space
a closed bounded convex subset
H into
C
o r into H. F o r the purposes of our present discussion we shall mention only those results from [9]
which bear direct relation to the iterative con-
struction of fixed points by means of the processes (1.5) o r (1.7)
.
As a first example we state the following proposition which for a Hilbert spdce i s an extension of Proposition 1.8
.
Proposition 1.9. Let - A be a contractive mapping of
-
that if
u = R Au, then u i s C C and the sequence
u 6 C and -if
fixed points in
a
-
kntl)
- C into - H. Suppose fixed point of A. Then - A has -
determined by the retraction-
iteration method
converges weakly to a fixed point of
A
in C.
-
C = B (0) then our second assumption reduces tp the r Lerady-Schauder condition (1.6) and in this case the retraction-iteration method Note that if
is practically constructible. Following
b]
we say that a bounded closed convex s e t
mly smooth with smoothing constant point
x
1
R > 0 if and only if for each boundary
in C
1) C has only one supporting hyperplane at and (x u) 1'
->
c
0
for all u in
C)
I
x (i.e., (xl, V) = c 1 0
I
1
in H such that u -x = R and u -u < R 0 0 1 0 all u(C (i.e., CCB (u ) and x 6 ~ B ( U )). R 0 1 0 Our next result i s the following extension of Proposition 1.4.
2) There exists
for
-
C is unifor-
u
11-
Proposition 1.10. Suppose that in addition to the conditions of Proposi-
- 318 tiion 1.9 we a,ssume that A
W. V. Petryshyn
i s a demicompact map of
R
is uniformly smooth with smoothing constant
{xntl)
C into
H, where
-
. Then the sequence
determined by the retraction-iteration method (1.7A) converges to a
fixed point of
A in C.
-
Let us note that the main difficulty in proving Proposition 1.10 the proof of the demicompactness of in
[9]
R A.
l i e s in
The r a t h e r tricky proof of this
C
utilizes the idea (which, essentially, was used for the first time
by the w r i t e r in [35]
) that outside some narrow band about the surface of
C the retraction is strictly contractive. Needless to say the chief difficulty in the application of Propositions 1.9 and 1.10 to actual problems l i e s in the fact that unless
i s a ball there i s no efficient way of construction
C
R,
the retraction mappings Following
p]we
L
.
s a y that the mapping
A
tly pseudo-contractive if there exists a constant k IIAX-A~
112 51x-yll
2
t kll (I-A)X
It has been shown in 1 9 3 that for every fixed t
of C into
H is stric-
< 1 such that
- (I-A)~II
f o r all x, y c C.
such that 0
< t -< 1-k the
A = tA + (1-t)I i s contractive, A has the s a m e fixed points a s t t A in C and (A ) = i I + ( I - ) ~ ) A =~ A where = 1-(I- )t with tA . t < 1-k for any fixedl\€(0,1) if and only if > k. Consequently we have mapping
7
the following extensions of Propositions C into
C
-
A
mapping
C. Then , for
any
A be a strictly pseudocontractive mapping of x 6 C and any 0
converges weakly to a fixed point of
A
T
(1.9) and (1.10) for
.
Proposition 1.11. Let C into
Y
i s demicompact, then
{" A
d
A
yI such
that
k
<
in C. If additionally we assume that
-
x converges strongly.
Similar result holds for the retraction-iteration method (1.7) when it i s applied to strictly pseudocontractive mappings which
on
A
of B (0) into H
r
S satisfy the Leray-Schauder condition (1.6) , r Remark 1.4. In [ g l t h e authors also discuss the construction of fixed
W, V. Petryshyn points of a pseudocontractive mapping
A
of
B (0) into
ing the Leray-Schauder condition on S , where A r t r a c t i v e if
1)
- ny[ 2
Ax
H
r
with A
satisfy-
is s a i d to be pseudocon-
--
x, y 6 C.
for all
The mothod used in the construction of fixed points of s u c h mappings i s e s sentially based on t h e s t r i c t contraction mapping principle combined with the radial retraction
R. The idea is that for each fixed
we construct the fixed points
in B (0) of the mapping s r + ( 1 - s ) uo f o r any uo in B r (0) by the method
u
n u = l i m (u ) = l i m ' n n
(1.10)
{ R(l -
( 1 ~ ~ ) w)
,
with 0 < s < 1
s
A = sA s
wo 6 Br(o)
w h e r e the r e a l number /A l i e s in a c e r t a i n interval ( s e e [9] show that
u +u,
where u
S
is
a fixed point of
A
+
and then
closest to u
0
. We
will not investigate the method (1.10) s i n c e this is outside t h e scope of o u r discussion. Let
X be a r e a l s e p a r a b l e Banach space. A p a i r of sequence ( X , n) { p n ) ) , w h e r e each X is a finite dime1:sional subspace of X and P n n i s a l i n e a r projection of X onto Xn , is s a i d t o be a projectionally
{
.
cqmplete systeln if P x +x f o r each x in X ( s e e [34] ) n Definition. A nonlinear mapping A of D(A) ( C X) into
gy -compact --
be called tly l a r g e dominating
n
y
and
if
(i. e.,
sequence {xn} with {P Ax
bn)
- pxn)
X will
P A is continuous in X f o r a l l sufficienn n t h e r e e x i s t s a constant Y > 0 such that f o r any p
p
if
2 7 if )c >
x$ Xn
n D(A)
0 and
p
> y if
= 0) and any bounded
the s t r o n g convergence of the sequence
implies the existence of a strongly convergent subsequence
:noanelement
x
in
D(A) s u c h
that
x +x n.1
and
W. V. Petryshyn
Note that our present definition of
PY -compactness is a generaliza-
tion of the concept of P-compactness introduced and studied by the writer in [31,32,33J and P
n
. Indeed,
when
Y = 0 . D(A) = X
and the sequences
{xJ
a r e such that X n C Xn+l(n = 1.2.3
,...). nU Xn = Xand 1 P n1 -< K , K>- 1, then the
Po-compactness i s just the P-compactness. In what follows we shall retain the the term P-compactness whenever
IX,
= 0 , Let us remark that upon a clo-
s e r examination of the proofs of fixed point theorems in L31,32,33J it becomes obvious that in those proofs we have only used the properties of A and pact
X which a r e embodied in the requirement that
A be P -corn1 (according to our extended definition) a s a mapping from B-(0) 1-
into X. However, in the applications of fixed point theorems in [32,33] to monotone operators we still have to use the definition a s given in 31,32 It turns out that our present extended definition of
Py
.
-compactness is
more suitable for theoretical and applicational purposes and, particularly, for the purpose of unification of various results. Returning to our discussion of iterative methods we would like to finish this section by stating the following result proved in [37]
for
P-compact mappings. Proposition 1.12. Let
-
C be a closed bounded convex subset of a real
uniformlv convex Banach space having a projectionallg complete system. If
A is a contractive P -compact mapping of C into C, then the 1 determined by (1.5) converges to a fixed point Picard sequence ('nt~
-
1
2.1 Convergence proofs. In this section we prove three theorems concerning the strong and weak convergence of the Picard sequences
( A;
x \ under certain conditions on A and
tive o r a strictly pseudocontractive mapping of
X,
where C
A is a contrac-
into C. We have seen
in the previous section that in each case the strong o r the weak convergen-
W. V. Petryshyn
ce of ( A x~) depended upon the existence of fixed points of turns out that in c a s e
A
is
A in
C. It
a contractive mapping (which is the case
that we a r e interested in this discussion) the properties and the additional A
conditions on
which insure either the strong o r the weak conver-
gence of the iterative method (1.5) admit at the same time a simple proof of an existence theorem which, a s will be seen below, i s general enough to cover all c a s e s surveyed in Section 1.1. Thus, there i s no necessity to r e f e r the reader to the m o r e general existence theorems proved in
.
[4, 19, 233 by rather complicated arguments
Unless explicitly stated otherwise in this section the set
C will al-
ways be assumed to be a closed bounded convex subset of a given space
be a general Banach space and let
Theorem 1. Let tractive mapping of closed s e t in
a subset
X. Then
-A
C of X into C such that (I-A)(C) -is a
has a fixed point in C.
Proof. Assume without l o s s -
r
Let
n
be a con-
of generality that the origin
be a sequence of numbers such that
0
0 ( C.
< r < 1 for each n
n
r -1
a s n+m. It i s obvious that A = r A i s a strictly contracn n n tive mapping of C into C. Hence, by the s t r i c t contraction mapping and
principle, for each
n there exists a unique
A u = u F o r the sequence {un n n n'
u since re
n
r -1
n of (I
- Au n = rn Aun and
{AUA
C
- A)(C) . Since,
and therefore
A
-Au
C is
} n
=
u
in C n thus determined we have
(r n
point
- 1 ) A un- 0 a s
bounded. Hence
by assumption,
has a fixed point in
0
such that
n+m l i e s in the closu-
(I-A)(C) i s closed,
0 t (I-A(C).
C.
Lemma 1. If in addition to conditions of Theorem 1 we assume that X
is
strictly convex, then the s e t
a closed convex set.
F(A) of fixed points of
A
in -
Cis
W. V. Petryshyn Proof. By Theorem -
F(A) #
1,
F(A) and z = tx t (1-t)y for and convexity of
Since
C,
z
x
and y a r e in
- Ay)
C and
Ax
Ax and
But we ;!so have that must have that Remark 2.1.
Ay
and, hence, on the line through
1) AZ- AX^ -< 1 z-xl
Az = z
and
. The closedness of
Lemma 1 i s valid under
I
- Az =
A z is on
f o r some real number a. Therefore, the vector
the line through
A
t 6 (0,l). Then, by the contractivity of
X i s strictly convex the above inequality implies that
= a(Az
we
6 . Suppose that
x and
< ~ I Z - ~ ~.I Hence
A Z - A ~ ~
F(A) is obvious
.
X
the assumption that
strictly convex and the contractive mapping
y.
is
A has a nonempty set F(A)
of fixed points.
-
a uniformly -
Lemma 2. Let
X be
C (cX)
a contractive mapping of
AA = A l t ( 1 - ))A the mapping n xO - A> xo-O a s n +m
A?'
-
points a s
for
x
0
x0 - u
1 An
x
-
I
in
0
C) and
- A>
F(A) = F(AA )
C and i s contractive. Now let
and, for any given
C. Then for any fixed
i s asymptotically regular
A.
Proof. It i s obvious that into
into
convex Banach space and
and that
u be a fixed point
C, let
A€
A
(0, 1)
(i.e.,
has the same fixed
AA of
maps A in
C C
be the Picard sequence. Then
1. 1 1 ~ : ~ ~
xo-~uA
xo- ul
for all n
.
converges to some d > 0. If d = 0, then 00 A x --r 0 and the proof is finished. Suppose now that d > 0. Altl Xo 0 0 Since Au = u and Thus
-
xO
n
u
- 323 and since
W. V. Petryshyn
I
( An t 1 x
-u +d O , ~ I ~xO-u ; J+dO and P A n x0-uI
5
1
A ~ X ~ - U I it /
follows from the uniform convexity of X that
since
A-
1 = ( 1 - ( I - A ) ; hence the proof of Lemma 2 i s complete.
We now prove the first main theorem (compare it with Proposition 1.6) of this section by the simple arguments analogous to those used in r8).
X be a uniformly convex Banach space and A a
Theorem 2. Let
--
C of --
contractive mapping of the subset satisfies condition
(I-A)
(d),i.e.,
into C. Suppose further that -
X
(I-A) -.maps every (bounded) closed
subset into a closed subset. Then the set F(A) of fixed points .-of C --in C i s a nonempty closed convex set and for --of A - each x0 in-- C An xO} converges (strongly) and 1( ( 0 , l ) the sequence ------ to a fixed --
1
point of
A -in
C.
(4) , (I-A) (C)
Proof. Since, by condition
-
from Theorem
1 that
F(A)
# fl . Since
u
is
a
fixed
1
point of
A,
i s decreasing
.
closed set it follows
uniform convexity of
its strict convexity, Lemma 1 implies that If
is a
F(A) i s
and thfrcfore of
X implies
a closed convex set. A
the sequence
It suffices therefore to show that there exists a subsequence of IA;
xO
1
which
converges strongly to a fixed point of
be the strong closure of the s e t n (I-AA ) (AA xo) -+0 a s n m (I-Al )(C)
[ A;
. Hence
xO}
. But , since for any AE (0, I),
(1-Al ) satisfies condition condition
(4) , the
set
( 4 )if
in
. BY ~ e r n r n a2,
0 lies in the strong closure of A
(
1 ( A ) we s e e that
and only if (I-A) does also. Hence, by 0 lies in
exists a strongly convergent subsequence of y
C. Let G
(I-AA )(C) is closed since G is closed (and
necessarily bounded) and therefore
an element
A in
G such that
(I-A
)
Y
[ A;
(1-AA )(C)
. Hencc there
xu} which converges to
= 0, i.e.
, y i s a fixed point
-
324
-
W. V. Petryshyn
We shall now prove that in addition to completely continuous o p e r a t o r s the two c l a s s e s of mappings which satisfy condition and
(4) a r e demicompact
P -compact mappings studied by the w r i t e r . 1
-
L e m m a 3. L e t
-demicompact -
X b e a general Banach s p a c e and
mapping
Proof. L e t
of
be a sequence in
Q
.
such that (I-A)u
a bounded sequence, and of
. Then fi
Q be a bounded closed s e t
-.-
tness
D(A)(C X) into X
1,
(I-A)un
n
+v
in
A a continuous
s a t i s f i e s condition-
D(A) and
a s n+m.
Since
) i s strongly convergent,
for some
u
in
and the continuity of A
Q
C
u
is
J
n by demicompacI
A, t h e r e e x i s t s a strongly convergent subsequence
Hence the closedness of
/ uk]
let
\
IUni}
imply that u n . + u 1
(I-A)u = v and L e m m a 3 is proved.
Q with (I-A)u,,-+ 1
Let -
-L e m m a 4. plete -
([xn
\,
D(A)CX -into
2 . Then -
be a sequence
ger
k
and shall
be a r e a l Banach s p a c e with a projectionally com-
pn}). L A A
Proof. L e t system
X
A
be a Lipschitrian
-
Q
1
is demicornpact and hence s a t i s f i e s condition P .
Q b e a bounded closed s e t
in
P -compact mapping of
s u c h that
in
(I-A)uk - + ~
D(A) a s k -c w
( 1 ~ , ~{Pn)) ) is projectionally complete in and
= 1 k a s s u m e that
and t h e relation
t h e r e e x i s t s a n integer n(k)
imply that with w
. Since the
X , f o r each inten(k) (which we c a n
> k) s u c h that Ilu k - P .(kIUk
(I-A)u -+ v k
and l e t
1
< 'k
-
n(k) - Pn(k)Uk
'
in X
we have
This n(k)
1 Pn(kfWn(k)-Wn(k) + v 1 -< 1Pn(k) Awn(k)-Pn(k) Auk 1 Since
{ P n ) is uniformly bounded,
Lipschitzian, s a y with constant
say,
L > 0,
by
K> 1 and
A
is
it follows f r o m the above inequa-
W. V. Petryshyn
Thus, since
/
P -compact, there exists a subsequence 1 ~n(j)I and an element u in D(A) such that w .*u and Wn(k) n(J) P . Aw .*Au a s n(j) -co Consequently, u.-uI/( -W ~ ( J I "(J) J j n(j)
I
+
is
A
1
.
H wn(j) ,-u II- t o
lies in Q i.e.,
A
as
n(j) --coo.
Since
1
IJu
+
{ uj ] , being a subsequence of
Q is closed we must have that u e Q
and
II
and Au = v,
i s demicompact. The last assertion of Lemma
4 follows from
the above proof and Lemma 3. Remark 2.2. In virtue of Lemmas 3 and 4 and the observations following Proposition
1.4, Theorem 2 implies the validity of all propositions
of the previous section asserting the (strong) convergence of the sequence
{A;
xO) provided that in these propositions
A maps
C of a uniformly convex Banach space
convex subset
a closed bounded
X into
C.
To derive the strong convergence of the retraction-iteration method, Remark 1.1 implies that we need consider only Hilbert spaces tue of Lemma 3,
H. In rip-
the strong convergence of the retraction-iteration me-
thod (1.7) o r (1.7A) will follow from the following lemma. Lemma 5.
Let -
bounded convex subset mly smooth
A be a contractive demicompact mapping of a closed C of r e a l space
with smothing constant R.
H onto
- C, then the mapping Note. In case C = B (0), r
. For
in[35]
H into H, where C is unifor-
-
If -
R
C
denotes the retraction of
RC A is demicompact. Lemma 5 was first proved
by the writer
C, which i s uniformly smooth with smoothing constant R,
Lemma 5 was proved in [9]
by the arguments which we use here. It is
essentially based on the following lemma whose rather tricky proof is given in
[9 1 .
-
Lemma 6. If
C i s ,uniformly smooth with smoothing constant
R
and
- 327 -
A& (0, 1). t < 1-k
if and only if
i s to hold for any
1E
ty
r
W. V. Petryshyn
> k. Indeed, if t < 1-k and 1 > Y > 0
(0, 1) it follows that
must satisfy the inequaliI ) > 1 - (1-k)(l-A) > k . On the other hand, if k<
= 1-t(1-
= 1-t(1-l\ ) is to hold for any) C (0.1) the parameter
the inequality k follows
< 1-t,
i.e.,
t
< 1-k. Now, since
that if A is demicompact then s o is
Theorem 2, the Picard sequence fixed point of
t
l=
must satisfy
A -1 = (1-f)(A-I)
Ab,
, it
and consequently, by
{ (A~); x O j= {A"
xO) converges to a
C,
A in
To justify the obeervation following Proposition 1.11, concerning the convergence of the retraction-iteration method for a strictly pseudocontractive mapping of
B (0) into H, by the second part of Proposition 1.11 and 'I
r
Lemma 5, it suffices to show that tion
A
satisfies the Leray-Schauder condi-
i. e., A] x -j'$ x # 0 for all x in S and any > 1. r ' r AI xO- F o x O = 0 for some x in S to the cbntrary, that 0 r
on S
Suppose, and some
Po
r
A~ => 1 , since > 1 , in contradiction to (1.6) 0 1Suppose now that we drop the additional assumption that condition
(d)
. Without this condition,
I.
{A;
l a r to those applied in
[a,
x o j we will
(
. Let -X
X such that
in
xmi
01 [xn]
/
with
J of
use the arguments simi-
.
-
xn4xo
such that f o r any
equality holding in
. In discus-
be a uniformly convex Banach space having a wea-
kly continuous duality mapping quence
xo)
. The two succeeding lemmas were proved
30,9]
in a somewhat different form in [30] Lemma 7
A satisfies
thus far it was only possible (see
[39, 8, 30 , 9 ) to prove the weak convergence of sing the weak convergence of
.
-
.
X into X*
-
.
Let {xn] be a se-
w
Then there exists a subsequence
in
X
(2.2) if and only if
x =x
0'
W. V. Petryshyn
Proof. such that d
It is obvious that there exists a subsequence
a r e some real numbers. Hence,
-x ) A0, the definition J(xm o quality
imply that
/1 (d0)d0 -< P
proof suppose t (I-t) l1xm
Lemma
d =d
o
- xO 1 for
.
it follows from the first part of d = d that 0 and [x -(txt(l-t)xg)ll m
Ilxm-x Jdd0
Hence, by uniform convexity of
1
-
in
7 and our assumption
+
i.e.,
x -x 0 implies m 0 and the passage to the limit in the ine.
(d )d, from which (2.2) follows. To complete the 0 in (2.2) Since llxm (tx+(l-t)xo) ) 5 t Jkm-x +
any t
I Xm-X~I
since
J
of
I
xm) [xn) a s m + m, where d0 and
1 xm-xOll doand 1 xm-x 1 +d
0'
11 ( x ~ - x ~ ) - ( x ~ - xo>) ~ ~ +
X, it follows that
x=x 0' An immediate consequence of
+d
Lemma 7 is the following result which
is essential in our discussion.
- X be
Lemma 8. Let
a uniformly convex Banach space with a weakly
continuous duality mapping
into -
tive mapping of
C
a closed convex s e t
of
is a contrac-
-
X into X, then the map-
ping - (I-A) is strongly closed. Proof. Let {xn to some
x
in X. Since
C.
lies in
0
in X and l e t
{xn-Axn
C which
converges weakly
] converge strongly to some
C i s closed and convex and hence weakly closed,
Hence it suffiees to show that
a subsequence of
(x
1 n
x -Ax -+ yo, there m m yo t zm and = xm
-
1 be a sequence in
1
such that
lim
exists a sequence
(I-A)x
0 - Yo
1 xm -x 0 11 exists. z
+0
m
x
. Let [ xm Since
such that
Ax = m
y
0 be
0
- 329 -
1) lim m
- 11-
W. V. Petryshyn
, It follows from > lm im x Xm-x~ m-y~-Axo~ AX this and Lemma 7 that x O-Yo 0' Remark 2.4. As was already neted, Lemma 8 was first proved in a which implies
that
[51 using the theory
more general setting in the assumption that
f
A is defined on all of
of
J-monotone operators under
X. In case of a Hilbert spa-
ce a somewhat different proof i s given in
.
191 Using Lemma 8 it is easy to show that if
Remark 2.5.
X has pro-
perties assumed in Lemma 8 and A is a contractive .mapping of a closed bounded convex subset i s closed
. Indeed,
let
X into
C of
X
, then the s e t (I.-A)(C)
( un\ be a sequence in
n --, co. We need to show that
.
(I-A)u +v as n n Since X, being uni-
lies in (I-A)(C) 0 formly convex, i s reflexive, we rnay replace un by a subsequence,
1,
v
C with
[
which
u AU for some u in X. Since n 0 0 C is closed and convex and hence weakly closed, u lies in C. Hence, 0 by Lemma 8, (I-A)uO = v Consequently by Theorem 1, if A maps C 0' into C, then A has fixed points in C. we again denote
by (un
such that
We now prove the second main theorem (compare it with Proposition 1.8) in this section by the arguments similar to those used in [39,30,1 in the case of Hilbert space.
-
Theorem 3. Let
C be a closed bounded convex subset of a uniformly
convex Banach space having a weakly continuous duality mapping into X* and let A be a contractive mapping of any x in C and any),E ( 0 , l ) the sequence {A" 0 to a fixed point of A in -C.
J
of X -
C into C. Then for
xO) converges weakly
Proof. By Theorem 1, Remark 2.4 and Lemma 1, A has a nonempty closed convex s e t Since, for any
u
F(A) of fixed points
in F(A),
the sequence
in
C and F(A) = F(A )
[1
we can define the nonnegative r e a l valued function
+
xO
- ] UJ
.
is decreasing,
g(u) from F(A) into
- 330 -
g(u) i s a continuous convex function and, thus,
It is easy to s e e that
weakly lower semicontinuous on r e exists an element u
Furthermore,
W.V. Petryshyn
0
in
F(A). Since
F(A) i s weakly compact, the-
F(A) such that
u
is unique. Indeed, suppose there exists another point 0 v in F(A) such that d = g(v ). Then, by the convexity of g(u), it fol0 0 0 lows that, for any t ~ [ 0 ,11 , g(tuo t (1-t)v ) < tg(uo)t(l-t)g(vo) = do. Hence, 0 a s n -+ m , we have
11Anx0-u0 1 +do, I~A;
x0-vOl(+dO and
xo-(tu 0t(1-t)v 0 1+d 0
X
from which, on account of uniform convexity of
1 ( A ~x0-u 0)-(Anxo-vo)ll +o,
IA;
, it follows that
i. e., uo = vo
.
]
n A x ~u If not, then since {An x is A 0 0' 0 bounded and X i s reflexive, there exists a subsequence [Ani xO) and i an element u in C such that A x \u and u # u Since, by I 0 O' ni Lemma 2, At\ i s asymptoticallji regular it follows that (I-AXA x d -,0 "i and A x a uGC. Hence, by Lemma 8, (I-A)u = 0, i.e. , u 6 F(A). 0 Finally, by Lemma 7 , We now prove that
This yields the contradiction of the definition of
d
Hence we must ha0' ve that u = u, i. e., every weakly convergent subsequence of 0 xol converges weakly to u Hence A x converges weakly to u 0' 01 0 ' Remark 2.6. F o r a somewhat different proof of Theorem 3 in c a s e
I4
n.
X is
spaces
a Hilbert
space s e e
[91
. We r e m a r k
that the class of Banach
X having weakly continuous duality mappings
Hilbert spaces a s well a s the sequence spaces a s was shown in
1
for
(71 , it does not include any of the
J
includes a l l
1 < p < a but, P L spaces except
W, V. Petryshyn F(A) has just one point, Theorem 3
, As was already noted, when
',I
was first proved in [8]
by using the theory of J-monotone operators. In
its present form it was proved
in [30]
.
We remark that the weak convergence of the retraction-iteration me(i. e. , pro-
thod (1.7A) (i. e., Proposition 1.9) and of the sequence {A;
position 1.11) follows from Theorem 3 and the observations following Remark 2 . 3 since, a s is not hard to prove the mappings
AY
RCA and
a r e contractive from C to C and satisfy the needed conditions on the boundary of
C.
For the sake of completeness we end this section by proving Edelstein's
extension of Proposition 1.2 to strictly convex Banach spaces (see
Theorem 4. If A is a completely continuous contractive mapping
-
of a subset
C of a strictly convex Banach spaces
X into C, then for
P
-
a fixed point of A in C. Proof. It suffices to show that there exists a subsequence x s i c h converges to a fixed point of
[A;
A in
of
C. F i r s t note that,
by Schauderts fixed point principle o r by Theorem 1 above, A has nonempty set F(A) of fixed points in F(A) = F(AA ). Next, let By Masurts
exists a subsequence If
rn'
he or em [2d
AA
M
C which
is closed, convex and
be the convex closure of the set A(C)u{x
M is compact. Since
I
xO and
u
in
{A:
1.
0
x o ) ~f i , there
C such that
A
xj u .
o
u F(A) , then Theorem 4 follows. Suppose, to the contrary, that
for each x in F(A) , the strict convexity of X /implies .thatThenfor, some x, a kiJ)-xl - I I A ~ u-x 1 >0. a > 0 , depending on u F(A)
Since
n.1 A x +u, it follows that, for all suft'iciently large 0
n. , 1
W. V. Petryshyn Since
XCF(A ) and AA is contractive, the latter two inequalities and the
strict convexity of X imply that
I I A ~ xo
ciently large p which contradicts the
-xll< Xu-x(- a/2 for all suffii
convergence A A x -+u. 0
3.1. Convergence of fixed points of strictly contractive mappings kA.
Let A be a contractive mapping of a subset C (closed bounded and convex) of a Banach space X into C and let
kn] be a sequence of real
numbers such that k - c l (n+m) and O < k
.
point u in C such that A u =u Our problem is to discuss the conditions n n n n' under which, a s n-cao, the sequence (u,] converges to a fixed point of A in C. In case X is a uniformly convex Banach space (in particular, a Hilbert space) with X* strictly convex and with a weakly continuous duality mapping J of X into X*
and A is a contractive mapping of X into
A maps C into
C , this problem was completely settled in an interesting
paper by Browder [63
X such that
. Browderts discussion rests heavily upon the theory
of J-monotone operators introduced in [53
and further developed in [7]
since, a s shown by Browder, there exists a nice nonnection between such operators and contractive mappings defined on the whole space. X
.
The purpose of this section is to rederive ~ r o w d d r f sresults [6] for a contractive mapping A which is defined only onthe subset A
C , i.e.,
maps C into C. It is worth noting that one cannot use the theory of
J-monotone operators when A i s defined only on C assertions of Lemmas 7 and 8
. Our proofs utilize the
. Let us add that our proofs thus become qui-
te simple. At the end of this section we consider a practically useful choi-
/
ce of the sequence k suggested by de Figueiredo [17] nl gle iteration process converges.
for which a sin-
We now reformulate and prove somewhat generalized versions of Browderfs results in
161 .
W, V, Petryshyn
-T h e o r e m
5. Let X be a uniformly convex Banach space having a
-
weakly continuous duality mapping J of - X into X*
--
-
t r a c t i v e mapping of a subset C of -X of C. F o r each kn in (0, 1) --
and l e t
A be a con-
into - C.-L e t v
be an a r b i t r a r y point 0 be the s t r i c t l y contractive mapping of
A n C into C defined by A x = k Ax+(l-k )v f o r a l l x in C. L e t u be the n n n 0 n unique fixed point of A in C. n Them f r o m each sequence n(j)+co we can
-
-----
- -
-
e x t r a c t a strongly convergent subsequence converging to a fixed point of
A
in C. Proof. It -
suffices t o a s s u m e f o r a given sequence k . = k . - t l J ~(JI as j-cco , that v . = u AVand to prove that v.-+v. Since C is weakly J n(j) J closed, v l i e s in C and s i n c e v . i s the fixed point of A . = J n(l) .= k At(1-k )v = k.A + (l-k.)v in C we have n(j) n(j) 0 J J 0
a s j+co
since
k,+l and J a fixed point of A in C , i. e.
and, since
v is
Av.
J j
is bounded. Hence, by L e m m a 8, v i s
, vgF(A) , Now f o r each
a fixed point of A
j we have
in C,
( 1 - kJ . ) ~t k j (v-AV) = (l-kj)v.
(3.2)
Subtracting (3.2) f r o m (3.1) and taking its inner product with J(v.-v) we get
J
Since v., v E C and J
definition of
J that
A
i s contractive on C, i t follows from this and the
W. V. Petryshyn
Thus, it follows from (3.3) and the definition of
J that
Cancelling the positive factor (1-k ), we obtain 1
v . 2 ~and J i s weakly continuous, J(vj-v) L O . Hence it follows J f r o m (3.4) that v.-v --to, i,e., v+v J J Theorem 6'. If in addition to conditions of Theorem 5 we assume that Since
.
1 1
there exists a point
u
0
in F(A) such that
-
> 0 for all v in 0 0 ,J(u0-v)) -
(3.5)
( V -U
then u -+ n
u
as k
-k
n Let u
0 -
1 (n
F(A) ,
ca).
be a point of F(A) such that (3.5) holds for all v 0 in F(A) Then, since u is a fixed point of A in C, by repeating a part 0 of the proof of Theorem 5 we get Proof.
.
-
and J i s weakly continous , ~ ( v ~ - u ~ ) - J ( v - u v.-u -v u J 0 0' 0) ' Hence, by (3.5), (vO-uO,J(v.-u ))--c (vO-uO,J(v-u )) < 0, Consequently, (3.6) J 0 0 implies that v.+u a s j +ca Since { v j was any weakly convergent s e J 0 quence gotten from [ u n i , it follows that u +u n 0' Note. Observe that we did not require for X to be strictly convex. Since
.
}
-
-A be a contractive mapping of a subset C of-a Hilbert space H into C. F o r any fixed point v in C and any 0C defined by A (x)=k Axt(1-k vo kn6(0, 1) let A be the mapping of C into n n n d for all x in C. Let u be the unique fixed point of A in C . Then --n n Corollary 1. Let
u +u as n n 0 -
-
a,
, where u 0 is the fixed point of A in C nearest to
W. V. Petryshyn
Proof. To obtain -
Corollary 1 from Theorem 6, first note that in case
X i s a Hilbert space H we can identify its dual with H by the inner product and the simplest weakly continuous duality mapping i s the identity mapping I
. Thus,
all that remains to prove i s that to any point
v
0 such that
in
C
there exists a (unique) point u in F(A) nearest to v 0 0 (v -u , u -v) > 0 for all v in F(A) This fact was proved by Browder [6] 0 0 0 (see also [9,27] ). But, for the sake of completeness we reproduce this proof -
~
.
-
here. It i s well known that, since F(A) i s a closed bounded convex set in H, to each point v
0
in C o r in H
there exists a unique point u
nearest to v for
If, for any other point v in F(A) , ' we put 0' 0< t
The above inequality implies that for any v in
F(A)
On the other hand, suppose there exists another point for all v in
in F(A) 0 u = (1-t)u +tv t 0
u in F(A) such that 1
F(A)
(3.8)
(vo-U1' ul-v)-> 0. Setting v = u
1
in (3.7) and v = u
0
in (3.8) and adding we get
i s uniquely characterized by (3.7). 0 Special choice of the sequence -(knJ Let kn be a sequence of real numbers such thah k -+ 1 a s n n + a, and O < k (1. The partial disadvantage of Theorem 5 and Corollan r y 1, from the computational point of view, i s that to obtain a fixed point i , e. , uO=U ,
and, therefore, u
u of A in C we need to go through two limiting processes, one to obtain 0 the fixed point u of A = k A in C (we assume heer, a s in 671, that OtC n n n and take v =0) for each given k the second to construct u a s the limit of 0 n' 0 u a s n 4 m (i.e.,k -+I). n n In [17] de Figueiredo, using Browderts Theorem 1 in [6] ( a s s e r ting the strong convergence of u to u in Hilbert space H) shows that by n 0 choosing k = n/n+l and, for each y in C , determining the sequence n 0 . by the process 2 n (n=1,2, An = n / n t l A) (3.9) Y, ' An
IYn\.
...,
to u Using our Theorem 5 we now n 0' prove the convergence of the method (3.9) for mappings A acting in certain we get the strong convergence of
y
Banach spaces. Theorem 7. Let X be a Banrch space having a weakly continuous duality m a p p i x J of - X into -
X*
and .- let A -
be a c o n t ~ a c t i v emapping of
C(CX) into C. Assume that
0 C and that there exists u (F(A) such that 0 (uo, J(uo-v))< 0 for all v in F(A) A =k A(k = n / n t l ; n = l . 2 , 3 , . .), n n n then for each y in C the sequence determined by the process (3.9) 0converges to a fixed point of A in C .
.
-
-
Proof. Let
.
be the fixed point of A in C Then since A i s n n n strictly contractive for any initial approximation x in C we have the 0 e r r o r estimate (3.10)
x
( 3 k H ~ ~ x ~ -ix ~ l l 1 ntl
1 o 011-
7A X - x
-
nk
< m
( ~ t n ) ~ - ~ ~
where m i s the diameter of C. Since, by Theorem 5, u EF(A) it i s easy to s e e that 0 (3.10) that
y n3 u0'
x --t u
n 0 Indeed, it follows from
and (3.9) and
W. V. Petryshyn ria -1
Since, a s i s not hard to verify, n n a / ( l t n )
-+O
as n.-+
(a
for any a
>1
and x--+ u it follows from the above inequality that y -+ u n 0 n 0' X is a Hilbert space, in virtue of Corollary 1, Theorem Note. In case
-
7 reduces to the result contained in
1171
.
W. V. Petryshyn References ---------1. L . P . Belluce and W.A. Kirk, Fixed point t h e o r e m s for families of contraction mappings, Pac. J. Math. 18(1966), 213-217. e t , Dokl. Akad. 2. M. S. Brodsky and D.P. Milma, On the c e n t e r of a convex sNauk SSSR 59 (1948), 837-840. 3a. F. E. Browder, Existence of periodic solutions f o r nonlinear equations -of evolution , P r o c . Nat. Acad. Sci, USA , 5 3 (1965), 1100-1103. 3b. space, -
, Fixed point t h e o r e m s f o r noncompact mappings in Hilbert -P r o c . Nat. Acad. Sci., USA, 53 (1965), 337-342.
4. , Nonexpansive nonlinear o p e r a t o r s in Banach space, -- Proc. Nat. Acad. Sci., USA, 54 (1966), 1041-1044.
, Fixed point t h e o r e m s f o r nonlinear semicontractive ---m a p 5. pings in Banach spaces, Arch. Rat. hlech. And Anal. , 21 (1966), 259-269. , Convergence of approximants to fixed points of nonexpan6. sive nonlinear mappings in Banach spaces, Arch. Rat. Mech. and Anal. (1967), 82-90. 7 . F. E. Browder and D. G. B e Figueiredo, J-monotone nonlinear o p e r a t o r s in Banach s p a c e s , Konkl. Nederl. Akad. Wetcnsch, 69 (1906) , 412-420. 8. F, E. Browder wna W. V. Petryshyn, The solution by iterattion of nonlinear functional equations in Banach spaces, Bull.Amer. Math.Soc., 72 (1966), 571-575.
9. , Construction of fixed points of nonl i n e a r mappings in Hilbert space, (to a p p e a r i n J. Math. Anal. and Appl.). 10. C.R. De P r i m a , Nonexpansive mappings in convex l i n e a r topological space s (to appear). 11. M. Edelstein, An extension of Banachfs contraction principle, P r o c . A m e r . Math., 12 (1961), 7-10.
, On predominantly contractiv'e mappings, J. London Math. Soc., 12. 37(1962), 74-79.
, On nonexpansive mappings in Banach s p a c e s , Proc. Cambrid13. ge Phil. Soc. , 60 (1964), 439-447. , A r e m a r k on a t h e o r e m of M.A. Krasnoselsky, Amer. Math. 14. Monthly, 13 (1966), 509-510. D. G. De Figueiredo, Fixed point t h e o r e m s ,for weakly continuous 15. -- mappings, Math.Res. Center, Univ. of Wisconsin, Technical Rep. No. 638 (1966).
16. ---- , Fixed point theorems---for nonlinear operators and Galerkin approximations, J. of Differenzial Equations, 3(1967), 271-281.
-
-
, Topics in nonlinear functional analysis, Inst. for 17. fluid Dynamics and Appl. Math. , Univ. of. Maryland, Lecture Notes Nn 48
, and L.A. Karlovitz, On the radial projection in nor18. med spaces, Bull. Amer. Math. Soc. 73 (1967), 364-368. -----
---
e-r kontraktiven Abbildun~, Math. Nachr. 30 19. D. Gohtie Zum Prinsip d (1965), 251-258.
--
--
20. ----,Uber Flxpunkte bei steligen Selbstabbildungen rnit kompakten Itericrten. Math. Nachr. 28 119641. 45-55. 21. R. 1. Kachurovsky, On some fixed point principles, Usen Zap. Moskov Reg. Ped. lnst. 96(1960),-215-219. nonlinear operators in Banach space and appli-22. S. Kaniel, Quasi-compact ---cations, Arch. Rat. Mech. and Anal., 21(1966), 259-278. -23. W.A. Kirk, A ----fixed -- point theorem for mappings which do not increase distance, Amer, Math. Monthly, 72(1965), 100471006. -of the theory of functiors24. A. N. Lolmo~oroffand S. V. F d r m i n , Elements -and functional analysis, Graylock P r e s s , 1957. -----of successive ap25. M.A. Krasnoselsky, Two ~ b s e r v a t i o n sabout the methodproximations, Usp. Math. Fauk 10(1955), 123-127. 26. M. Lees and M. Ii. Schultz, A Leray-Schauder principle for A-Compact mappings and the numerical problenis, - solution of----nonlinear two-point boundary value ------John Wiley and Sons, Ioc. , New York 1966, 167-179. 27.
J. L. Lions and G. Stampacchia, Variational inequalities, (to appear).
28.
R.De Marr, Common fixed points for commuting contraction -- mappings,
29.
S, Mazur, Uber gegebene kompakte, ------die - kleinste konvex Menge, die eine----Studia Math. 2(1930), 7-9.
Pat. J . Math. 13(1963), 1139-1141.
Menge enthalt, ----
Z . Opial, Weak convergence of the sequence of successive approxin:ations -30. for nonexpansivekappings (to appear).
-
W. V. Petryshyn, On a fixed point theorem for nonlinear P-compact ope31. r-a t o r s in Banach space, - Bull. Amer. Math. Soc. 72(1966), 329-334.
---
-
---
32. -----, On nonl.inear P-Compact operators in Banach space with applications to constructive fixed point theorems, J. Math. Aaal. Appl. --15(1966), 228-242.
--
33.
--- ,
Further r e m a r k s on nonlinear ------ P-compact operatop-
in Banach space, P r o c . Nat. Acad. Sci. USA, 55(1966), 684-687 (extended v e r sion in J. Math. Anal. Appl. 16(1966), 243-253).
, On the extension and the solution of nonlinear o p e r a t o r equa34. tions, Illinois J. Math, , 10(1966), 255-274.
-
, Construction of fixed points of demicompact mappings in Hil35. b e r t space, J. Math. Anal. Appl. 14(1966), 276-284.
, R e m a r k s on fixed point t h e o r e m s and t h e i r extensions, T r a n s . 36. Amer. Math. Soc. 126(1967), 43-43. , and T. S. Tucker, On functional equations involving P -compact 37. o p e r a t o r s (to appear). J . Schauder, D e r Fixpunktsatz in Funktionalraumen, Studia Math. 2(1930), 38. 171-180.
H. Schaefer, Uber die Methode s u k z e s s i v e r Approximationen, J a h r e s b e r . 39. Deutsch. Math. Verein. 59(19579, 131-140. M. Shinbrot, A fixed point theorem and s o m e applications, Anch. Rat. 40. Mech. Anal. 17(1964), 255-271. E, Zarantonello, The c l o s u r e of the n u m e r i c a l range contains the spectrum, 41. Bull. A m e r . Math. Soc. 1 0 ( 1 9 6 4 ) , 781-787.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
W. V. PETXYSHYN
ITERATIVE CONSTRUCTION O F FIXED POINTS OF CONTRACTIVE TYPE MAPPINGS IN BANACH SPACES
C o r s o tenuto ad I s p r a dal 3-11 Luglio 1967
ON THE APPROXIMATION-SOLVABILITY O F NONLINEAR FUNCTIONAL EQUATIONS IN NORMED LINEAR SPACES by W, V. Petryshyn
1.
1. Introduction. The purpose of this part of my talk i s to outline some
of the recent results concerning the projection-and the approximation-solvability of nonlinear functional equations in normed linear spaces. We first state a general theorem on projectional-solvability proved in [15]
which at the s a -
me time unites the e a r l i e r results on linear equations obtained in
Fl, 18, 10,
6, 131 (*) with the recent results on nonlinear monotone o p e r a e r s obtained
in [7,12,2,3, Let
ce na
19, 141
.
X a finite dimensional subspan of X and P a linear projection of X onto X Let --+n and n n "denote respectively the strong and the weak convergence in X. The
sequence P_x+x
X be a separable Banach space,
.
{xn) i s
-
called projectionally complete in
for each x in X.
Let
A be
X
if and only i f
a (possibly) nonlinear mapping of
I,
X into
X and
In [ld the author
{ A ~ ]a sequence of mappings defined by considered the problem of giving a constructive proof
of the existence and the uniqueness of solutions of the equation
a s strong
limits of solutions
x
n
in X n
of approximate equations
a complete and self-contained theory of the projection method s e e 1151 ; for a more complete l i s t of references and contributions to the theory of projection and Galerkin methods s e e rl 1, 18, 15, 51 -1
.
(+)1n [IS] the author also conside:ed the problem of constructing solutions of Eq. (1) a s wkak limits of solutions x._ of Eq. (2) in case the operator A I1 is demicontinuous o r weakly continuous and monotone o r K-monotone.
W. V. Petryshyn
Definition 1. Eq. (1) i s s a i d t o be strongly projectionally-solvable if and only if t h e r e e x i s t s an integer N each f and
x
in
> 0 s u c h that f o r each n >- N and
X Eq. (2) h a s a unique solution
is t h e unique
solution of E q . ( l )
One of the r e s u l t s obtained in
-
r+ m
f o r each
A
n
n
such that
x +x n
> 0 and a continuously s t r i c t l y incr?
0 )into R
function c( ( r ) of R
l i m d ( r ) = co s u c h that
.
in X
n
15 w a s t h e following t h e o r e m .
T h e o r e m I. Suppose t h e r e e x i s t s N sing
x
i s continuous in
t
with d (0) = 0 and
X
n-
for n
-> N ,
A x ---c Ax n
x in X and
.II n
IIA X-A y
(3)
> q ( x-y
-
I
) f o r each n
-> N- and - -a l -l
x and y in X
n'
Then Eq. (1) is strongly projectionally-solvable provided A s a t i s f i e s the following condition (c): If
is a n a r b i t r a r y
sequence in
{x,] i s any Xwith x m
subsequence -- of Xm such that
x
m
2
x and
A x + g, then Ax = g. m m We omit h e r e t h e proof of T h e o r e m 1 s i n c e below we s h a l l prove a much m o r e general t h e o r e m
. Theorem
I w a s then used in(15]
to e s t a b l i s h
t h e s t r o n g projectional-solvability of various c l a s s e s of l i n e a r and nonlinear equations. T o illustrate the generality of T h e o r e m 1 we now deduce f r o m it the following two important c o r o l l a r i e s . Corollary 1. If X i s a Hilbert space H and A = T t S, where T and S a r e l i n e a r bounded o p e r a t o r s s u c h that S is completely continuous and
- -
T is definite. i. e..
then Eq. (1) i s
T
is such
strongly projectionally-solvable provided that N(A) = 0 ,
where N(A) i s the null s p a c e of Proof. -
that
Since
i s continuous in X P n it suffices to show that 'A s a t i s f i e s (3), i. e., that t h e r e e x i s t s
and A x - t A x , n N>O and (in fact,
@
A
A.
s a t i s f i e s condition (c),
(P) -<
o r ) such that
A
w.v. x
(3')
d (x
) ( cx
Assume, to the contrary, that quence m -+a
{ x m ] c H with
W. V. Petryshyn
)
for all x in X and n > N. n
0
(3 ) does not hold. Then we can find a s e -
l l ~ ~= !1 such
that
x -x and A x -0 as m m m S implies that S x -+ Sx m m
. First, the complete continuity of
and
But, T x Tx. Hence, m m nite, we have
+ Ax = 0. Thus,
s i ce
-
Tx = -Sx o r
(SX,x) x
Ax = 0. Now, since
- (Tx, X ) - (Tx, X) + (Tx, x)
T
i s defi-
= 0
= 1 and Ax = 0 in contradiction to
N(A) = (0)
.
.An independent proof for satisfying slightly weaker conditions was given in [lo]. gee also [lq . Remark. Corollary 1 was proved in [6]
T
F o r other applications of Theorem I to linear bounded and unbounded opera-
.
t o r s s e e 1151
Corollary 2. If X = H and A is continuous nonlinear complex mono-
-
tone
-
-
operator, i. e.,
-then Eq.
(1) i s strongly projectionally-solvable.
Proof.
The continuity of A implies that
A
n
i s continuous in X
and n the Buniakovski-Schwartz
A x-+ Ax. Since P x=x for all x in X n n n' inequality, when applied to (5) shows that (3) is satisfied with d ( r ) = cr.
that
Hence by Theorem I, it suffices to show that Suppose Amx,+
{ xm]
C H
with x 6 X m m and g. Then, by (5) ,
I
A satisfies condition (c). = 1 such
that
x 2 x and m
W. V. Petryshyn
-
(g, x) + (Ax, x)
= 0.
.
---c x and A x + Ax = i. e., A satisfies condition (c) m m m Remark. Corollary 2 i s certainly true when A is strongly monotone,
Hence, x
i. e.
112 . When
, Re(Ax-Ay, x-y) -> c 1 x - ~
the result was first proved by Minty
A=I-F , where
F i s monotone,
[lq (see also Kachurowski [7]) . F o r
general continuous A, it was proved by Browder [2].
For complex monoto-
ne, operator A with the additional assumption that
A i s bounded, Corol-
lary 2 was proved by Zarantonello [19] and by Browder [3] with the boundedness assumption dropped. All the above proofs
a r e nonconstructive and
rather complicated. In his forthcoming paper [4] Browder notes that for certain classes of nonlinear opertaros (e. g. J-monotone operators) a direct verification of condition (c) seems sometimes to require additional hypothesis, which thus might in general restrict the class of equations to which Theorem I is applicable. He then shows that if instead of condition (c) we assume that Eq. (1) possesses a solution, then somewhat weaker assumptions ensure that Eq. (1) i s strongly projectionally-solvable.
In his subsequent paper [57 Browder
exploits further the solvability conditions (s) by deriving general results on the relation between solvability condition and the approximation-solvability (to be defined below) of Eq. (1). In [16] the speaker showed that at least when X
is a reflexive Banach space then, under the assumption of the other
hypotheses of Theorem I, the condition (c) and the solvability condition (5) a r e equivalent. In his subsequent paper 1173 the speaker showed the equivalence between the solvability condition (S) and the modifed condition (c) (the so-called, condition (H)) without the assumption that the underlying
W,V,
Petryshyn
s p a c e s a r e Banach spaces. Some of the r e s u l t s mentioned above a r e given in the next section.
2. Approximation-solvability. -.-s p a c e s and l e t
[XAC X
and {Y
ICY
Let X
and Y
b e two normed l i n e a r
be two sequences of finite dimensional n subspaces. L e t P be a mapping of X into X and Q a mapping of Y into n n n Yn L e t A be a sequence of nonlinear mappings of X into Y defined by n n n , where A is a nonlinear map of X into Y A = QnA
.
.
we give a constructive proof of t h e existence and uniqueness of solutions of the equation
a s s t r o n g l i m i t s of solutions x CX of equations n n
To make the nature of o u r problem m o r e p r e c i s e we need the following definitions ( s e e a l s o F 6 , 1 7 3 and Browder [4,5]
)
.
.
Definition 2. A quadruple of sequences ( X -nJ / ~ n ~j p n '(Qnf f I
{
called a n approximation s c h e m e f o r Eq. (6) if and only if dim X =dim Y n n for each n , P and Q a r e continuous, and P x j x f o r each x in X and n - n n Q n y + y -for y in Y. Definition 3. Eq. (6) i s s a i d t o be uniquely approximation-solvable with r e s p e c t to the given s c h e m e =
-integer
N
solution x
n
>0
(x
,
{
PQ
) if t h e r e e x i s t s an
such that f o r each n>N and each f in Y , Eq. (7) h a s a unique
in X such that n --
x -+ x in X and x is the unique solution of Eq. n
Note. --. We emphasize that t h e mappings Pn and Qn a r e not a s s u m e d to be projections o r even l i n e a r though f o r most of t h e direct methods (7) f o r the approximate solution of Eq. (6) they a r e l i n e a r mappings induced by any one of the the following groups of methods: the projection o r Galerkin type m e -
-
348 -
thods, and the finite difference methodsp) -
[ 83,
W. V. Petryshyn
(assuming, a s in Lax-Richtmyer
that exact and approximate solutions belong to the s a m e space). The main result of this section i s the following theorem. Theorem 1. Let
X and Y be two normed linear spaces and let
rn - be an
approximation scheme for Eq. (6). Suppose.there exists an integer N>O -and a i s continuous in X for each n n>N
, Q A P x -t Ax for each x in X and n n n>N and all -x and - y in Xn
(8)
Then Eq. (6) is uniquely approximation-solvable if and only if sfies the following condition (H): Iftion scheme
rn
rm
for Eq. (6) and
such that
-X
g
, then there exists a
such that
x
-+ x and A x
m.
1
each u
and v in
-
is any bounded sequence in X with
and ana element-x in
Proof. (Uniqueness. ) -
A sati-
i s any subscheme of the approxima-
for some
oftm]
.
+
m. m , 1
1
Suppose that uf v and Au=Av. Because, for
X, P u and P v lie in X (8) implies that for each n n n'
U-HI, P V-+V and Q AP x -+Ax for each x in X, the passage n n n n to the limit in the above inequality implies that 0 >o( (llu-vlb. The latter
Since P
-
inequality contradicts the properties of& ( r ) and shows that
A i s one-to-
one.
' '1f we do not assume that X,
and Yn are subspaces of X and Y respectively (as is usually the case when Eq. (7) i s a finite difference analog of Eq. ( 6 ) ) , then all results outlined in this section remain valid without any substantial chahge in the proof provided that, a s in Aubin [I] (see also the papers discussed during this meeting by Lions, Reviart) we introduce the mappings >,+X , s :Y+Y and pn:X-X subject to certain conditions (for details s e e n n and b e paper of Raviart) guaranteeing that the operators A uefiiled by n A = s Ap have the needed properties. n n n
1' :
W. V, Petryshyn
A i s a one-to-one continuous n n>N, and he,9ce, by the Brouwer Theorem on -
(Existence.) By (8) and our assumptions,
mapping of X into Y for each n n Invariance of Domain, the range R(A ) i s an open set in Y . furthermore, it n n' follows from (8) and the continuity that R(A ) is also a closed s e t in Y for n n in Y then there exists a each n>N - Indeed, if { Y ~ ) C H ( Aand ~) y m n' sequence {x,} C X such that y = A x and, by (8) flyl 2 n m n m > (I~x 1 f p ( r ) denotes the continuous strictly increasing function of
.
-+ R
-tkJI).
.
lnto R
with!
(0) = 0 which i s the inverse of
d (r), then since y m is
a convergent sequence
(lxt -
1 5 p ( IIy
Xk
(1,k
- ~ ~ l-OJl )
- ~ )
such that x + x and A x +A x= n m n m n for n> N, i.e., R(An) is closed. Since
Thus there exists an element x in X
-
y
R(A ) by the continuity of A n n R(A ) i s a nonempty s e t in Y , which is both open and closed in Y n , it n n follows that R(A ) = Y n n Hence for each n > N and for each given f in Y there exists a unique
.
{
in X such that A x = Q (I). F o r the sequence xn thus n n n n n determined, (8) and the fact that Qn A ( O ) ( ~KO < for some constant K imply 0 that element x
I(
whence, in view of
( 7 ) and the fact that
(11 xn 11 ) 5 i. e.
{ xl,
,
a s n+m.
{xn I
A
n. I
x
n. I
0
which implies lllat
is a boundpd seque~lce with
= Qn(f)+ f,
quence
KtK
11 ~ ~ , ( f )-( ( K<
for some constanl K
I( xll 1 z p ( K O t ~,)
x , ~ X I , and such that
x =
n n Ilence, conditio~)(11) implies that there exists a subse-
f
of [xi1 and a.) element x in X sucll that x x n.
+ Ax, a s
A
n.+ I
ce
.
and
I
Because A x = Q (f)+C n. n. n. I
1
:>htai~ithat Ax = f, i . ~ . , x is a solution of Eq. (6)
1
. Sincc, a s
as
n-
I
a, we
was shown
W. V. Petryshyn
i s the unique solution of Eq. (6) for a given f in Y we conclude
above, x a posteriori
that the selection of the subsequence was not necessary. Conse-
quently, the entlre sequence ( a n / converges strongly to uniquely approximation-solvable. Converse
. Let
I'm= ({xmI , x
sut;schenle for Eq, (6) and let x
m
in X
m
Y(),
,(P,],
x,
i. e., Eq. (6) i s
kmj
) be an a r b i t r a r y
be any bounded sequence
in
X with
such that
m
for some g
in
Y. To prove that A satisfies condition (H) note f i r s t that,
by our conditions on
Q
Since, by hypothesis,
m'
Eq. (6) i s uniquely approximation solvable, for every
m> N and every g in
Y,
there exists a unique element
y
m
in X
m
such
that (11) Ym-t
Amyrn = Qm(g)
Y
in X, a s
m+w,
and y i s the unique solution of the equation
Ay = g. Substracting (9) from (11) we obtain the equation
whence, in virtue of (8), for all
P
This, (10) and the properties of
m> N we derive the inequality
( r ) imply that
Since y
W. V. Petryshyn
-
-
y, (12) implies that x y, a s m d m , and Ay =g. Consequenm m tly, (9) implies that A x + Ay and thus shows that A satisfies condition m rn (HI
.
The following theorem gives the relation between the solvability condition (S) and condition (H)
.
Theorem 2. Under the hypothesis of Theorem 1 the following equivalent assertions a r e valid: satisfies condition (H)
(A1)
A
(A 2)
Eq. (6) i s uniquely approximation-solvable.
(A3)
F o r any given f in
Y, Eq. (6) has a solution x in - X.
Remark. Looking over the proofs of Theorem 1 and 2 we note that nowher e did we use the continuity property of the mappings P and Q assumed in n n Definition 2. Consequently, Theorems 1 and 2 remain valid without the assumthat P and Q a r e continuous. This fact may on occasion prove to n n be important and useful particularly in the application of these theorems to ption
the collocation and finite difference methods. Let
us r e m a r k in passing that combining the results of Theorem 2 with
those of Theorem 2 in [16] we have the following useful theorem Theorem 3. If additionally we assume that
X and
-
.
Y a r e separable
reflexive Banach spaces, then under the hypotheses of Theorem 1 the following equivalent assertions a r e valid; (A1) A satisfies condition (H). (A2)
Eq. (6) is uniquely approximation-solvable.
-Y, Eq. (6) has a solution (A3) F o r any given f in (A4) A satisfies condition (C) : is an arbitrary sequence in 1and A x + g, then Ax g.
rem
m m Remark. 3 (A4)
.
-
x in X.
rmis any subscheme of rn and
with
x
m
in
- Xm
such that
.=
Theorem I follows a s a corollary (sufficiency part) of Theo-
W. V. Petryshyn
An immediate and practively useful c o r r o l l a r y of T h e o r e m 2
is the fol-
lowing theorem. and Y -b e two n o r m e d linear s p a c e s and l e t e t X -T h e o r e m 4. L be a n approximation s c h e m e f o r Eq. (6): Suppose that mapping of uniformly --
X
for
--
an integer N>O
into Y y
A
is a continuous
and that
Q (y) converges to y strongly in Y and n in a compact subset of Y. Suppose f u r t h e r that t h e r e e x i s t s
and a function
( r ) a s in T h e o r e m 1 s u c h that the inequality
(8) holds. Then a s s e r t i o n s (A ), (A ) and (A3) -- , under the above hypothesis, --1 2 of T h e o r e m 2 a-r e equivalent.
-
Remark.
Under the hypotheses of T h e o r e m 4 with d ( r ) satisfying sli-
ghtly weaker conditions, the equivalence of a s s e r t i o n s (A ) and (A ) was f i r s t 2 3 obtained by Browder in [5]
.
3. P e r t u r b e d problems. O u r next t h e o r e m s deal with the problem of constructive solutions of p e r t u r b e d equations
a s s t r o n g l i m i t s of solutions
xn€ Xn
of approximate equations
where B i s a completely continuous mapping of X into nuous and m a p s bounded s e t s in
X
Y(i. e., B i s conti-
into relatively compact s e t s in
Y) and
A s a t i s f i e s c e r t a i n conditions t o be specified below. The or e m 5. Suppose that
-converges ---
Xand Y
a r e normed l i n e a r s p a ce s , Qn(y)
y strongly ----- in Y -and ---- uniformly f o r y in a compact s u b s e t of Y , B i-------s completely co n t i n u o-~and ~ ~ A sa t i sfiescondition (11). Suppose a given f in Y andeach n N the solutions of Eq. -x n exist and a r e uniformly bounded by aconstant independent of n. ---
-further ------1% -
tu
that f o r
in h a s a t most one solution - Y Eq. -(3 -- x in- X, then Eq. (8) i s approximation-solvable (i.e., Eq. (14) h a s a solution x in n If f o r any given ------
f
W. V. Petryshyn
X for each n > N nlution of Eq. (141.
-
Let
such
that
x + x, as n*, n
and x is --the
unique so-
us remark that though the second hypothesis of Theorem 5 i s rather
restrictive, nevertheless it is not artificial and often can be verified in applications. Condition (a) which was introduced by Browder
X is a reflexive Banach space,
of the same strength. Furthermore, if
Y = X* and
L51is essentially %
P
and Q = P* a r e linear projections in X and X , ren n n spectively, then the second hypothcsis of Theorem 5 i s implied by the coer-
civeness condition
which has .been successfully used by a number of authors ple, Browder [5, 3 3 and Lions
[93 j
(see, for exam-
.
Remark. For the proofs of Theorem 2 to ----
5 a s well a s for other r e -
sults and the relation of condition (H) to the concept of P-compactness see
References ----.nonlinear operational equa1. J. P. Aubin, Nonlinear stability-appro~imation~of tions (to appear).,
-
F.E. Browder, On the solvability of nonlinear functional equations, Duke Math. J. 69 (1963), 557-566.
'2.
, R e m a r k s on nonlinear functional equations 11, Illinois J. 3. Math. 7 (1965), 608-618. 4.
, Nonlinear a c c r e t i v e o p e r a t o r s in Banach s p a c e (to a p p e a r ) .
, Approximation-solvability of nonlinear functional equa5. tions in n o r m e d l i n e a r s p a c e s (to appear in Archive Rat. Mech. Anal.). 6. S. Hiltlebrandt and E, Wienholtz, Constructive proofs of representation theor e m s in Hilbert space, Comm. P u r e Appl. Math. 17 (1964) , 369-373. R.I. Kachurovski, Monotone nonlinear o p e r a t o r s in Banach spaces, Dokl. Akad. Nauk. SSR, (1965), 679-698.
3.
8. P. D. L a x and R.D. Richtmyer, Survey of stability of l i n e a r finite difference equations, Comm. P u r e Appl. Math. 9(1956), 267-293. 9. J. -L. Lions, Sur c e r t a i n e s equations paraboliques non l i n e a i r e s , Bull. Soc. Math. F r a n c e , 93(1965), 155-175. 10. V. E. Medvedev. On the convergence of the Bubnov-Galerkin method, P r i k l . Mat. Mekh. 17(1963), 1148-1151. 11. S. G. Mikhlin, Variationsmethoden d e r Mathematischen Physik, AkademieVerlag, Berlin, 1962. 12, G. J. Minty, Monotone (nonlinear) o p e r a t o r s in Hilbert space, Duke Math. J., 29 (1962), 344-346. 13. W. V. Petryshyn, On a c l a s s of K-p. d. and non-K-p. r a t o r equations, J. Math. Anal. Appl. 10(1965), 1-24.
d. o p e r a t o r s and ope-
, On the extension and the solution of nonlinear o p e r a t o r 14. equations, Illinois J. Math. 10 (1966), 255-274. , Projection methods in nonlinear n u m e r i c a l functional 15. analysis, (to a p p e a r in J. Math. Mech.) , R e m a r k s on t h e approximation-solvability of nonlinear 16. functional equations (to a p p e a r in Archive Rat. Mech. Anal. ) , On t h e approximation-solvability of nonlinear equations 17. (to a p p e a r in Math. Ann. )
N. I. Polsky, Projection methods ----- in applied mathematics, Dokl. Akad. Nauk, SSSR 143 (1962), 787-790. 18.
E. H. Zarantonello, The closure of the numerical range contains the spec19. trum, Bull. Amer. Math. Soc. 7 0 (1964), 781-787.
--
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
P.A. RAVIART
NAPPROXIMATION DES EQUATIONS DIEVOLUTION PAR DES METHODES VARIATIONNELLES"
C o r s o t e n u t o a d I s p r a d a l 3 - 1 1 L u g l i o 1967
-
-
359 APPROXIMATION DES EQUATIONS DtEVOLUTION PAR DES METHODES VARIATIONNELLES Par P..A. Raviart (Universite de Rennes) Introduction
.
Le present cours e s t une introduction A ltapproximation p a r l e s differences finies des solutions des equations aux derivees partielles dtCvolution. On y traite des equations lineaires du l e r ordre et du 26me o r d r e en t A ltaide des methodes variationnnelles : cette etude reprend avec quelques ameliorations techniques un certain nombre de resultats de 1101. Dans chaque cas, aprks un bref rappel theorique, on examine deux schem a s classiques, l'un implicite et ltautre explicite, et on donne des theorgmes de stabilite et de convergence. En appliquant c e s resultats a un c e r tain nombre dtexemples, on obtient en particulier des resultats de stabilite pour des equations paraboliques et hyperboliques 3 coefficients m e s u r a ~ bles et bornes dans des domaines cylindriques. L e s methodes utilisees peuvent s e generaliser A ltCtude de ltapproxi-
1 , dtequations
mation dtbquations dtevolution couplees 17 non bornes
[8]
B coefficients
, ou dlequations p r i s e s dans des domaines non cylindri-
. Elles permettent egalement dtetudier dans des cas generaux l e s methodes de directions alternCes et B pas fractionnaires 16 , 3 . ques [9]
Enfin
, elles peuvent stappliquer avec succks B certaines equations non
1 .--E q u a t i o n s d t e v o l u t i o n d u l e r ---o r d r e en t.
-
1.1. -Formulation abstraite. -Soient V et est separable, que
H deux espaces de Hilbert s u r C. On suppose que VC A
avec injection continue et que
V
V est dense dans
- 360 H. On designe p a r
11 11 l a
norme dans
P. A. Raviart
V, p a r
venent le produit scalaire et l a norme dans H fort de
V, de norme
11
11* s i on identifie ;
. Soit
H A
Vt
ve l e produit scalaire dans Soit
u, v
VXV
sur
{
( I *2,
Soit
v 6 V, (f, v) e s t
le
en antidualite. Si
f
H, on retrou-
t 6 [o,
TI,
T
<
a,
, avec l e s pro-
, l a fonction t -+ a(t;u, v) e s t mesurable et
a(t;u, v) < h i,uII IIv,,,
K = constante independante de
t ;
il existe deux constantes reellles et d > 0 telles que 2 2 , V V E v pep. en t. ~e a(t;v, v) + /\lv/ >Qt/,vl
.
~ ( t ) ( E [ ( v ,V1)
que l'operateur A(.) 2 dans L (0, T;V1) PI
ltoperateur defini pour presque tout
donne dans
t
par:
V et de lthypothese
(t--cA(t))
(1.1) , on deduit 2 e s t lineaire continu de L (0, T ; V)
Sous l e s hypothBses precedentes et pour f 0
on
:
De la separabilite de ltespace
u
Ht ,
une famille de formes sesquilineaires continues
dependant du parametre
V u, v 6 V
1
V
V1 llantidual
.
H
-+a(t;u, v)
prietes suivantes
(lo
et
respecti-
son antidual
a V C H C Vt avec injections continues. Si f & Vt et produit scalaire qui met
I
( , ) et
H , il existe une fonction
donne
u et
dans
~ ' ( 0 ,T;V'),
une seule verifiant
4 .
Si X e t Y sont deux espaces vectoriels topologiques, (X, Y) e s i ltespace des applications lineaires continues de X dans Y ( 2 ) ~ iX est un espace de Banach de norme 11 I! LP(0. T;X) , 1 < p < o designe llespace des (classes de) fonctions t --ru(tf fortement mesuFables s u r (0,T) pour l a mesure de Lebesgue dt B valeurs dans X et telles T I/P < to Modification usuelle s i p = o U ( ~ ) I I dt ) 0 X x
4
I
.
.
P. A. Raviart
Pour la demonstration de ce theoreme
, voir 1 4 3 , [5]
.
Remaraue 1.1. Dtapr&s[5] l e 2 une fonction
, toute fonction u satisfaisant continue de LO, TJ dans
un sens. Notons egalement que s i sant
2
u et
(l.4), on a l a formule de Green
Dans la suite, on decomposers l a forme
oh
et l a condition
H
.
a(t;u, v) en
0
v , la fonction t ao(t;u, v) < KO 11 ulllvll
$!
-t
(1.6) a
v sont deux fonctions satisfai-
a (t;u, v) est une forme sesquilineaire continue s u r vu, v
(3)
i (1.4) e s t p.p. 6ga-
a (t;u, v) 0
V X V verifiant
est rnesurable et
, K 0 = constante independante de t ,
designe la derivee de u au sens des distributions sur10, T[ i valeurs dans V1 (cf. [15] ) Autrement dit, Ithypothese (0, T;V1) signifie qutil du 2 eniste une fonetion - = g E L (0, T;V1) telle que (g(t), v ) r ( t ) dt = dt
.
g & ~ ~
[
P. A. Raviart
tandis que
a (t;u, v) e s t une farme sesquilineaire contlnue s u r 1
V %H
avec v u E: V , 'dv € H , la fonctlon
1
(1.
a (t;u, v) ( Kl J J u I/
et
valentes
.
a (t;u, v) e s t mesurable 1 , K2 = constante indipendante -t
de t l,
ll est clair qutune telle decomposition proprietes
I VJ
t
( 11 , (1.2
et
(1.8) est toujours possible et l e s
(1.8) , ( 1 1
, (1,2)0 , ( 1 1
sont equi-
1.2. Exemple I. Slit
R un ouvert de l'espace euclidien
On designe p a r sur
R~
de f r o n t i e r e y =
L (R) l'espace des (classes d e ) fonctions definies (p.p)
R 5 valeurs complexes et de c a r r e sommable s u r
de Lebesgue dx. On munit
IIu
3 R.
L
1
= (
R pour la mesure
LL(R) de l a norme hilbertienne
j'
(x),
dxl1I2
.
~~(f-4 1 H (R) l'espace de Sobolev des (classes de) fonctions
On designe p a r 2 311 u € L (R) dont l e s derivees premieres i ) ,~i = 1,. , n , sont dans 2 L (R) , c e s derivdes &ant prises au sens dek distributions s u r R (cf. [14] ). 1 On munit H (R) de l a norme hilbertienne
..
Si la irontiere de
r
e s t .assel r6guli6ren , on peut definir l a trace
u sur rlorsque
u & ~ ' ( 0 ): cgest un element de
des (classes de) fonctions complexes de c a r r e sommable s u r la mesure superficielle 1 tinue de H (R) dans
d6
. En outre l'application
L2(r)
.
"
r o L 2 ( r ) , espace
---lo
pour
est
P. A. Raviart
On note
1 1 H (0) 11adh6rence dans H (R) de c W ( R ) , sous espace 0
0
.
des fonctions indefiniment differentiables B support compact dans R Si 1 l a frontigre e s t nassez r6guli$ren , H (0) e s t exactement lfespace des 0 1 1 fonctions u 6 H (R) telles que F o u = 0 On definit H (R) comme &ant
r
ltantidual fort de
.
-
,HI (R)
; il e s t aisC de voir (& ltaide du theorgme de lo Hahn Banach) que H (R) s e compose des distributions T s u r R de l a
-
for me
Ceci etant pose, on prend H = L~(R) (1.12)
1 1 1 V = sous espace ferme de H ( 0 ) avec H (R) C V C H (0). 0
La famille de formes sesquilineaires a(t;u, v) est donnee p a r
0s
les
a3
a . ., ai, a. € L (QT) 1J
, QT
~ $0, T1 avec
P.P. dans Q
T*
On considgre l a decomposition suivante de la forme
a(t;u,v)
:
- 364 -
P. A. Raviart
+J;ao(x. I1 est clair que
oh
6
t) - a ) u(x) i ( x ) dx
a (t;u, v) verifie les hypoth&ses (1.1) 0
et
0
(x, t) est la norme euclidienne de la matrice
(1.2)
. avec
0
(a..(x, t)) ,tandis q i
que a (t;u, v) verifie llhypoth&se (1. 1) 1 1' Nous sommes dans l e s conditions dlapplication du theorkme 1.1. suffit maintenant dlinterprCter le probleme resolu par ce theoreme minons deux choix simples pour l'espace
On prend
2 f E L (0, T;H-'(R) )
Il
. Exa-
V.
2
, u E L ( 0 ) . Soit 96 c:(Q~) 0
une fonc-
tion
complexe indefiniment differentiable B support compact dans Q T' Si u est la solution de (1.4) , (1.5) et (1.6) , on a
Ainsi au sens des distributions s u r
La condition
QT
2 1 u L (0, T;H (R) ) signifie
2 tient A L (QT) et
0
que
ro
, l a solution u verifie
que
u,
3 u (i = 1,. ..,n) appar5;;: 1
u = 0 p.p. en t
aassez reguli&re\ En resume la solution
s i la frontiere
r
u = u(x, t) de (1.4) , (1.5) , (1.6)
verifie (i) llequation (1.17) au sens des distributions s u r Q T' (ii) la condition initiale u(xj 0) = uo(x) , (iii) l a condition aux limites
de R est
u(x, t ) = 0
, x & E pep. en t
.
P. A. Raviart
On a ainsi
resolu l e problgme de Cauchy avec conditions aux limites de
Dirichlet pour lloperateur
1 b) V = H (R). On prend ici
2
2
f C L (QT) , uoE L (R)
solution u = u(x, t ) de (1.4) des distributions s u r
QT
. Comme au c a s precedent, la
, (1.5) , (1.6)
.
vkrifie l'equation (1.17) au sens 1 On en deduit que pour tout V EH ( 0 )
=i f(x.t)T(x) dx
et ceci doit e t r e egal
Formellement,
pep. e n t ;
A
on en deduit que pour tout
v
1 H (R) et p.p,
en
t
dloh en utilisant -toujours formellement la formule de Green (1.19)
n 3 1~ - (x, t) = 2 24 ) i, j = l
a . .(x,t) cos (n, x , ) 1J
1
3u 3 x.J
(x, t ) = 0 ,
x E ~ , P . P .en oh
n desipi~e l a normale extdrieure
en x
P
et
t
cos (n, x.) designe 1
P. A. Raviart
le i
Erne
solution
cosinus directeur de cette normale. En resume, dans ce cas, l a u = u(x, t) de (1.4) , (1.5) et
(1.6) vCrifie
(i) ltequation (1.17) au sens des distributions s u r (ii) l a condition initiale u(x, 0) = u (x) , 0
---
(iii) l a condition aux limites formelle (1.19)
QT
.
Ceci resoud l e probleme de Cauchy avec conditions aux limites de Neumann pour ltopCrateur (1.18)
.
Signalons que l e s raisonnements formels effectuCs peuvent &tre justifies avec des hypoth&ses de regularites (cf. [6] )
.
1.3. Exemple I1 On dCsigne par 11
u =
2
H(b ;R) l'espace des fonctions u E L (R) telles que
2 L (0)
$4 i = l 3x.
. On m;!:iit
H(b ;R) de l a
norme hilbertienne
On prend alors par exemple
H
2
= L (R)
, V = H(L\ ; n )
.
I'
oh
a = a(x, t ) 6 L*(Q T) avec
(1.22) Les formes
Re a(x, t ) a (t;u, v) et 0
->
d >0
p.p.
dans
a (t;u, v) dCfinies p a r 1
QT
.
- 367 verifient l e s hypotheses (1.1)
0
, (1.2)
P. A. Raviart
et (I. 1)
0
1
avkc en particulier
On interprete l e problGme resolu p a r le theoreme 1.1 de la meme facon 2 2 que precedemment. Soient f E L (Q ) et u EL (R) ; l a solution u = u(x, t) T 0
(i) (1.25
%+A
( a d u) = f au sens des distributions s u r Q~
(ii) l a conditioq initiale u(x, 0) = u (x) 0
(iii) l e s conditions aux limites --formelles (1.26)
Oh
Au(x,t)=o.
a
5;;
designe l a derivee pormale B
a u ) ( t )
x6.f.
. .
On pourrait bien entendu multiplier l e s exemples (cf. f4] )
2. F o r m u l a t ion-a-b s t r a i t e d e l ' a pp r o x i m a----------tion des equations du l e r ---
o r d r e e n t. a
soit
une partie bornee
lrorigine ; on notera
li
h (4)
.
ItP -
101 dont ltadherence
l e point generique
s e r a un parametre destine Sgit
de
5 tendre vers
dea&
rontient
. Dans la suite,
0.
V
un espace de Hilbert s u r C dependant du parametre h On note ( , )h l e produit scalaire dans Vh et 1 I h la norme
correspondante. On s e donne une autre norme Cquivalente
B
l a norme
I
J
sur
11 11 h '
'
; elle verifie done
C(h) = constante (5)
----------
(4)En pratique V s e r a une espace de dimension finie tel que dim V +CQ h 11 lorsque h +O (5) Le .as interessant e s t celuj oh C(h) -+W lorsque h -0
.
.
h
P. A. Raviart
On designe p a r
* l a n o r m e duale de )I )Ih
1) 1)
, i.e.
On a trivialement
lfhlh
(2.3) Soit soit
S un
k = (ko;kl,.
5
IJfhl/fh
C(h)
param6tre entier
..,kS-l )
R'
max
VfhcVh > 0 destine
satisfaisant
k
S
& tendre v e r s t c o e t
B
= 0
On va c o n s i d e r e r deux s c h e m a s type dtapproximation de
et nous degagerons s u r c e s deux s c h e m a s l e s m6thodes g e n e r a l e s dt6tude de lfapproximation Pour
.
chaque couple
{ h, k )
on s e donne
P. A. Raviart
On considgre l e schema implicite 5 un pas de temps variable. Trouver (s) Uh,k=(Uh,k(Vh ; S = o , l , S solution de
...,
I
Comme deuxiBme schema, on considgre l e schema explicite egalement a un pas de temps variable. Trouver u
; s = 0,1,.
. ., S
tion de
On va etudier successivement l a stabilite de c e s deux schemas et l a convergence des solutions (1.6)
u
vers l a solution u de (1.4) , (1.5) , h, k dans des s e n s qui seront precises. Bien entendu, on peut conside-
r e r des schemas dtapproximation plus elabores. (cf. [ll]
p a r exemple)
3.1. Quelques lemmes. Donnons dlabord deux l e m m e s faciles qui seront d'un usage constant dans la suite. Lemme 3.1. Soit d'adjoint
X B*
Alors pour tout
un espace de Hilbert s u r tel
que
a x 6 X , on -
C
et
soit
B
(X;X)
.
P. A. Raviart
B ~ B * -112 Demonstration. Dtapr$s (3.1) ltop6rateur (-) existe et appar2 (X;X) On a done tient
.
d
d'ob
l e resultat puisque
Lemme 3.2. (Lemme de Gronwall discret) Si l e s (3.3) avec -
7
, s = 0, 1,
7 s
.
..., S, sont des nombres ->
0 qui verifient
6 2 * k r y r , r = I,...,s .? -< c r =o
-
0
-
C > O , g > O , ona
Demonstration. On montre par recurrence que s-1
'fs On obtient
-
2
r=o
k
r
< C
r -
l e resultat en remarquant que
(I* r=o l t $kr
-<
6kr) . exp ( S k r ) ,
3.2. Stabilite du schema implicite. Pour Itetude de l a stabilite, on va faire s u r les operateurs l e s hypothkses suivantes
:
A") h, k
P. A. Raviart
{
(3.6)
fi
> 0 independante de h, k, s 2 v ) , > h h -@lIvhflh ' VVh6'h '
il existe une constante
telle
I
que Re (A h,k,o
V
h'
( il existe une constante P 3 0 independante de h, k, s telle que
Ltoperateur A peut e t r e considere comme la .partie principalen de h, k, 0 A(') Considerons dtabord l e schema implicite (2.6) h, k
.
.
Sous l e s hypothgses precedentes, l e schema (2.6) admet une solution unique -
uh,
E v r l lorsque max k
Demonstration.
u
h, k
G .
s
< p2
Connaissant
u(s) h,
s
Vh , le schema (2.6) donne pour
Itequation
Pour que cette equation admette une solution unique, il suffit dtapr&sl e theoreme de Lax-Milgram
Mais en vertu de
que
(3.6) et (3.7)
11 suffit donc que l'on ait pour
s = 0, 1,
..., S- 1
P.A. Raviart
dloh l e resultat.
On fait l e s hypotheses precedentes, Pour tout choix des constantes E
5Y
0 <E < 2
f i , 0 < )1
du schema u h, k implicite (2.6) vkrifie une i d g a l i t 6 de llbnergie discrgte lorsque k est ?DL essez petit (1 6 ks 1 ) telles que
-
Yl
"2"
Le schema implicite (2.6) (3.9)
< 1, l a solution
7
est donc inconditionnellement stable au sens de
.
Demonstration. En multipliant scalairement (2.6) par u + ) (et en h, k supprimant l e s indices h et k pour alleger 1'Ccriture) , on obtient
On prend deux fois la partie reelle de l'equation obtenue et on remarque
On en dkduit donc que pour
dloh pour tout
& >0
s = 0,1,
..., S- 1
P. A. Raviart
En sommant
Si k
(3.10)
e s t choisi
de 0
& s-1 , on trouve
de facon que
2P2
-
s-1
x
r=o
kr/u
1
(r+l) 2
2 s-2
-x
< 2P - t r=o k r I u (
Llinegalite (3.11) devient a l o r s e n divisant p a r
pour s = 1 ,
rtl) 2
I
+(l-?)lU(
I .
s) 2
Y
..., S .
I1 e s t a l o r s possible d'appliquer l e l e m m e de Gronwall d i s c r e t : l e resultat sten deduit trivialement.
3.3. Stabilite du schema explicite
.
ConsidCrons maintenant l e schema explicite (2.7)
. Nous
allons etudier
l a stabilitC sous l e s m e m e s hypotheses que precedemment. Fixons dlabord
P. A. Raviart
d
une notation. Si B E (V ;V ) h h h suivante de 110p6rateur B h
, nous designons p a r
l a norme
Sous l e s conditions de
On fait l e s hypothhses (3.5), (3.6) e t (3.7)
> 0 e s t une constante a r b i t r a i r e m e n t petite e t independante de h, k, s ,
ot~ -7 (3.15)
&
) Bh 1
kSc(hl2 (
fl
f
,
s = 0.1,
.... S-1
,
> 0 e s t une constante a r b i t r a i r e m e n t grande e t independante de
h, k, s, l a solution u nergie d i s c r g t e
h, k
du s c h e m a explicite (2.7) verifie llinegalite de 1'6-
~[r~t\I:C
s=l
(r)
s-1
2
t L r=o k~Ifh,kIIh
eup (LP'
2 r =o kr)
avec
L e s c h e m a expiicite e s t donc conditionnellement stable. Demonstration. pour
s = 0,1,.
On multiplie s c a l a i r e m e n t (2.7) p a r
.., S-1
( a p r 6 s suppression d e s indices
4:;
h e t k)
; i l vient
- 375 -
P. A. Raviart
E n prenant deux fois l a p a r t i e r e e l l e de cette equation e t e n remarquant que 2Re (u on obtient
Mais d t a p r & s ( 3 . 7 )
1%
(s)u(s), u(s))
+ ( J s ) , u(S))
I 1 +\I tk 2(p u(s)l
f(s)ll
.(~)n <
-
4 (PI U(')l t llf(~)II )2
-
*
> 0 , c e qui donne e n utilisant (3.6)
On obtient a i n s i pour s = 0, 1,
...,S- I
(3.18)
u(stl)-u(d)I 1
-< p--t I1 faut rnaintenant r n a j o r e r l u
1 u ( ~ + ' ) - u(') J -< kS
k
s
(2-6)ks Re(A(s)u(s) 0 u(s)j <
'+
1
(P 1 ~ ' ~+ )l l f ( s ) ~ ~ * ) 2,
- u ( ~l2) . D f a p r e s (2.7) on a ( 1(0) ~ ~ I JA(;) U ( ~I )+If(') I +
d ' o t ~ e n dtilisant l e l e m m e 3. 1 avec B = A( s ) 0
P. A. Raviart Remarquons maintenant que d t a p r & s (2.1) e t (2.3)
De (3.19) e t (3.20)
oh
3
est
on deduit que pour
une constante
1
>0
s = 0,1,.
. ., S-1
a r b i t r a i r e . Reportant l a majoration (3.21) d e
dans llinegslite (3.18) , on trouve que
1
stl) 2 . I U ( ~ ) 2t(2-,5 ) - ( i t b ) k s IA:)( lu(
*
(s) A(~)+(A, 1 O
)-1'212)ks R~(A?)U'), u"')
Sous l e s conditions de stabilite (3.14) e t (3.15) e t en choisissant p a r exem2 ple , on a pour s = O , l , S-1
5
=r,t=r
En s o m m a n t
(3.22) de 0
...,
s-1,
(s < S) , on obtient
I1 suffit dlappliquer l e l e m m e de Gronwall d i s c r e t pour obtenir l e resultat.'
P. A. Raviart
R e n a r q u e 3.1. Les conditions de stabilitd ne dependent que des parties principales des operateurs A ( s )
is:,
h, k
.
Corollaire 3.1.
s
Sous l e s hypoth&ses (3.5), (3.6) , (3.7) et lorsque pour tout
* ) (i. e.
l a partie principale de chaque operateur (s) (Ah, k, 0 %,k auto-adjointe), l e schema explicite (2.7) est stable sous l e s conditions
Eh, k,
=
0
Demonstration.
Il suffit de r e m a r q u e r que lorsque
auto-adjoint, on a
%, k, 0
est un opdrateur
avec
Faisons maintenant deux hypoth&ses .dont l a p r e m i & r e ne fait que p r e c i s e r llhypoth&se (3.6)
(3.26)
i
Pour tout s = 0, 1 , .
. .,S-1
telle que vh'vh)h pour tout s = O,1,.
'h' 'hih
, il existe une constante
ps
>O
?P~II~~~~E
Yvh6Vh
..,S- I,
il existe une constante
M~ /l"hlh ,,vh,/h
' v"h h'
Ms
>0 Vh
'
P. A. Raviart
Bien entendu, on
lorsque (3.6) et (3.26) sont verifies.
Corollaire 3.2.
-S ~ u sl e s
hypoth&ses (3.5) , (3.6)' (3.7) , (3.26) et (3.27) ,Je schema
explicite (2.7) est stable s i 2 ks C(h)
(3.28)
<
2Ps 2 (1-1) M
*
, s = O , . . .,S-1
S
Si de plus A ) pour tout S, la stabilite est assuree s i h, k, o - (Ah, k, o -
7 -
Demonstration. -
.
On part de l a condition de ~ t a b i l i t e ~ ( 3 . 1 4 ) On deduit de
llhypoth&se (3.26) que
et de llhypoth&se (3.27) que
Les conditions de stabilite (3.14) et (3.15) sont donc entralnees par (3.28). Lorsque
A(') h, k, (3.24) et (3.25)
0
.
=
jc
(Ah, k, 0
) pour tout
S, l a condition
(3.29) entralne
-Remarque 3.2. En pratique, on aura toujours assez grand independant de h, k, s.
2 2Ps/Ms
< f (resp. 2
P
/ ~ ~ )<pour
P
P. A. Raviart
Nous allons appliquer l e s resultats precedents A ltetude de l1approximation par des methodes de diffkrences finies de problemes evoques aux n
.
0
1.2
et 1.3. Precisons dtabord quelques notations. On choisit
et on considere l e reseau
A tout
point
~
b
9
des points M de la forme
$on ~associe l e pave
wM
h
et llensemble
6
definis
Par
,M la fonction cararteristique du pave w A h duit lloperateur aux differences fillies hi
On designe par
oMh
v .
. Enfin on intro-
P. A. Raviart
avec
h
x
- i -+T
(xl,.
..,
Xi-l
h -y i ' Xitl"..'xn)
' Xi+
'
4.1. Exemple I-a. 0
On considere llexemple a du Naux limites de Dirichlet)
et on designe p a r
1.2. (correspondant aux conditions
. On pose
'(Rh) llespace des suites
de c a r r e sommable s u r
Rh
. On choisit
vh = { v h ( ~ 6) C ; M 6 flh
alors
muni du produit scalaire
On definit ensuite l'operateur
q h e ( ( ~ h ; ~ 2 ( R)) par
11 est clair que
On considere enfin
Calculons l a constante C(h) intervenant dans (2.1)
. On a
f
P. A. Raviart
puisque l e s fonctions dans
R
x 4 q v (x
. 0,, e n deduit que
h h
7 h.12 ) ont. 1
l e u r s supports contenus
Definissons maintenant l e s o p e r a t e u r s A S
, s = O,l,.
. ., S-1,
sont d e s n o m b r e s
e t A(') h, k, 0 h, k, 1 > 0 t e l s que
v fi6c, p.p.
on pose
. ~i l e s
dans R%(t t
s' s t 1 )
P. A. Raviart
Il est clair que toutes l e s hypothCses (3.5) ,(3.6), (3.7) , (3.26) et (3.27) sont verifiees avec
p
=
d et
Q(x, t) designant toujours l a norme euclidienne de la matriee (a. ( x , t)) l
-
Notons aussi que l e s operateurs A
sont auto-adjoints s i l a matrice h, k, 0 precedente est hermitienne p.p. dans Q T* Il nous reste idefinir l e s elements f(S) et u!~) Pour cela, consih, k h, k 2 derons 110p6rateur r € (L (R) ; Vh) : h
.
11
2 -1 Si f 6 L (0, T;H (R)) est donne sous la forrne 2 fi 6 L (QT) pour
i = 0, 1,
...,n, on choisit
n f=f t 0
s=O,l,. On prend enfin
f i-
i=l
x. 1
..,s-1.
avec
- 383 P.A. Raviart
Tous l e s rCsultats obtenus
au N
0
3 peuvent alors stappliquer. En
particulier, en utilisant l e corollaire 3.2, il y a stabilite du schema explicite pour l e s
ob de
k
S
assez petits lorsque :
est (comme ) une constante > 3 arbitrairement petite independante
7
h, k, s. Lorsque la matrice
(a. .(x, t))
l< i , j<- n QT, on 0btier.t l a condition de stabiiite pour k
est hermitienne pep. dans
1.l
S
assez petit
Remarque 4.1. Si R
est borne, on peut prendre au lieu de (4.9)
L e s normes (4.9) et (4.9)' sont alors equivalentes uniformehent etl h. -n ( h= ( ) I 2 et l e s conditions de stabilite (4.21) et (4.22) ont lieu sans i=l h la restriction nks assez petit". 4.2. Exemole I. b. Passons maintenant
0
i llexemple b du N 1.2. (correspondant aux
conditions aux limites de Neumann). On pose cette lois
On choisit toujours
V,, = / 2 ( ~ , 1 ) muni du produit scalaire
P.A. Raviart On pose encore
La relation (4.8) ne subsiste plus mais on a
Lemme 4.1. L8application vh+]Jvh
est une norme
SIJr
vh
11
definie par
.
Le seul point non trivial est de verifier que vh h=O # entralne v = 0. En effet JR lqhvhIs = 0 irnplique q v = 0 s u r R de sorh h h te que vh(M) = 0 pour tout M t ROh = ( M ( M L ~ iy ~ , n i l # fi) n ne reste Demonstration.
-
.
.
plus qu% examiner l e cas oh M 6 R -RO Pour un tel M , on a h h -M 21flQ./ et w fl R = 6 ; il existe alors un i tel que -M+(hiI w fl R # h Choisissons par exemple l a premikf yl ou que w R# h - ~ + ( h 2) ~/ R # 6 et M + ~ R~; E dloh v ( M + ~ ~ ) = o . r e 6veLltualit6 : alors w h Puisque A P-(hi/2) p+(hi/2) I ) = 0 s u r R, -Oh 1 P E R, hi
fir
n
.
-
11
nous en deduisons que l e coefficient de 1 (V (Mthl) hi h
-
-
vh(M)) , est nu1 dtoh
M+(hi 12) , cfest-&dire Oh
vkM) = 0
.
Il est facile de verifier que l'expression (4.10) de C(h) reste valable. 0
On definit ensuite l e s operateurs A
exactement comme au N 4.1. h, k, i seront donnes comme suit. On considbre
Les elements f(S) et u(O) h, k h, k lloperateur rh&&(~2(R), Vh) 1
( rh v )(M ) =-hl. :.hn
6,
;(x) dx.
MC Rh
P. A. Raviart
ob
5
v est le prolongement de v par 2 et u 6 L (Q), on pose 0
0 dans Q(h)-Q
. Alors si f
2 L (91)
1
0
11 est clair que les resultats de stabilite du N 4.1. restent valables
(inegalites (4.21) et (4.22) pour
k
S
assez petit), la remarque 4.1. etant
evidemment mise A part. 4.3. Exemple I1
. 0
On considere ltexemple du N 1.3. On pose
On choisit
puis
Vh
&(v 11
qh
=e
'(Rh) avec l e produit scalaire habitue1 2 ;L (R)) de la maniere standard et on prend
. On definit
P.A. Raviart
avec
On verifie aisement, comme pour le lemme 4.1, que (4.32) definit une norme s u r
V .D'autre part la constante h
C(h) e s t donnee par
i
On definit l e s operateurs
A(')
h, k, i
en posant
svec
puis (4.38) S
2 2 Si f 6 L (Q ) et s i u 6 L ( Q ) , on definit T 0 (4.28) , (4.29)
.
I1 y
f ( s ) et h, k
do) h, k
comme en (4.21),
a stabilite du schema explicite dans l e cas general oh a est com-
plexe lorsque
- 387 -
P.A. Raviart
inf ess.
Re a(x, t)
s = O,l, pour l e s
...,s-1,
k a s s e z petit. Dans l e c a s oh a e s t reel, l a stabilite a lieu lorsS
que
toujours pour
k
a s s e z petit.
S
On comparera l e s resultats obtenus ici avec ceux de [is]
. On pour-
rait ici a u s s i multiplier l e s exernples. Nous renvoyons & 1 2 3 pour l a discrdtisation de formes de vue
5. Etude
a(t;u,v) plus gendrales et &
[ 11
pour un point
un peu different.
de l a convergence.
Nous allons donner seulement un resultat de convergence faible pour simplifier. Nous nous restreindrons au c a s du schema explicite, l a methode et l e resultat s e generalisant trivialement au cas du schema implicite. Soit X un espace de Hilbert s u r ce ferme de
X et soit
h
C tel que
H soit un sous espa-
lloperateur de projection orthogonale de
X
s u r H. On considere ensuite une famille dloperateurs p J(v;'~ ; 2 h, k L (0, T;H)) tels que 'h, k = 'h, k (V:~;L~(O, T;H)) avec l e s p r o p r i e
5
(5' 2, pour tout
11 qh, k vh,
'h, k = {vk l:
11 f ( 0 ,
T;H)
< C2
max ('1 ( Ivh,k h s=o, ,S
..
...
St 1
6 Vh ; s = 0, I, , ,s)6vh
C2 &ant independantes de
h
et
k. Soit
.
'
l e s constantes
C1
et
w un operateur d e d (V;X).
Soit de
un s o u s espace dense d e
19 dans
V h
p
. Nous faisons l e s hypothPses
H. 1 Si une -
famille
Ph, k Wh,k
(5.3)
V et soit
une application l i n e a i r e suivantes :
[wh, k ) e s t telle que
-
2 W dans L (0, T;X) faible l o r s q u e
h, k
0,
on a 2
a)
(5.4)
w=?TWCL(O,T;V),
b)
pour tout
H. 2. -
1
v C o e t toute fonction .f 6 C (0, T )
L e s donnees f(') e t u (O) h, k h, k
satisfont
et C sont independantes de h e t k. De plus, on a 3 4 1 v 6 g e t toute fonctlon C (0, T)
oh l e s constantes pour tout
W=ww,
C
Sous l e s conditions d'application du theorhme de stabilite (3.3) et s o u s ------
-
du s c h e m a h, k explicite a l e s p r o p r i e t e------s de convergence suivantes --lorsque h et k 0
l e s hypoth&ses de consistance 1-1.1 e t 11.2 , l a solution --
u
P.A. Raviart (5.11)
Ph, k Uh, k qh, k Uh,k
(5.12)
Demonstration.
-
L -+ wu dans L (0, T;X) faible,
u dans LCO(O,T;H)-. faible.
.
Soit
u la solution du schema explicite (2.7) h, k Lorsque l e s conditions de stabilite (3.11) et (3.15) sont verifiees, on a en vertu du theoreme 3.3 et des hypotheses (5.7) et (5.8)
oh
C
est une constante independante de h et k. Ensuite, on deduit de 5 2 reste dans un borne de L (0, T;X) tandis (5.1) et (5.2) que p h, k Uh, k reste dans un borne de LCO(O,T;H) On peut donc extraique qh, kUh,k r e de l a famille {h, k] une sous famille, notee toujours h, k telle que
.
Ph, (5'. 13) qh, uy Soit
m uh, k-t%=XU* dans L (0, T;H) faible.
est la solution v r A t soit
mant l e s indices
Mais pour
,U dans L (0, T;X) faible,
2 L (0, T;V) et
D1apr&s H. 1, que
2
Uh, k+
6
de (C:
4
= wu
.%'
11 suffit maintenant de demontrer
(1.4) , (1.5) et (1.6) 0, T )
. Nous
.
deduisons de (2.7) (en suppri-
h et k)
k assez petit
puisque y ( 0 )
=0
et que
(tS,l)= 0
. On a donc pour
k assez
petit
- 390 -
l a limite gr'ace B H. 1 et H.2 , on trouve
En passant
$I
pour tout
v(
V , on obtient butions s u r
Mais puisque
(5.17)
P. A. Raviart
fret toute fonetion ye cW(0, T
). Comme
0
(5.15) pour tout
10, TIA A(. )
v6V. I1 e n resulte que, au s e n s des distri-
valeurs dans V1 , on
% (.)
L
et f(. ) 6 L (0, T;V1)
I
29 e s t dense dans
a
, on dCduit de (5.16)
2
~ $ L6 (0, T;V9) , u*( t )
I1 r e s t e 3 montrer que voisinage de T. Alors pour
+
$(O)
A(t) %(t) = f(t) dans V1 , p.p. en t. =u
0
. Soit 'f) e C1(0, T) nulle dans un
k a s s e z petit e t pour v g
En passant B l a limite, on obtient pour tout
vEV
L16quation (5.18) et l a formule de Green (1.7) entralnent
on a
P. A. Raviart
V dans H, on dCduit que
De la densite de
Enfin de l'unicite de que clest toute la famille
u+ (0) = u
0
.
la solution u du problhme continu, il resulte u
h, k
qui converge v e r s
u au sens indique.
Remarque 5.1. Sous des hypothhses raisonnables, on peut obtenir L p a r t i r du theo2 r e m e precedent des resultats de convergence forte dans L (0, T;X) et m
.
L (0. T;H) ; nous renvoyons pour ce point & dans [lo]
1167 et 131 Notons que 2 on demontre la convergence forte dans L (0, T;H) p a r un argu-
ment de compacite : ce genre de raisonnement est utile pour l e s problem e s non lineaires
(cf.
[i~]) .
Remaraue 5 . 2 . On peut utiliser differemment llinCgalitC
de stabilite (3.16) pour pasune approxiu h, k
s e r & l a limite. Dans [I] , on construit L p a r t i r de mation
Ph, k Uh, k
de la solution
u dans
l'espace
muni de la norme hilbertienne naturelle et on obtient directement la convergence
de
PhJ
Uh,k v e r s
u dans cet espace
@.
6. Retour s u r l e s exemples.
Nous reprenons maintenant l e s exemples du
0
N 4 et nous allons
montrer que l e s hypothCses H.1 et H.2 sont verifiees dans tous l e s cas. 0
6.1. Exemple du N 4.1.
On choisit
P. A. Raviart 2 X = ( L ( Q ) ) ~ " muni d e l a s t r u c t u r e hilbertienne produit.
(6.1)
2 2 A tout F = (f, f , f f n ) ~( L ( R ) ) Y 1 , on a s s o c i e t F = f C L (R) 1 2""" 2 ntl definit l t o p e r a t e u r U~&(H;(R) ; (L (R)) ) p a r
w = (v,
. On
gl,...,z) 3v
1 v6Ho(R).
,
n
On prend ensuite
(6.3) (6.4)
(Phv)(M) = v(M)
.
.
V M € Rh , v d c:(R)
Definissons l a famille d10p6rateurs
Ph, k
'
Dtabord s i
v
h
eV
h'
on
pose
puis
si
v
h, k
(6.6)
Ph, k Vh, k
. .., s
; s = 0, 1,
={v(;lk€vh (t) = Ph
(s)
Vh,
, t
S
-<
1, on pose t
s=O,l,
st1 '
..., s-1.
I1 e s t a l o r s c l a i r que l e s hypothbses (5.1) e t (5.2) sont satisfaites. P a s s o n s maintenant
l a verification de llhypoth&se H. 1. a . Si 2
W = (w,w
w ) e s t l i m i t e faible dans L (0, T;X) d e p monh, k Wh, kJ 1' " " n 3w a0 t r o n s d t a b o r d que wi = , i = 1 , . ,n. Soit 9 6 S (QT) ; on a pour
..
h a s s e z petit A
J Q A i
- JQT q h , k W h , k(x, t). v h p .t ) dx dt
'h, k Wh,k ( x , t ) . y ( x , t ) . d x d t = A
Mais q
h,k
w
h,k
(resp
Oh, qh,
Wh,
1
faible e t i l e s t facile de v e r i f i e r
) *W
2 ( r e s p . w.) dans L (QT) 1
n
que
v hi?+ 2 3 x. dans
En passant 2 l a limite, on voit donc que pour tout
1
00
2
L (Q ) fort. T
96 Co (QT ) , on a
P. A. Raviart
w.(x, t).y(x, t). dx dt =+ 1
-( QT
QT
=e
ce qui signifie que w au s e n s des distributions s u r QT. On en 2 1 3xi 2 deduit que w C L (0, T;H](R)). Pour montrer que W E L (0, T ; H ~ ( R ) ) il , suffit de remarquer que prolongement Ph, k k de ph, k Wh,k p a r n dans R (0, T) converge vers W prolongement de W par 0 dans
3,
o
R ~ X (0, T ) dans
( L ~ (0, ~ TX) ) ) ~ " faible. On voit alors comme precedem1 n ment que w L (0, T;H (R ))). Si R est "assez reguliern, ce que nous supposons, il en resulte que r o w = o d1oh W ~ 2L(0, T;H;(R)).
-
2
Consacrons nous maintenant
A la verification des hypotheses H. 1.b. 1
Introduisons dtabord une notation. Si (4 6 C (0, T) , posons
On voit aisement
que
remarque alors que
b
fk
+?
et
vkgk
dans
2 L (0, T) fort. On
- 394 -
P. A. Raviart
-
2 ntl v wv dans (L (R)) fort (verification immediate), la h' 4 h conclusion resulte de l a convergence de p v e r s ww dans h, k Wh,k ( L ~ ( Q ~ ) ) ~faible. *' Puisque
Il ne nous r e s t e plus qutB examiner l e s hypoth&ses H. 2. Nous allons pour cela demontrer l e s deux lemmes suivants. Lemme 6.1. Sgit -
2
-
uo€ L (0); alors
Demonstration. On a
Dtautre part (rh
U
~ hV)h ' =~ @lo> qhf
et (6.12) resulte de l a convergence de 2 L (R) fort, Lemme 6.2. Si -
f
2
L (0, T;H
hv) qhfh
- 1(R) e s t donne sous l a
V
vers
forme f = f
v
dans
afi
0
+C i=l 3 ~ :
- 395
P. A. Raviart
O r il est immediat de voir que
JGw Gh. 1
M f . ( x , t ) d x Z - / f,(x,t)V Oh(x)dx 1 R 1 hi
de s o r t e que A
A
I1 en resulte que
On en deduit que S- 1
s-0
< D'autre part
2/
1fi(x. t)12 dx dt i = o QT
.
- 396 P.A. Raviart
Puisque
p
2 nil wv dans (L (R)) fort, on trouve que
h PhV
Mais la derniere expression n'est autre que e s t donc demontre.
l
(f(t),v)? (t) dt. Le lemme
On a ainsi verifie les hypotheses H. 2. On peut donc appliquer l e theor e m e 5.1. Si u est la solution d'un schema stable, on a h, k 2 2 --tu dans L (QT) faible et dans L*(o, T;L (R)) faiblt, qh, k Uh, k (6*16) hi
qh, k uh,
k+g
dans L ~ ( Q T faible, ) i = 1,
..., n.
Reniarque 6.1. Conformement B la remarque 5.1, on peut demontrer que dans (6.16) toutes l e s convergences sont en fait des,oonvergences fortes. 0
6.2. Exemple du N 4.2. Ce qui i ete fait au 6.3. Exemple du
NO
0
N 6.1. s e transpose triGialement dans ce cas.
4.3.
On choisit ici
et on definit l'operateur
~ $ ( H ( A ; Q ) ; ( L ~ ( R ) p) ~a r)
On prend ensuite
qui est bien dense dans H(A ;R) lorsque R e s t "assez reguliern. La verification des hypotheses de consistance s e fait comme au
NO
6.1.
P. A. Raviart
7. Equations devolution du 2eme o r d r e en t. Formulation abstraite.
I
On considere l e triplet d'espaces V, H, V'] comme au u, v
s
NO
I. 1. Soit
a(t;u, v) une famille de formes sesquilinkair-es continues s u r
V% V dependant
du parametre t
([o, T]
,T
< m , avec l e s proprietes suivan-
tes : a(t;u, v) = ao(t;u, v)
(7.1)
+ a1
(t, u, v) ,
a) l e s formes u, v + a (t;u, v) sont sesquilineaires continues s u r 0
t / u , v 6 V chaque fonction t
-c
VX V et
ao(t;u, v) est une fois continhment differen-
tiable dans 1 0 , T] , (7.2)
Vu,veV,
b) l e s formes u, v -a et
(t;u, v) sont sesquilinkaires continues s u r
u, v c V chaque fonction
On designe par
Vx H
1
t
-c
a (t;u, v) e s t mesurable, 1
A(~)&c((v;v~) llopkrateur defini par
a(t, u, v)
.
On a alors l e theoreme suivant : Theoreme 7.1. Sous l e s hypoth6ses precedentes et pour u
-0
donne dans V, u, donne dans I
2
f donne dans L (0, T;H),
H, il existe une------fonction
u vkrifiant
- 398 -
P. A. Raviart
Pour la demonstration de ce theoreme, nous renvoyons
L [4] , i 5 ]
.
On trouvera egalement dans 151 des proprietes supplernentaires de regulariu
t6 de la solution u. Ltunicite de l a solution
ntest connue qutavec des
hypotheses supplementaires, par exemple l a fonction t +a l (t;u, v) est une fois continiment differentiable danslo,~], \/UCV ,VVEH.
(7'9)
Les exemples s'obtiennent de la meme facon que pour l e s equations du
.
l e r ordre : nous renvoyons
[I] Dans la suite on supposera pour simplifier un peu que t
depend pas du parametre general
A(t) = A ne
mais nos methodes sladaptent aisement au cas
.
(cf. [lo])
8. Formulation abstraite de ltapproximation des Bquations d16volution
du 2&me ordre en t. Resultats de stabilite. 8.1. Les schemas dtapproximation.
.
2 l e s espaces de Hilbert Vh On choisit T ensuite un pas de temps k constant (pour simplifier): k = ,- avec S m. On introduit comme au
Pour chaque couple
{ h, k]
NO
U
, on s e donne
A h6$(Vh;vh)
(independant de k) ,
(s) uh, k ={uh,
On definit alors un schema implicite. Trouver
Vh ; s =O,
verifiant : (8.2)
(stl) 2(uh,k
- d Sh, )k
-
t U ( s - l \ + ~ hUh,(s+l) (s)
h, k
-fh,
.
, s = I,. , ,S-1 ,
= De la memefacon, on considere l e schema explicite : trouver u h, k (s) Vh ; s = 0, 1, ,S) verifiant = juh,
. ..
...,S}
- 399 -
P . A . Raviart
On va etudier maintenant l a stabilite de c e s deux schemas. 8.2. Stabilite du schema implicite. On va faire s u r les operateurs
il existe une constante
P>
il existe une consta::te
P
A
h l e s hypotheses suivantes :
0 independante de h telle que
> 0 independante de h telle que
Sous l e s hypothese precedentes, l e schema implicite (8.2) admet une solution unique
uh,
~i?,~' lorsque
k<
.
Demonstration analogue ?I celle du theoreme 3.1.
Sous l e s hypotheses precedentes et lorsque s t e une constante K
k
e s t assez petit, il exi-
> 0 independante de h. k. s. telle que l a solution u
du schema implicite (8.2) verifie ltinegalitC de 18&nergiediscrete pour
h, k
P.A. Raviart
Le schema implicite (8.2) est donc inconditionnellement stable au sens de
DCmonstration. Introduisons dlabord une notation : on pose
Lie schema implicite (8.2) peut alors stCcrire (en supprimant l e s indices h et k ) p o u r
r = 1,
...,S-1
$ 2 U ( r + l )+ A ,(r+l) = f( r )
(8.10)
Multiplions scalairement (8.10) par
-v u(
.
) et prenons deux fois la partie
reelle de l'equation obtenue ; nous obtenons
On remarque que
ce qui entraine
En majorant l e second membre de cette inegalite, on trouve
+ (Ao U
En multipiant par
,
k et en sommant de r = 1 ?I r = s-1 (s < S) , on en de-
duit compte tenu de 11in6galitC de coercivitC (8.6)
- 401 P.A. Raviart
(
+ 2k dfoh pour s = 1 , .
..,S
S
1 +
i ~ ( ~ )p2k
r=2
2
llu(r)~/ , r=2
I1 suffit maintenant dlappliquer l e lemme de Gronwall discret P (8.11) (ce qui est loisible s i k assez petit) pour obtenir llinegalite de stabilite p. 8)
.
8.3. Stabilite du schema explicite. Passons maintenant B 116tude de la stabilite du schema explicite sous l e s m&mes hypothhses que precedemment.
On fait l e s hypothBses (8; 4), (8.5), (8.6) et (8.7)
. Sous l e condition de
stabilite
o~ y
> 0 est une constante arbitrairement petite et independante.de h, k, la
solution u
h, k
du schema explicite (8.3) verifie ltinCgalit6 de llenergie discrete
Dour s = 0.1. . . .. S-1 et Dour k assez wetit
oh K est une constante > 0 independante de h, k. s et /Y
L
.
P.A. Raviart
Demonstration. L e schema explicite (8.3) s'ecrit avec l e s notations (8.9) et en supprimant l e s indices
vu(')
(8.14)
+
Multiplions scalairement deux fois la
h et
A u(r)= f ( r ) (8.14) p a r
k = 1,. ..,S-1
v u( r ) t v urr) >
; on obtient, en prenant
partie reelle de lt6quation obtenue et en remarquant que 2Rs(
V V u('),
fu(r)
v l ~ ~ ( ~, ) /
t. V I ( ~ ? ) = ~
+ 2 Re On en deduit que 2
.
(A u"), 1
.
PU(~))
(8.15) peut s'ecrire
1
t P ( A u(r), 0
=
2Re (f
,
t
V(A
0
u(r), U(r))- k2
p ~ ( ~~u ()' )t) - 2Re ( A ~ U ( ' ) ,
p ( v ~~( ~ vu('))= ~) , fu(''
+ vu(r))
ce qui donne en majorant le second membre de cette equation de la fapon habituelle
- 403 P.A. Raviart
Multiplions llinCgalitC (8.16) p a r k
et s o m m o n s d e r = 1 B s ; on trouve
.., S-1
pour s = l , .
si bien que (8.17) peut s l C c r i r e
D1apr&s l e l e m m e 3. I., on a 2Re (A0u(')
, VU")) < 2
/ A J / ~ (Ao u ' ~ ) , u ' ~ ) ) " ~~ q u ( 1~ -<)
Dans c e s conditions 11in6galitC (8.18) devient
P . A . Raviart
Sous l a condition de stabilite (8.12) 11inCgalit6 (8.19) e n t r a i n e pour petit de facon que
ou
2k
5
,
2p2k
k assez
c p,~
K1 e s t m e constante independante de h, k, s. I1 suffit maintenant d1appli-
quer l e l e m m e 3.2. Qemme d e Gronwall d i s c r e t ) pour obtenir l e resultat. Remaraue 8.1. La condition de stabilite (8.12) ne depend que de pale de l l o p e r a t e u r A
A
h, 0
p a r t i e princi-
h' F a i s o n s maintenant llhypoth&se
,
il existe une constante
M
> 0 telle que
Corollaire 8.1. Sous l e s hypothhses (8.4), (8.5) , (8.6) , (8.7) et (8.21) , l e s c h e m a explicite e s t s t a b l e (au s e n s d e-8.13) s i
La demonstration e s t immetliate.
P.A. Raviart On pourrait, comme pour l e s equations du l e r ordre, illustrer l a theorie faite p a r de nombreux exemples. Ce point ntoffre aucune nouvelle difficult6 et nous renvoyons, comme au scretisation au N
0
5
de ltop6rateur
A(t)
NO
4, i [ 2
1 , [lo]
, [I] pour la di-
. L16tude de l a convergence s e fait comme
. On obtient avec des hypoth&ses convenables des r6sultats de con-
vergence faible dans
~ ~ (T;V) 0 , (cf. [lo] )
gence forte nous renvoyons A [IO]
, i161,
. Pour des resultats de conver131.
-
p. AUBIN Approximation des espaces de distributions et des operateurs differentiels, Memoires Soc. Math. France, 12, 1967.
1
J.
2
J. CEA Approximation variationnelle des problemes aux limites, Ann. Inst. Fourier (1964), 14, p. 345-444.
3
J. LIEUTAUD
4
J, L. LIONS
5
J. L. LIONS
6
J. L. LIONS E. MAGENES Problemes aux limites non homogenes (11), Ann. Inst. Fourier (1961), 11, p. 137-178.
7
-
- These (a paraftre),
-
Equations differentielles operationnelles e t problemes aux limites, Springer Verlag, Berlin (1961)
.
- Equations differentielles operationnelles dans l e s espaces de Hilbert, Cours CIME (1963), Editions Cremonese, Rome.
J. L. LIONS - P. A. RAVIART - Remarques s u r la resolution et l'approximation dtequations dtCvolution couplkes, I. C. C. Bull. (1966), 5, p. 1-20
-
.
-
8
J. L. LIONS P.A. RAVIART Remarques s u r l a resolution , exacte et approchee , d'equations d~evolutionparaboliques a coefficients non born& ,Calcolo (1967), 4, 2, p. 221-234.
9
A. MIGNOT - Methodes dlapproximation des solutions de certains problems aux limites lineaires, These, P a r i s (1967).
10 P. A. RAVIART - Sur llapproximation de certaines equation dlevolution lineaires et non lineaires, Jour. Math. pur. et appl. (1967), 46, p. 11-183.
-
11 P.A.RAVIART On the approximation of weak solutions of linear parabolic equations by a class of multistep difference methods Technical report CS 31, Stanford University (1965)
.
-
12 P.A. RAVIART Sur la resolution et ltapproximation de certaines equations paraboliques non lineaires dkgenerees, Arch. Rat. Mech. Anal. (1967), 25, p. 64-80. 13 R.D. RICHTMYER - Difference methods for initial value problems, Interscience New-York (1957)
-
.
14 L. SCHWARTZ Theorie des distributions, Hermann, Pariz, t. I (1951) , t. 11 (1957)
-
.
15 L. SCHWARTZ ThCorie des distributions 2 valeurs vectorielles, E r e partie Ann. Inst. Fourier, (1957), 7, p. 1-139. 16 R. TEMAN - Sur la stabilite et la convergence de la methade des pas fractionnaires, Th&se P a r i s (1967)
.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
H. BREZIS
" METHODES D'APPROXIMATION
et M. SIBONY
E T DtITERATION POUR LES OPERATEURS
MONOTONES1t
C o r s o tenuto a d I s p r a dal 3-11 Luglio 1967
INTRODUCTION
Soit V un Banach sur iR ; V' son dual. On s e propose de r6soudre numQriquement certaines Qquations de l a forme (1)
Au = f
p u r f do~~ic!dans V'
03 A e s t un op6rateur monotone non ngcessairement l i n e a i r e de V dans V ' , Nous donnons un thbrsme d'existence e t d'unicitb.
On c o n s i d b e ensuite l16quation : (2)
AuX + A BuX = f
avec B monotone born6 de V dans V'. On montre, sous certaines hypothzses que uX=
u dans V f o r t 03 u e s t l a solution de (1).
A l'espace V, nous associons un espace V
h
de dimension finie. Dans
V n m s avons a l o r s 1'6quation discr6tis6e :
h
(3)
$ Uh
.- fh.
Sous certaines hypothzses l a solution
B de
( 3 ) converge fortement
dans V vers l a solution u de (1). Enfin nous donnons une mdthode i t e r a t i v e permettant de rgsoudre explicitement (3).
Le plan e s t l e suivant :
- un theorerne d'existence e t d'unicit6, -1 2 - h o p r i 6 t 6 s de 1'opi;rateur A Z
- PropriEtS de l a solution de l'equation AuA+ BuX=f 4 - Applications 5 - dthodes dlapproximtions numeriques i rh) 6 - Application aux famillrs dqappmximtion { v1 ~ ' ~Vh,, ph, 7 - Resolution du probldme discretise : dthode iterative 8 - Applications & l a r6solution nm6rique de certaines equations aux 3
d6riv6es partielles
- Un th6or;me
I
d'existence e t d1unicit6
~ 6i nf i t ion 1.1 On d i t qu'un espace n o d V e s t uniform6ment convexe si, Y
o
< E < 2
,3
E
t e l que
6 ( ~ )> 0 t e l que l e s r e l a t i o n s :
IIuII:1
9
--
IvI121
IIu+vII<2-6
et
,
Ilu-VILE Vu,vtV
Soit V un Banach sur lR uniformkment convexe de nome
)I 1).
Soit V1 son
dual. Soit A un op6rateur non nkcessairement l i n 6 a i r e de V .dam V1. On cherche u
c
V verifiant :
(1.1)
Au = f"
pour f donn6 dans V'.
Th6or$me 1.1. S i l1op6rateur A v 6 r i f i e l e s conditions
pour tout u e t v I\.+
V,
B
O;
y
e s t une applicatior, strictement croissante de
IR t e l l e que
Les r e s t r i c t i o n s de k aux segments de V sont continues
(1.3) dans V' faible. &or$
f
E V'
, i'
u
c
V unique t e l que Au
=
f.
Exists~cc! L3. condition
\u
strictement croissante entraine :
-
(Au Av
(1.4)
, u-v) 2 0
D'autre part, si on fait v = 0 dans (1.2) il vient (~u,u)2 (A(O) ,u)
+
-
(~((lul() ~(0))llull
et l'on sait qu'avec les hypoth2ses (1.3) (1.1) admct une solution. cf. 131 et
(1.4) et (1.5) le problsme
[43.
Avant de dgmontrer 1 'unicit6, nous avons besoin de deux lemmes Leme 1.1. On suppose (1.2) et (1.3). Aiors l'ccsemble 15 des solutions de 116quation Au = f est un convexe fern6 de V. DEmonstration : L'Cquation ?.u = f est gquivalente 2 la relation
En effet : Au = f + (Au
- f , V-u)
0 et
(Av-f,v-u) = (6u-f,v-u) + (AV-AU,V-u)
0
d'aprss 1s monotonic de A. RCciproquement pasons v = u + t w (:,(u+t
?)-f ,'f)
2 0 et l'on fait t
-4 0,
, t; > 0
:
(1.6) donne
E'OU
Au = f. S
I
v = {U (avo-f,v0-u) 2 O! est un dcmi enpsce fern6 de V. 0
V Y EV
Donc S = ensemble des solutions de Au = f e s t un convexe fern6 de V, c z
Lemme 1.2. L'ensemble des solutions de Au = f e s t s i t u d sur une sphe're de V, !@nnstration
: Soient u e t u2 6eux solutions de Iq6quation Au = f .
1
En reportant dans (1.2) il vient :
(~(nq) - L f ( ~ ~ u 2( ~I U) )~ -I II I U ~ I=I o) ~ I o i i: llulll =
lb2i
UnicitE L'ensemble S des solutions e s t un convexe ferm6 de V s i t u 6 sur une sphbre. Comme V e s t uniform6ment convexe,
s est
r6duit 2 un point, dl06
llunicit&.
Leme 2.1, Soit r
c
8, e t r A6 8+
croissante de R+
Soit y une application strictement
-+8 t c l l e que (9(ri)
Alors rA-r
, V 2 0.
quanC X
J!+h,o"!+h~_r$j~o
=,
-
~ ( r ) ()r i
=-
r) -3 A o 0
0.
: On montre facilement que r
X
tenant quc r A-rj r. U o r s on pourrait e x t r a i r e r
e s t borne. Supposons main-
u
, p # r e t V(rll) +$r)
Dew cas : 1) S i p < r S o i t r t e l que ;; < r < r & p a r t i r dsun c e r t a i n rang r < r ce qui 1 1 u- 1 donne : y ( r p ) 5 't'(rl)
< q(r)
e t & l a limite :
ce qui e s t absurde. 2)Sip>r Soit rl t e l que r < rl < p 2 p a r t i r d'un c e r t a i n rang r > r ce rlui u- 1 entraine : y>(rp)2 y(r1) > k ( r ) e t 5 l a limite
ce qui e s t absurde.
C.Q.F.1).
On suppose (1.2) c t (1.3): !!I-ors , e s t nonotone, born; ( c ' e s t 2 '31
d i r e transforme l e s e~sembiesborn& en des ensembles bornks) e t continu de V' f o r t dms V f o r t . Dgmonstration : I1 e s t 6vident que A-' -u-"-.s-------
I l f l 11. ~ On a (nu
- A(o),u) = (f
t .
n(o),uI
c s t ~ronotone, Soit Au = f crvec
,
( Y ( ~ U I J )~ ( 0 ) l)l ~ l l
-
.p(/lu(J)2 a + lla(o)ll + d o )
D ' O ~:
Ceci montre que *I-le s t borne. Montrons que A-'
e s t continu de V' f o r t clans V f o r t . I1 s u f f i t dc
m n t r e r que s i f n = Aun e t s i f n --+ f = Au dans V1 f o r t , a l o r s un -+ u dans V f o r t . La s u i t e f e s t born&; donc u aussi. Suivant un u l t r a f i l t r e plus f i n n n un
f dans V' f o r t , Par s u i t e A{
5 dans V f a i b l e , Aun
-j
Dl03 u = 5 d'aprss l t u n i c i t 8 . I1 en r g s u l t e que un --+u Dl
=
f = Au.
dans V f a i b l e .
autre part (Aun
- t~u
(Aun
- AU , un -- U) = (Aun
J
Un =*
- ~ ( I I u I I ) ) ( I I u-~ ~[ ~ i l )
U) 2 ('f'(lunll)
O
et f
, un
- U)
-4 0
Donc (~(llunll)" ?( llull) (llunll DVapr6sl e lemme 2.1.,
[IUh!
--+
[(uII.
"
11211) +0
D'ou un --+u
dans V f o r t .
Th6orbme 3 $1. Soit A un op6rateur de V dans V v ir6rifiant (1.2) e t (1.3) e t s o i t 73 un op6rateur monotone borne de V - - + V h t l
que l e s r e s t r i c t i o n s
ments de V s o i e ~ tcontinues dans V v fzible. Llors 1)
v f c- V'
,
V
A 2 0 l'eqmtion
Ftia
see-
.
admet une solution unique uA
0 02 u e s t l a solution de lq6qua-
2) uA4 u d m s V f o r t quand A
t i o n Au = f. Gmocstration : 1 ) Pol,r l'existcnce e t 11unicit6 on applique l e thko-
" . . " I -
rsme 1.1. h 1'Qquation (3.1). 2) bntro:lo quc ul e) 3 N >
en e f f e t r or
o tel
-
u d m s V f o r t , quand
que ! u A l 11, ~ POUP t o u t
A
~ S S O Zp e t i t .
h +0
NOU.
avon.
- ~ ( 0,uX) ) 1 (Y( I(uA(( ~ ( 0 ) [uA(I AuA = f - A BuA ( f - h BuA - > i ( ~ ) , ~ A )('f'(((uA(1)- 'do)) nuA[( (Au1
"
Dqautre part
-
(BuA B ( O ) , U ~ 2 ) 0
=+
(f
- ~ ( 3,uA) ) 2 (Y(!U,,(() .- ~ ( 0 )CAI( ) k(B(O1 ,uA) +
I~r(llu,ll)
- r ( o ) ! lu,ll 2 llf
,-
A ~ ( o.-l A ~(o)ll,, IluAl,
D'oG (3.2)
V ( ! ~ U ~c! )Y ( 0 ) .+
[If
- n(0) -
A B(O)~,,,'
ce qui entrainc Ilu A [ bornE, car sinon il e x i s t e r a i t u LI t e l l e que
ll~+,ll
; , +a,
e t on aurnit d o r s l ~ ( J l u ~-+ J ) ) +a, ce qui e s t contraire
(3.2)b) u h
h effet
+
L
u faible~.eat.
CO~BLL' 3 C S born6, ~
1 ah()2 ;! e n t r d n e I(BuhII( W' . Donc quand
2
-
Soit % un u l t r s f i l t r e plus f i n . Suivalt ?A, uh Vt,uh
.
-
faiblement e t
f fortement I1 en r 6 s d t e que Av = f = Au. Donc v = u. En e f f e t ,
on va montrer que s i u
X
e t ,4ui
v faiblement
-3
(iu,ll ( !I)
f f ortene~lt
9Av = f.
-
- v) 2 0 e t & l a l i m i t e ( f - Av , v - v ) 2 O
Nous Rvons (AuX A w , uh
v v
6
V
d v
L
V,
or ceci e s t 6quivalent ? liv = f. Donc uh c ) Montrons que u
X - r F -+
>
u faitlemcnt.
u fortemcnt , On a
- fiu , uA - U) 2 ( Y ( I ( u ~ I-( ) 'f'(bD)(IIuAll - IIuII) ( f - h Buh - Au,u X - u) 2 (Y'(l(uXll- v(lull))(llu,ll - !UII). (AUA
ce qui donne
- X ( B U ~ , U ~ -LU() d / l u X [ j.= Q ( I U I ( ) I ( I I U- ~II:1II) ~~ Come u -u e t Pu, sont bornks ; quand 1 -+ 0 l e l e r mcmbre tend vers 0. X
Donc :
11
rgsdte
d.u lle&me 2.1. que [ u
h
1 4 IIu 1 e t
convexe on en dgduit que uA ---t, u d m s V fcrt
.
come V e s t unifom6ment
IV
- Applications
Exemple 4.1. Soit Y = \ i 9 P ( ~ () I ) muni de l a norme
$2 &ant un ouvert born6 fie
V est un Banach unifom6ment convexe pour c e t t e norme. L10p6rateur
- i=l rs D. 11
AU
=
1
( I D ~ U ~D P~ -U~applique ) <'P(iI)
dans W - ' ? ~ ( Q )avec
-P1 + -Y1 = 1.
Montrons que A v 6 r i f i e l e s hyyothsses du th6or;me 1.1. avec' ~ ( r =)
F1, r 2 0.
En e f f e t :
( I ) On d6signe par d9'(9), W p c W llespace des (classes des) fonctions
u LP
(a)t e l l e s
que D e w = s C ~ P ( ~ ) , 1
des d i s t r i b u t i o n s sur S.
axi
$yp(Gi). ~ ' e i ~ j uc -c' ' ~ ( Q )
, l/p
11s dd$riv6es C t a n t prisen au sen3 e s t ltadh6renke de 3 ( R ) lane
l.tlv(fi) +;/,I
= ! "fsignc l e dual f o r t de
I&"("). 0
-
Donc l a condition (1.2) e s t vgrifike. I1 e s t evident que l'application
t
(A(uttv),w) e s t coiltinue v u,v,w
E
V, dl05 (1.3).
Par consequent pour f donne dans w-~"(G), il e x i s t e u c d0t P ( n )
unique t e l que ku = f.
Exemple 4 - 2 . Plus &nGralement l e s problsmes
admettent une solution unique u e w~'I'(R) pour f donn6 dans ~ ~ ~ ' ~ ( 0 ) . 0
Soit V un espace de Banach de norme h
#
0 un p a r a ~ ? t r sdestin6
D6finition 5.1.
2 tendre vers
I/ 1,
e t s o i t h = (hl,.
..,hn)
0.
(cf. [ l l )
Nous appellerons h-approxireat ion de 1'espace V , l e quadruplet {V,Vh,ph,rh} d e f i n i par l a donnee de 1) Un espace V de dimension f i n i e e t de normc 11
11 1) h'
e
n Wt
2 ) Une aynlication p h
3) Un,
;iplication r
li
C-
Z(7 ,V.) appelge prolongement de Vh dans V. h
$(V,V!,)
ap?elge r e s t r i c t i o n de V dans Vh.
Dgfinition 5,2, La f x n i l l e {V9ilh,phsrh}e s t d i t e consistante s i dans V f o r t
Soit V,' l c uual de Vii. IOUS notons par ( , )h l a dualit6 entre Ul c t VA.
V"h e s t muni dc l a norme duale
Dgsignons par r[ l'adjoint de ph e t par p*h l ' a d j o i n t de rh. Nous svons
ucf'nition On d i t que f,
1.
c
5.3. V k ?onverpe discrdtemcnt vers f
6
V' s i
Soit V un Emach sur il?, w i f o n r C ~ c n tconvcxe, de norme
11 1.
Soit V'
son dual. Soit I: un opirateur non nc:ct:ssoirement l i n k a i r e de V d:ms V'. On se donne uno f m i l l c d7apyoximatior. tV,V:+,pk.r h ) de l'espace V *t I'on f a i t l e s h j ~ o t r ~ c s esuivantes r : (11 1 ) .? c s t ur, op6rateur born6 v g r i f i a n t (1.2) e t (1.3).
H 2 ) La familie {V,Vh,ph9rh} e s t consistante e t ph e s t injective.
On pose 2% = r: A ph : Vh
-9
Vi.
Nous avons a l o r s l e sch6ma suivant :
On munit Vh de l a norme
D1 autre p a r t s i f
llUhlh= Ilphylb.11 en r 6 s d t e que 1
converge discrstemcrit vers f a l o r s 1 I e s t born6. h h h
li
En eeffet :
Th6orSme 5 v l . Pour t o u t f
De plus si f
h
h
E
VA, .i.1 e x i s t e uh
E
Vh unique t e l que
converge discretement vers f
6
V P e l o r s phu, converge dans
V fort vers 13 solution u de (1,~).
Lemme 5.1. ~ ' 6 q u a t i o n (5.1) admet we solution unique. Dkmonstration ..-"-------"--
: Apnliquons l c thgorime 1.1.
1 ) Montrons que Vh mi Ee l a norme
2 190p8rateur i\h'
I(u,llh= I ( "Uh I( e s t
uniform&
memnt convexe. En effet : t/6
>
O
,
O
c E < 2
3 6(c) t e l que s i
d ' a p s s l'tmiforne convexitk de V, e t ceci cntreinc
3) L'applicizt ion t -,(:,,,(tu,,+v5) ,wh)h = (-lph(tuh+vh),phwh)
est continue d1spr8s (1.3). 2roG lakxi~tuncee t lq*micit63.e de (5.1).
Leme ---
5,2.
S i fh converge discrGtement vers T dors
ce qui donne :
% solution
(fh
- % ( o ) , % ) ~ (~(Ily,Hh) , - ~ ( 0 ) Il%llh ) .i(lly,l'? 2 ~ ( 0 +) Ifh .. ~ ' 0 , I l ; ,
D'ot
La convergence discrbte de f h vers f
+ IlfdlL born6 e t
l vhll h e s t born6 car sinon, suite cxtraite 3: t t e l e que l l ~ ~ l, l ~+co e t on aurait
@r. a done ~ ( l l t j l / , L~ )cte. i e c i entraine que existerait une
il
ukI k) .-+ +a ce qui e ~ inpossible. t
Leme 5.3.
Si f 11 converge discrGtement vers f : alors ph uh converge vers u dans V faible e t Aph
y, converge vers
f dans V' faible.
D6monstration : D'qrbs l e lemme prdc6dent
.s..---"-<*-".,m-
vant un u l t r a f i l t r e Aph
; ph
5 + II dans V' Iiontrons quc
rl
y,-3
Au = f.
2 ~ t e .Done
5 &ins V faible e t comme Aph
faible. :z
Il p h %(I
Soit v
6
V
D'autre part (fh,rhvlh -+ i f ,v). En eff5t :
sui-
5 e s t born6,
Ifh - $fl: Ilrhvllh
I(fh-rlfYrhv)hll
I (fh-$f
9rhv)h 1
2
8 fh-rif/l,* IIphrhvl1
O
(d1apr8s l a convergence discr?te dc f ver7 f ) . h et
(r;f,rhvlh
.,
v
IOU
e t par cons6quznt :
= (fy$rhv) -+ ( f , v )
= v 0
vv
C
v
= f = Au.
Par x i l i e u r s : ( L ~ ~ , =~ (%,uhlh ~ % ) = (fh-rip,uh)h + (r;f ,uJh et (fh-rif
,%Ih
)
0
( x i f > u h ) h= ( f ,ph%)
(f
D'oii
Rappelans cP.
k] qw si un
f i l t r e x. converge vers x dm? V f a i b l e , 1
Axi - 3 y dans V' f s i b l e e t linl
sup (LK;,x~) 2 (Y,x)
I1 ?n rdsultc qu* A( = f = tion u = 6. Cl'0i.i l e lcjme.
fi:d
e t donc dtnpr8s l'unicitk de l a solu-
(fh,uJh
= (f "#f,y.,)h h
+
h
(f,ph~)
on en d6duit que
l i m ( A ~ ~ - A U ~ ~ ~ I=+ 0, - U )
-
h+o
D oii
l i m (9 h+b c t par consEquent
lP,lhP
soit do
iR"
(
9
~
1
- ~ ( l l ~ l l ) X I l ~ ~ ~=~0l l - l l ~ I I )
--311ul.
On en d64uit que ph\
VI
(!P<+#
-3 u clans V fort.
- Application aux f k l l e s d ~ a p p r o x i m t i o n{\Ii1",vh
= ( v ~ . ~ ~ L P,( Div Q)
LWI.Q h
E
i
,Ph ,rh1
t un ouvert born6
-
e t ~ ~= ' f prig? au sens des distributions. axi Choi;~2e Qh
Nous dksignons par
M = (yhl,.,.,m
h ) n n
, mi
% l e r6senu dcs points c Z , i = 1,n
\
, !I
= (hi
Nous posons :
1 = (qi)l
--
, qi entier p o s i t i f
lq (
- EL
r$ = 0 1 1 11 e % , ,>; hfin :
i=:lqi
Q}
it de l a forme
,...,hn) , h
n 6%
,h #
0.
Choix de Vh Vh = espace des s u i t e s de nonbres r b l s
s o i t qi$.,.On pose ( y l ( x ) =
>I
I+, s i
x
a
.
t,oVh
f i n i e , que nous mettons en dualit6 evec
Posons vi4(x) =
g(x +
V;1
e s t un espace de dimension
par ? e ~ r o d u i tsciiiaire :
h.
.1 T - hi ~ - )-q
h.
1
Choix de
d
Yotons
m 0 = fonction caraetgristique de [(mi hi Xhi
,t e0 hi hi
f = fonction
((1)
1 - B)hi
caract6ristique de
{,o
n
, (mi
1 + i)hi]
m.
= i=l8hi' @
Want liopCrateur de ;onvolution), e s t un prolongement de Vh dans
On a i d e n t i f i g
%
l a fonction $ =
-7
-" 4I' %.
<7P
V e s t muni de l a norme
Choix dc r Soit une s u i t e de fonctions tendant vers 1 dans
y
h
6
dymsupport
dans lh =
M
&-$
I;Oh
Posons a l o r s
rh u(U) =
,
1
On =it ( e l . [I]) que l e a a,proximztions (1;"
1
Y
Vh ( a )
6
d 1' P
, q,i ,
,ah)
sont
consistantes. Application 6.1. s o i t A : \ i r P ( n ) - 9 ~ - ~ ~ ' ( ! ? ) ddfini psr
ktant un ouvert born6 de ZZn Be fronti6rc
r
"trss rggulisre
de Dirichlet : (6.1)
A U = E
ulr=o y o u r f r ~ ~ ( ~ )
nous nssocions l a formulation variationnelle
avec
. Au probl&ne
avec
D'apr;s
l a dgfinition ie
1%'
nnus avons :
e t f h converge discr2tcment vcrs f . D'd~r&il e thboramc 5.1, il e x i r t e
'i 1'on 3 ph%
u dnns
d*P(Cl) fort 1
% c:
Vh unique vgrifinnt (6.18) e t
; u i t i n t l a solution du problzifie
(6 .I).
Appl.ication 6.2. On cherche
(6. 5
u
c
$yp(,2j v6rifiant 0
Au
-
SAU
=
f
s20
f don& dens L ~ ( ~ ~ ~ )
ave c
On vgrific ctisfmnt q,ue l e tl:.iorGroe 1,l. svapplique e t l q o n a donc i'exis-
tcnce c t l * u n i c i t 6 dc l a solution u &c ((6.5). Corn dans lVexempleprgcttent
(6.5) discrbtiske se met sous l a forne
S i f h converge discrdtenegt v f r s f sn peut appliquer l e t h 6 o r h e 5.1, qui a s s u r e l P e x i s t c n c e c t i 9 m Z c i t 6 de l a sclutior.
% de
(6.6) c t de plus nous
avons : ;
ph
VXI
B 2-
sans
u
17) f o r t .
?;P(
- H6oolution du pro5l&e itGrat ive.
d i s c r 6 t i s 6 par une mgthode
On s e propose dans cc paragraphc de dancer w e m6thode i t g r a t i v e per-
mettant cle r6soudre dms Vh (de dimei~sionf i n i e ) lsEquation All ul1
fi
Plus ~69Eralemcnt s o i t laire (
,)
fll
;
f h donn6 dans V' h
un espace de Hilbert s u r 53 muni du produit sca-
e t de l a nolme
I/
. On
se propose de rksoudre dans % i q 6 q u -
t i o n Au = f 4 pour f dom6 dans I!.!, oG A e s t .un op6rateur non-lingnire de
1;dms .X3. On sc donne ?me qq.-4ication S
11 & s t e
(7.2)
6
2 (&:,g ) verifiant :
w e c o n ~ t u i t cc > 0 t e l l e que (Su,u)
On posc [ul2 = (Su,u). [u] d e f i n i t
SLU.
$ >mi.nome kquivdente
2
2
G
11 (1.
On s u ~ p o s eque Isop&rateurA v g r i f i e :
Qttelque s o i t N > 0
('7.41 U,V
a ?I>, avec
(?.u-:Lv,W
[?1]
_t
)(c(:!)
,3
un? constante C(M) t e l l e qut
, [v] -< N [u-V] [w]
alors d W E %
l l ~ 2l l
Nous avons avons l e Th60rsme :,1 Avec I e s hypotheses (7.1), (7'21, (7.3), (7.4) llQquation 1Zu = f adnet une solution unique u
e
'2 e t l a s u i t e un dgfinie par l f i t 6 r a t i o n
- p(aun - f )
S U ~ =~ Sun + ~
converge fortemcnt vers u pour p > 0 c t u
0
-gmonstration -*.-."--.-
c
% c~nvenablementchoisis.
: L9existence e t i 9 u r r i c i t 6 r6sultent du t h e o r h e 1.1.
Dfautre part, l f i t 6 r a t i o n a un sens puisque S e s t bijective.
En retranchant menbre 5 membre :
.- Su r Sun
.-. Su
- p(Aun - Au)
Posons an = un-u il vient :
Soit No t e l que [u]
2 N0/2
; on f a i t llhypoth?se de recurrence
[un-u] = [rn] 2 No/2. Ceci entraine que
pJ
No. Or
( A U ~ - A U , S ~ ~ ( ~) ~=- A [sW'(hn-~u U) )] 2
e t d7aprZs (7.4) e t 19hypothZse. de r6currence on a pour C(N ) =. C 0
ce qui entraine
On c h o i s i t
p
t e l que 9 < 1 e t l'hypoth~se 2c rrcurrence e s t vgrifi6e. ?ar
consdquent c --+O. Ce qui achsve l a d6monstration. n
En p a r t i c u l l e r l a s u i t e u convcrae vcrs u pour uo = O , n 2 N = , C=C(No) e t 0 =k/c 2 c e q u i d o n n e 0 = 1 - - b < 1. 0 kG opt c2
VIII
- ."pplicztions 2 l a rEsolution numerique de certaines Bqwtions aux dEriv6es p s r t i e l l e s .
Dans ce qui s u i t nous nous proposons d'appliqwr & quelques exemples l e s d t h o d e s prdcedentes
avec
.
-n Au =-id 0. ( I 1=1
I
pour f aonn6 dans ~ ~ ( 0 )
D ~ U ~Diu) ~ ' t
su
s > o
~onsidgronsl c probldme discrEtis6 associ6 5 (8.1) : On cherche
~1
E
Vh
t e l que (8.2) avec
nh '-'ti
=
n r, Ah%=-{giVi(I~i\l
P-2
Vi\)+s\
0 0
T?16or2me 8.1 i ) l l l q u a t i o n ((HI) adnet une solution unique u r t y p ( n ) i i ) Le probldme (8.2) admet une solution unique
% E Vh.
De plus nous
avons
%% --+. u
dans I~$$*(Q) f o r t 0
i i i ) L'iiquation (8.2) peut s e r6sou&:!
$' = .;:
*-
avec
p
convc~iablemntchoisi e t
D6~ons~,ratior, : Lrs --"-"-"..-----.-
par 1 7 i t k r a t i o n s u i v m t e :
n .(,\i,%-.fn)
~10 = 0.
points i ) c t i i ) ont &j?i6t6 d6montr6s aux para-
graphes 4 e t 6. Pour dcmcntrer i i i ) , upnliquons l c s r d s u l t a t s du th6orEmc
(%,vh)h =
(~~,q!)$
Posons
Come : ( \ u ~ ; - ~ $ v ~ , u ~ -2. ~s ~lly;vhll )~
2
L
on a avec l e s notations du thkorsme 7.1. k = s
Calcul de C(N)
[%I
S u ~ ~ s o nque s
5
rvhl h]l lo
9
on a :
[%12 = hl
h2...hn
T
lUII(~)12
ce qui entraine
-> hl...hn
[\j2
sup lu,,(H)I 2
De msme nous avons v x
1
(8.3)
%(XI I
2
2 h.I 4 lo"'n 6
Par a i l l e u r s
[%I
2 hi
n
( % % - q h ,wh)h =
I
I vivh lP2vivh.
& ( I viYr
!J!
T T ; h n 1
viwh
+
' s(%'vh5wt,)h ce qui donne n
2 (&(I vi%lp2viuh-!
I (%\-$'hs~hU
'ivhlPZpivh[
!whll
' orpour
lul2ti9
,
(61(ii1
I lulPs2a En posant a = vi\(x) n
,
,
- 161 ~
n
act 6ri.elsona:
1,
~ 5f ( lp ll)
6 = vivh(x)
, il vient
- 81
compte tenu de (8.3)
n
1
Donc :
1 (L\\-:'hvh,wh)h
1 5 (p-1)
.
.n 411:'~~'
!i.L+ 1
e h.
5
!i=l 4 IIwhII 2 1 1 2 hi
l l ~ ~2112 l ~ ~ a
1
Donc on a avec l e s notations &u tth6or&me 7.2.
ce qui donne
Donc
'1
0
=2
11~+11 ,,._,e s t
me constant^^: inde72ndante de h. En posnnt :
I
1.t
P
s
=C
2
1 i t g r a t ion
n {+' =%
,-
f(3
11
?A%-
fh ) e s t convergente d 1 ap&s l e th6or;me 7.1.
Remarque 8.1. 2 3 i f E L (II) on pourrait arendre aussi No =
IlfhlL2 ir
Remarque 8.2. 0
posone 0 = 1 .- 2-
uh = 0, nous avons a l o r s l a formule de majora-
c2
tion de l s e r r e u r suivante :
Le nombre d.'it6rtions n6cessaires pour avoir
n o
..
2 Log lull 2 Log Log 0
[<-%I
<
E
p u t &re estim6
E
bemple 8.2.
On cherche u (8.6)
r
W;*P(Q) v g r i f i a n t 116puation Au = f
pour f donn6e dans L (Q) P
avec
Consid6rons l e probldme discr6tis6 s?ssoci6 : on cterche
avec
\
c
Vh v 6 r i f i a n t :
Th6orSme 8.2. i ) ~ l & c l u a t i o n(8.6) admet une solution unique u i i ) Le problzme (R.7) admet une solution unique
r
\tvp(n)
~1 r
Vh. De plus on a
u aans u:'P( Q) f o r t PhUh -hG3 i i i ) L1&quationd i s c r e t i s 6 (fi.7) ?cut s e r6eouare & l ' a i d e de l a formule i t b r a t i v e suivante :
avec Shy, =
-
P f
2
.
iamonstration : Les points --."-----..---
Viy,
, p
convenablement choisi et,
yl0 = 0
i ) e t i i ) ont 6t6 d6montrC dans l e s p r a -
graphes 4 e t 6 . Pour i i i ) , nous appliquons l e s r e s u l t a t s du theorzrne 7.1. sn posant : "ab = Vh (.% )
6u produit s c l a i r e (uh,vh), = (I+,,V~).~
(% il cn r6sultc quc k
: :
'
h,Yh
9
y,.'vh)h .f
n
' ElI / v i ( v v h ) l 2g
s avvec l e s riotations 2u t h e o r h e 7.1.
L
Calcul - de B On a
o r L c L, w e c injectisn continue :)our y 2 2. Donc F
L
Donc Yo =
cte
S
ind6pendente rle i?. En yceznt i. = -c"(w) 3 n+1 3 n n hUF, " h"h :)(:k~i'fk*)
-
e s t converceate. D' oh l e thbor?rne.
1) s i f
e
2
L (Q) on pourrait prendre aussi
1' i d r a t ion
ncus avons alors la formule de majoration de llcrreur suivante :
o; C (Q)est une constante nc &pendant que de ilouvert Q. 1 3) Si n w
Log 0
alors on
;
Remrque 8.4.
On suppse que f
t
L2 ( a ) . Si l'ouvert Q = (0,l) * (0,l) et hl =
!i2
=h
alors on a :
Cas particulier
Si n
-
(~1~1); nous swns ~ l o r s
Pour ceux qui elinteressent aux resultats numdriques de Brezis-Sibony
B paraftre dans Arch.Rat.
Math. and Mech,
cf llarticle
BIBLIOORAPIIIE
--
[I]
AUBIIt
~ p ~ r o x i m a t i o&es n eespaccs de Cistributions e t des op6rateurs d i f f g r e ~ t i c l s ( T h h e 1966)
[2]
H. BREZSS
0p6rateurs rnonctonor. C.R. Ac. Sc. de Faris ( d v r i l
[3] F. RR3WI)ER Existence anu uniqxcn?cr sheorems f o r solutions
1967)
02-
non l i n e a r boundarj valuc problems.
.
Proceding of symposie i n q l l i e d mathematics , vol 17, p. 24-49, h e r . h t n . Soc. 1965.
[k]
J. LEKAY e t J.L.
LIONS
Quel,;ues r 6 s u l t a t s de Visik sur l e s pro-
blemes e l l i p t i q u e s non l i n g a i r e s par l e s m6thodes de
E!i~ity-Browder . Bull. Soc. Math. France 93, 1965, p. 97-107.
t6] R.S. VARGA
Matrix i t e r a t i v e analysis, Prentice Hall.
[5J PETRYSHYN 1 ) On the e x t e n s i o n and t h e s o l u t i o n of non l i n e a r equations .- I l l i q o i s J o u r n a l of Mathematics vol. ION0 2 , June 1966
2 ) Cours C.I.M.E.
1967
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )
V. THOMEE
"SQME TOPICS IN STABILITY THEORY FOR PARTIAL DIFFERENCE OPERATORS
Corso tenuto ad Ispra dal 3-11 Luglio 1967
'SOME TOPICS IN STABILITY THEORY FOR PARTIAL DIFFERENCE OPERATORSW
.
by Vidar Thomee (University of G0teborg) 1 . ----------Stability in L
P
d < p < o, and let W W denote L (R ) for 1 03 P P of bounded, uniformly continuous complex-valued functions on Let
W ; P
1< p<
-
03,
is
=6be the set R~
. Then
.
a
Banach-space with norm (x = (xl > . . J d )
,
n
m of distribuF o r l a t e r use we shall also introduce the Sobolev space W P P dd t i o n s s u c h that D ~ ~ = ~ 1 - ~ ~ ~t wPs f o rM/ d l =I ~ tJ-~( . <m~ j ,q d ) ) This i s also a Banach space with norm (d= ( a
...ax,
..
4
~ e t
=W :
.
, W; =
myo Wm ,d *
= W*m
. By Sobolev's
imbedding
w
consists of all infinitely differentiable functions with theorem, W P Ddu C Wp for a11 0( For 1 < p < co , W* i s dense in W When P P u is a N-vector u = (ul, , , we use again the above definitions 2 U ~ ) where in ( l . l ) \ u / = ( ) '12 j Consider the initial-value problem
.
.
...
El~jl
(I.2]
(1.3)
.
-
- P(x, D)u 2 3t/dl5 u(x, 0) =
V(X)
,
Pd (x) D~ u,
t> 0,
~
~
d x C R , u = u(x, t), and v = v(x) a r e complex
where
Pd (x) a r e ty to be in
N-vectors and
NxN matrices with elements which we assume for simplici-
4
The initial-value problem (1.2), (1.3) i s said to be correctly posed in
W
P
if P = P(x, D) (considered a s a densely defined closed operator
in W ) i s the infinitesimal generator of a C semi-group of operators P 0 E(t) on W for t > 0 , that is the family of bounded operators P . E(t), t > 0 , satisfies E(0) = I = the identity operator ,
The operator with
E(t) i s r e f e r r e d to a s the solution operator connected
the initial-value problem. The definition of correctness thus depends on
p ; the c a s e most
discussed in the literature i s the case of correctness in
L
In the ca2 s e of constant coefficients, using Fourier t r a n s f o r m s one easily finds that the initial-value problem i s correctly posed in for any
T> 0 ,
F o r a NxN
matrix
A with
we define
A (A) We
L 2 if and only if
= max Re j
eigenvalues
i J. ,j = 1,. ..,
N ,
A. . J
also set ReA =
Clearly Re
A is a
-21
(A t A*)
.
hermitian matrix. F o r hermitian
NxN matrices,
A
-<
B
means N
(Au, u) < (Bu, u) f o r
a l l N-vectors
A n e c e s s a r y condition f o r (1.4) to hold i s that
u, w h e r e
(1.2) ,(I. .- 3) is c o r r e c -
tly posed in Petrowsky's s e n s e , namely (1.5)
sup [ A ( P
(I) ) ;
~ E R <~ m)
.
N e c e s s a r y and sufficient conditions have been given by K r e i s s [I47 not o u r a i m
h e r e to give a
. It i s
complete ac'count of his work, but we s t a t e
one s u c h r e s u l t which we shall need in Section 2 : The condition (1.4) holds if t h e r e a r e positive constants d C1, C2 and for e a c h / g R a h e r m i t i a n m a t r i x H({) such that
T h e o r e m 1.1. 7
and Re(H( 5 ) P (
1
5
On t h e o t h e r hand, if (1.4) folds we have
a
(1.5) and t h e r e a r e positive
, C and f o r each 16 R~ 1' C2 3 H( ) satisfhying (1.6) and
constants matrix
C2 I
C
a positive definite h e r m i t i a n
We shall consider the c a s e of a hermitian (hyperbolic) s y s t e m
In the c o n s t a ~ l tcoefficient c a s e we obtain the c o r r e c t n e s s in
L
at once 2 f r o m the fact that e x p ( t P ( 5 ) ) = exp (ti A . j .) is then a unitary n a J J trix. The c o r r e c t n e s s in L 2 in t h e c a s e of variable coefficients w a s proved by F r i e d r i c h s l 7 ]
. As f o r the c a s e of
following t h e o r e m by R r e n n e r
[4
:
Wp, p
/
2, we have the
Theorem 1.2.
Let 1
-<
p
m,
#
2 , Assume that the system (1.7) has
constant coefficients. Then the corresponding initial-value problem i s correctly posed in
W
P
if and only if
This condition is the necessary and sufficient condition for the simul-
.. .
taneous diagonalizability of the matrices
A j = 1, ,d, and s o the j' the case when the system can be brought
result means that apart from into the forin
with real diagonal matrices not correctly posed in
W
D . , the initial -value problem for (1.7) i s J p# 2
.
P' For the approximate solutioli of the initial-value problem (1.2) , we' consider operators of the form
(1.3)
where
h
is
a
small positive number,
and
A, 1,
rence operators A. v(x) = 1,h
Z p
a
1p
(x. h)v(x+/h)
where the summation is over a finite set of
h
a r e explicit diffe-
,
p = ( p ...,,jd),,$
integer, J with coefficients
and a.p(x, h) a r e NSN matrices which a r e polynomials in h 1
in'&
. If A1,
= I, the operator i s explicit, in other c a s e s implicit.
Assume that (1.2) , (1.3) small positive number tied to
i s correctly posed in W
. Let
h by the relation
1= constant.
The idea is then to choose the operator ximate
E(nk)v by E ~ V: Eh
there is a dense s e t of u(x, t) = E(t) v
SO
v in
W
P
k
a s to be able to
i s said to be consistent
initial-values
wm for t -> 0 and P
Eh
P k/hM =
with
such that
be a
appro-
E(k) if
u(x, t t k ) = Ehu(x, t) + o(k)
(1.9) The operator racy /L
Eh
is said to approximate
o(k) in
if
in W when P E(k) with
(1.9) can be replaced by kO(hF )
order
h+O
.
of accu-
. In applications,
condition (1.9) turns out to be a purely algebraic condition on the coefficients. The operator
E
h
We have the well-known
i s said to
be stable in
W if for any T P
Lax1 equivalence theorem
[16]
> 0,
:
Theorem 1.3. Assume that the initial-value problem (1.2), (1.3) is correc-
.
tly posed in
W and let Eh be consistent with E(k) Then the staP bility condition (1.10) is equivalent to convergence ; for any v g Wp ,
and nay pair of sequences {hj);' when j -t oo, one has t> 0,
.
inj):
with hJ . 4 , n.k.+t J J
It should be noticed that stability is necessary for convergence only ; for individual P EWP One can have convergence even without stability. Generally this depends on the
v 6W
if one demands (1.11) for all
regularity of v; for an interesting example with analytic initial data and highly unstable difference operator, see Dahlquist
151 . Although
in
principle one may thus have convergence without stability, in practices round-off e r r o r s then cause problems. We will return to these questions in Section 3
.
Again the case most discussed in the literature is the case of stability in
L2
finds that
. F o r constant coefficients, Eh
i s stable in
L
2
using Fourier transforms one
if and only if for any
T
> 0 ,
where
E ( I ) is h
the symbol (or amplification matrix) of E
h'
A necessary condition for stability i s the von Neumann condition, i s the spectral radius of A)
Setting
(?€(A)
,
'1 2
lulH = (~u,u)
. IAJ~
= s u p J ~ u l / u~ l H /
one has the following analogue of Theorem 1.1
.
for difference operators:
Theorem 1.4.
The condition (1.12) holds if there a r e positive constants d ho, C1, C2, and f o r each S C R and h < h - 0 a hermitian matrix Hh ( ) such that
and
On the other hand, if
'(1.12)
holds wd have (1.14) and if
there a r e positive constants
ho, C1. C2 , and f o r each
h
Hh({ ) satisfying (1.15) and
-<
h
0
a hermitian matrix
Proof. F o r y = 0 , see Kreiss to Widlund
[26]
.
[13]
. The variant
Consider the case of a hermitian system (1.7) sistent with such systems,
Kreiss [15]
0< Y
<1
if R~ and
with Y > 0
i s due
. F o r operators con-
defined the difference operator
to be dissipative of o r d e r u ( v even) if (with the natural generalizah tion of (1.13) to variable coefficients) , E
V. Thomee
(1.16)
p(Eh(x,{))
< l - C l l h l 1v +C2k.
5
d R,h
lj L n .
and has been able to prove : Theorem 1.5.
Let v be an even natural number. Assume that
consistent with the hermitian system (1.7)
, has order of accuracy
. Then it i s stable in
and is dissipative of order v
Eh
L
2
is I,
-1,
'
Certain results have been proved also in the case of accuracy of order v -2
.
Consider now a hermitian system It
(1.7) with constant coefficients.
follows from the proof of Theorem 1.2. that
if
p { 2, except for
the case that the system can be brought into the form (1.8) not exist difference operators consistent with
, there do
(1.7) which a r e stable in
.
W We therefore do not essentially restrict the generality by consideP ring the scalar equation
a u =p-a a t
We then want to discuss
a x ,
the stability
rators
with constant
p real
in W p P'
. f 2, of consistent ope-
.
a . In the case of an implicit operator, the sum may be J infinite. Introducing the function
we have stability in p
#
L
2
if and only i f
2 we need :
Theorem 1.6. Let
1
-< p -<
co
a({)
(1 for
I I-
5 6 Rd
, For
, p # 2. The operator Eh in (1.18) i s
if and only if one of the following Mo conditions i s W P satisfied, natnely stable in
a ( { ) = ce
a)
la( { )I
b)
.
ij t
1;
< 1 except for at most a finite number of points
q=l.. 9 there a r e constants q
Ic ( =
1
.., Q ,
i n 1 { 1 ~ n where
pq.
oq,
5
where
(Y
i s an even natural number such that
Proof, See -
[22] [23]
case
m
p =
is real,
Re
..Q,
p > 0 , and 9
. Here we only want to sketch the proof in the
and
(1.19)
la({) < I
When p
q
a ( { ) = 1. F o r q = l . . .
for
0
I I-
a(O)=l.
= m we have
where
J
To prove the sufficiency of the conditions in the theorem, let
a real,
(Notice that 1.5
(1.19)
in this particular
estimates,
> 0,
(1.21)
a r e exactly the assumptions
and
v even
.
Re p
case. ) By (1.20)
of Theorem
and (1.21) we get by simple
the second of which after two integrations by parts ,
and hence we obtain easily
To prove the necessity, assume that
. Because of
(1.21) (1.22)
a({) does not satisfy
(1.19) we then have
a ( < ) = e x p ( i a j + i ~ P q ( 5 ) - v J V ( 1 ~+ ( l ) ) ) , 5-0, cu real, q({ ) polynomial
, q(0) f 0, 1 < P <
even,
,
v
R e p > O . It i s then easy t o prove
and using a lemma by van d e r Corput , max j
1 anjl 5 C2n-
1 1
and s o
: .1 lL
1/P
J max j
= fJanjJ
which contradicts the stibility since p As an application, the Lax-Wendroff Ehv(x) =
ti
with
2
anj
[17]
a
I njl
-1 - -1
C1
>F 2
n
P
JJ
.
consider for the solution of the equation (1.17) operator,
z1 ( P2 I 2 + p l )v(x t h) t (1 - P 2 .I2) v (a) t
( P 2 .I2 - p I ) v ( x
- h)
a ( { ) = p 2 I 2 cos
I
+
iplsin j t l
- P2 A 2
.
This can be described a s the most accurate explicity operator for (1.17) based on the thlcee poilits
and s o E hand, if
h
i s stable in
x+h
L
2 0 < l p l . A < l v e have
and'x. Here
if and only i f J p l .I < 1. On the other a(/) C l f o r
0 < 1 1 1< n and
1 4 a ( ( ) = e x p ( ~ l 'i6 { - - ~ ~ ( l - p ~ t ~o (~[ ) )i ) l, ~/ + o ,
It follows from Theorem 1.6 that
E
h
is unstable in
#
W P
for p
2.
[9]
have, by using
By the above proof we have 1112 Serdjukova
[19]
, PO] and Hedstrom
more refined techniques of estimating the ve more precise estimates of the growth of Eh
is stable in
183 ,
a . above, been able to gin.l JIE& for the case when
.
L
but unstable.in the other W In the particular 2 P case of the Lax-Wendroff scheme, the exact result is
more generally, when
a ( 5 ) has the form
(1.22) one has
The instability present here i s of course quite weak; we shall return to its influence on convergence in Section 3. The proof of the sufficiency part of Theorem 1.6. due to John [ll] and Strang [21] r 2 3 J for p
for
p =
. The proof of the necessity part I
< oa uses results about Fourier multipliers on
L
P sufficiency part i s a trivial consequence of the result for p = oa Theorem 1.6
.
u3
is
in
; the
has also been generalized to variable coefficients in 1243
,. P a r a b o l i c d i f f e r e n c e o p e r a t o r s . ................................ Consider a s in Section 1 an initial-value problem
u(x, 0) * v(x) ,
(2.2)
with coefficients the sjrstem
in
4 . With the notation of Section 1,
we say that
(2.1) is parabolic in Petrowskyts sense if
In this case we have correctness in
$
; one can actually even estimate
the derivatives of the solution : Theorem 2.1.
Let (2.1)
be parabolic in Petrowskyts sense. Then the
initial value problem (2. l ) , (2.2) is correctly posed, and for any differential operator constant
Q of order q and any T
>0
there i s a positive
C such that
-
(2.4)
O < t < T.
Proof. See e.g. Friedman [6]
.
For difference operators
E
h
of the form discussed in
Section 1,
Widlund [26] has defined a concept of parabolicity which can be considered a s a discrete counterpart of (2.3) , namely ,
(Compare the definition (1.16) of a dissipative operator Eh). It can then be proved : Theorem 2.2.
Assume that
Eh
is consistent with (2.1) and satisfies (2.5),
i s a difference o p e r a t o r consistent with a differential opeh r a t o r of o r d e r q. Then f o r any T > 0 t h e r e is a positive constant C
and that
Q
s u c h that
Proof.
See
[27]
. The proof
goes back to John [li]
and Aronson [2]
,
[3] , and depends on e s t i m a t e s of a d i s c r e t e fundamental solution.Actuall y the r e s u l t is .valid in much m o r e general situations, and p e r m i t s multi-step s c h e m e s , variations in the coefficients a l s o in t, and low r e gularity. It might b e considered natural to make inequalities like ( 2 . 4 ) and ( 2 . 6 ) definitions of parabolicity. In o r d e r to investigate where such definitions lead we shall r e s t r i c t o u r s e l v e s in the r e s t of this section to the c a s e of constant coefficients in P
and Eh
, and we shall consider only
s o that F o u r i e r - t r a n s f o r m s can b e conveniently applied. We then s a y L2 that the s y s t e m ( 2 . 1 ) with constant P, (x) = Pa is parabolic if the initial value problem ( 2 . 1 ) , ( 2 . 2 ) is c o r r e c t l y posed in differential o p e r a t o r
Q and any positive
Similarly, we s a y that the o p e r a t o r
x i s parabolic in operator
L2
Eh
if it i s stable in
T
L2
and if f o r ally
, T ,
with coefficients independent of L 2 , and if for any difference
Qh consistent with a differential o p e r a t o r Q ,
We have the following analogue of T h e o r e m 1 . 3 : Theorem 2 . 3 .
Assume that
Eh
. Then
i s parabolic if and only if for Eh Q Qh consistent with a differential o p e r a t o r
( 2 . 1 ) (with constant coefficients) any difference o p e r a t o r
i s consistent with the parabolic equation
of order
q, any v c L,
p a i r of sequences we have
11% Proof
{ h.)'J 1 n.
j
. See E5] .
Eh j
, any t -> 0 which i s > 0 if q > 0 , and any ,{nj) f , with h --r 0 , n.k,+ t when j + CO, j
v
- Q ~ ( t ) 1v
--r
J J
0, when j +m
.
We now want to characterize algebraically systems (2.1) and operators Eh which
a r e parabolic in
the present sense. By Parsevalts relation
conditions (2.7) and (2.8) a r e equivalent to d sup{l Q ( l ) e x p (t P ( I ) ) ~ ; 0 < T < t< T,lcR ] < w 3
(2.9) and
s u p ( I Q ~ ( I ) E ~ (;~ O) <~ T-< n k < T , I € R ~ }
(2.10)
respectively. Here E similarly for
h Qh( 5 )
<m,
( 5 ) is the symbol of the difference operator E h and
.
Consider first the case of a system. We then have : Theorem 2.4. Assume that the initial-value problem (2.1) tly posed in L 2
. Then it i s parabolic in
sitive constants
C
2
if and only if there a r e po-
such that
1' C2'
Proof. F o r details see -
L
, (2.2) i s correc-
25
. We give a short sketch . The suSf'~c~~c~~ir.). of
the conditions follows easily from the inequality N- 1
exp (t A)
-<
exp(t A (A))
j=O
(2t A )' ,
which holds for any NxN matrix A. To prove the necessity of conditiori (2.11)
, we notice
that by the parabolicity condition (2.9) , we have
and s o with
-
A(r) = max t=r
-
A(r)
(2.13)
-<
A
log C
(P(t) ) ,
- log (1 t r)
-+ -03
when r
-. 03
Using the Seidenberg-Tarski elimination theorem one can prove that is algebraic in
r for large
r and thus by (2.13), there is a (rational)
positive number p and a positive
- -
( r )= This implies (2.11)
2
-
A(r)
C such that 1
(1 t 0 1 ) when
r
-
m
.
.
In general it i s necessary in the formulation of Theorem 2.4 to explicitly assume the correctness of the initial-value problem. There are, however, some cases when the correctness follows from (2.11) case is when P ( [ )
is a normal matrix, in particular if
. One such
P ( ( ) is scalar,,
An example of this is offered by the equation
where 2 3 Re P ( F ) = R e ( i t ) + ( i t )
=
-5
2
.
One other case when correctness i s automatic is when (2.8) holds with = M ; this i s the case of parabolicity in Petrowsky's sense. An
example where
(2.11) is satisfied without correctness is given by (N=2)
where
Systems which satisfy (2.11) a r e called parabolic in Silovts sense; the present parabolicity concept is thus more restrictive than Silovts.
V, Thomke we can give a characterization of parabolic
Using Theorem 1.1
equations which contain at the same time the correctness and the condition
(2.11):
Theorem 2.5.
The initial-value problem (2.1)
only if there a r e positive constants a hermitian matrix
H(
1)
C
, (2.2) is parabolic if and
, C 2, C 3,
1
p and for each r e a l
such that
and (2.14)
Re (H(I)p({) ) (
(-c21~I"+ C 3 ) I -
Our aim is now to similarly characterize parabolic difference operators. We have : Theorem 2.6. Assume that
the operator Eh
is consistent with the equa-
.
Then E is parabolic in L if and only 2 h 2 if there a r e positive constants C1, C2, ho , v such that
tion (2. l ) and stable in
Proof. See [25] p
r e complicated
L
. The proof
is similar to that of Theorem 2.4
. The inequality
( P = P (A)) , which is analogous
cy
but mo-
of (2.15) for
(2.10)
to
. To prove
(2.12), is used to prove the sufficienthe necessity one uses again the
Seidenberg-Tarski theorem, but this time trigonometric polynomials take the place of ordinary polynomials. Also in this case the stability has to be explicitly assumed; it is easy to give examples where (2.15) is satisfied by a n unstable operator
.
Again, the case of a normal matrix Eh({) , in particular a scalar h Eh({) , and the case v = M in (2.15) a r e exceptions ;-in this latter case E, E
is parabolic in the s e n s e of Widlund. Using T h e o r e m 1.4 one c a n prove the following counterpart of T h e o r e m 2.5 :
is parabolic in L if and only if t h e r e h 2 C1, C2, C3, ho, D and f o r each h < h and
Theorem 2 . 7 . The o p e r a t o r a r e positive constants
5
E
-
2
R~
E
a positive definite m a t r i x H
0
h (5 ) with
and
The l a r g e s t possible
p
and
u
in (2.11)
(or (2.14)) and (2.15)
(or (2.16)) , respectively, a r e r e f e r r e d to a s the o r d e r s of parabolicity. It can be proved that i)
if (2.1) is parabolic of o r d e r p , then t h e r e e x i s t s a n o p e r a t o r consistent with (2.1) and parabolic of o r d e r p
ii) if u
Eh
Eh
;
is consistent with (2.1) , and (2.11) and (2.16) hold, then
i p ;
iii) if (2.1) is parabolic of o r d e r
p
, then t h e r e exist o p e r a t o r s
Eh consistent with (2.1) which a r e unstable, o t h e r s which a r e stable but not parabolic, and s t i l l o t h e r s which a r e parabolic, but of o r d e r
F o r details, s e e
[25]
.
3. T h e r a t e o f c o n v e r g e n c e . .......................... Consider again an
initial-value problem
u
.
In the sequel we shall demand not only that the initial-value problem be correctly posed in
W but that it satisfies the stronger requireP' ment of the following definition. We say that the initial-value problem is in W if for any m > O . , v E wm implies P P E(t) v € wm and there is a constant C such that for all v c w m , P m, T P
strongly correctly posed
w
.
E(t). W C wW P ' P It can be proved that if P = P(x, D) has constant coefficients, o r if
In particular, this definition implies that
it i s
of first order, then strong correctness
consequence of correctness.
in
W i s an automatic P Further , systems which a r e parabolic in
Petrowsky's sense a r e strongly correctly posed in
-
l l P < W .
W for any p P
with
We shall also introduce a more general parabolicity concept than in Section 2. We shall say that the system (3.1) i s strongly parabolic of o r der b
in W if the initial-value problem (3.1) ;(3.2) i s correctly P posed in W if v € W implies Da E(t)v 6 W for all a when P' . P P t > 0 , and if
One can prove that the o r d e r equal to the o r d e r Petrowsky's p
strong parabolicity i s at most
of the system. Systems which a r e parabolic in
M in W for any P , and systems which a r e parabolic of o r d e r b in
sense a r e strongly parabolic of o r d e r
-
with
M
b of
1< p<
oo
L2 in the sense of Section 2 a r e also strongly parabolic in order
b
.
L2 of
Consider
difference operators of the s a m e form a s previously, na-
mely
F o r the sake of simplicity we shall assume here that that the summation i s over a finite number
i s explicit, s o Eh of t e r m s only. We have pre-
E with E(k) to mean that for any h sufficiently smooth solution u(x, t) of (3.1) ,
viously defined consistency of
more precisely, racy
y
Eh
i s said to approximate
if for any such u(x, t
(3.4) When (3. I), (3.2) to assume this Theorem 3.1.
+
E(k) with ortler of accu-
u, k) = E h u(x, t)
+ kO(hP
),
h +0
.
i s strongly correctly posed in
W , it i s sufficient P condition locally to obtain the following global estimate :
Assume that the initial-value problem (3. I), (3.2)
gly correctly posed in order of accuracy M+P any v€ W ,
.
i s stron-
W and that Eh approximates E(k) with P Then there exists a constant C such that for
P
Proof. See -
1181
. The proof consists in expanding
E(k)v = u(x, k) and
F v in Taylor s e r i e s around the point (x, 0) , using (3.4), and estimating v the remainder t e r m s in integral form. In doing so, it is sufficient to consider v in the dense subset
W= of
.
W P We now easily obtain the following estimate for the r a t e of conver1:
gence : Theorem 3.2.
: Assume that the initial-value problem (3.1) , (3.2) i s
strongly correctly posed in
W and that E in stable in W and apP h P proximates E(k) with order of accuracy y Then there i s a constant M+ C = CT such that for v W , nk
.
Proof. We -
have
- E(nk)) v = and s o by the stability of
n- 1 j=O
$-I-'
(Eh - E(k)) E (jk) v ,
Eh , Theorem 3.1,
and the strong correctness,
which proves the theorem. In special cases, this theorem appears in many places. Thus, the situation i s that for initial-values in
W we have (by P Lax' equivalence theorem) convergence without any added information on its rate, and if the initial-values a r e known to be in conclude that the rete of convergence i s
O(hy ) when
wMtPWe can P h+O It i s natural
.
what one can say if the initial data belong to a space "intermedia-
to ask
and wMtq To answer this question we shall introduce some W P P spaces of functions which a r e interpolation spaces between Wand P wm in the sense of the theory of interpolation of Banach spaces (cf
ten to
.
P
1;8]
and references) Let
0 of
<
-< 1.
0
u
E
W P
s
be
Set
.
a positive r e a l number and write T, u(x) = u(x+ T )
. We then denote by
such that the following norm
s = S + a , S integer,
B~ the space P i s finite, namely
Thus
BS
w
i s defined by Lipschitz
type condition for the derivatives of
o r d e r S ; these spaces a r e sometimes called Lipschitz
spaces. F o r the
Heavyside function o , x < 0 (3.6)
1,
we have for
(d = 11,
0,
X L
1< p < ca ,
and it follows that
ca
if
(o E
Co
l/p B P
, then
4
One can prove that
s
and that for integer
and
E
> 0 arbitrary ,
The main property of these spaces that we will need is then the following 1
interpolation property; assume that and
s
is
a r e a l number
with
0<s<m
C such that any bounded 1i n e a r operator
rr
whe have
<
P
'
. Then
A
is a
natural number,
there i s a constant
in W
P
with
Theorem 3.2
and (3.8)
with
A=E
- E(nk) proves immediately
n h
the following result : Theorem 3.3.
Assume that the initial-value problem (3.1) , (3.2) i s stron-
and that E i s stable in W and approximaP h P E(k) with order of accuracy y Then for 0 < s < M + P there i s
gly correctly posed in tes
W
.
a constant
C=C such that for any s , "I'
Notice
-
that Y = P ( M t i u )
1
S
v
E
B , nk P
-<
T ,
lim V = '1
grows with y and
.
This means that the estimate (3.9) becomes increasingly bette? for fixed
s when y grows sed initial-value
. In other words,
if for a given strongly correctly po-
problem one can construct stable difference schemes of a r -
bitrarily high order of accuracy, then given rates of convergence arbitrarily close to initial-values in
and let (3.6)
v=
,U
(P
O(hS) when
h
-,0
for all
L -stable operator E with order 2 h for the hyperbolic equation
X where
. By above we have
F o r dissipative operators Apelkrans [I]
s > 0 one can obtain
.
BS P As an application, consider
of accuracy
any
cp
m
F
CO
an
and X
is
the Heavyside function
in this case
Eh
, stronger results have been obtained i n
, where also the spreading of discoundinuities i s
discussed.
It is natural to ask if for a parabolic system the smoothing property of the solution operator can be used to reduce the regularity demends on
the initial data in Theorems 3.2
and 3.3.
This is indeed the case; the
result on the r a t e of convergence i s then the following. Theorem 3.4.
Assume that the system (3.1)
i s strongly parabolic of
W and that E is stable in W and approximates E(k) P h P with order of accuracy p Then for any s > 0 , T > 0 , there is order b
in
a constant
Proof. For -
.
C=C
ST
such that
details, see
[18]
-
for
-
v ~ ~ ; , n
. Here we will only sketch a proof for the .
v 6 B' where M + p b < s < M t P When b = M the other P cases can be treated similarly ; if b < M the proof in [18 uses slightly case
1
more sofisticated interpolation theory. We shall use (3.5)
. For
and s o by (3.7) and (3.8)
For
j
>0
j = 0 we have by the stability and Theorem
, since' s
>p
,
we have by Theorem 3; 1 and the strong parabolicity ,
and hence by (3.8) ,
where
Adding over
j
we get
which proves (3.10) in the case considered. see Junco-
For an earlier particular result in the same direction, s a and Young [12
]
.
We shall complete this section with result by Hedstrom [lo]
. Consider a s in
a simple case of a recent Section
1 the initial-value
problem for the equation
a u ,,& at
ax
P real
'
and a consistent difference operator of the form Ehv(x) = j
Set again
and assume that we have the case that in
W
P'
p
#
.
a .v(x ijh) J
2; in particular assume that
Eh
i s stable in L
2
but unstable
ct. real,
q ( J ) real polynomial,
V even,
Re
y >
0
/
q(0)
0, 1 <,.U
+ 1< v ,
.
(In Section 1 we used ,u instead of
p
t 1 ; here ,U
i s a s above the
order of accuracy.) We then have : Theorem 3.5. 0 <s
-<
p
+1
Under the above assumptions, for any there i s a constant
C
s with
> 0 and S
such that for v 6 Bco , nk = t , P S-
-
ltP P t l , 2
h
B
t
-P2
00
log-l.
<s< p t 1 ,
-
S
=- P t 1
2
'
2s( v -I)-( v - p -11
-
v 6 B:
-
where P + l < . S < P + l , 2 the estimate for the r a t e of convergence i s the same a s the one we would < P 2f l , have obtained by Theorem 3 . 3 , in the stable case F o r 0 < s however, the rate of convergence becomes slower. In the extreme case In particular this means that for
.
s = 0 , we recognize the growth in (1.23)
.
-
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2,
D. G. Aronson, The stability of finite difference approximations to second order linear parabolic differential equations. Duke Math. .J. 30 (1963), 117-128.
3.
D. G. Aronsons, On the stability of certain finite difference approximations to parabolic systems of differential equations. Numer. Math. 5(1963), 118-137.
4.
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5. 6.
A. Friedman, Generalized functions and partial differential equations. Prentice-Hall, Englewood Cliffs, New Jersey, 1963.
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K.O. Friedrichs, Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7(1954), 345-392.
8.
G. W. Hedstrom, The near-stability of the Lax-Wendroff method. Numer. Math. 7(1965), 73-77
9.
G. W. Hedstrom, Norms of powers of absolutely convergent Fourier series. Michigan Math. J. 13(1966), 393-416.
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G. W. Hedstrom, The rate of convergence of some difference schemes. To appear.
11.
F. John, On integration of parabolic equations by difference methods. Comm. P u r e Appl. Math. 5(1962), 155-211.
12.
M. L. Juncosa and D. M. Young, On the order of convergence of solutions of a difference equation to a solution of the diffusinn equation. J. Soc. Indust. Appl. Math. 1(1953), 111-135.
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P.D. Lax and R.D. Richtmyer, Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9(1956) , 267-293.
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