Domain Decomposition Methods for Partial Differential Equations (Numerical Mathematics and Scientific Computation)

N U M E R I C A L MATHEMATICS A N D SCIENTIFIC COMPUTATION

*P. Dierckx: Curve and surface fittings with splines *H. Wilkinson: The algebraic eigenvalue problem *I. Duff, A. Erisman, and J. Reid: Direct methods for sparse matrices *M. J. Baines: Moving finite elements *J. D. Pryce: Numerical solution of Sturm-Liouvilleproblems K. Burrage: Parallel and sequential methods for ordinary differential equations Y. Censor and S. A. Zenios: Parallel optimization: theory, algorithms and applications M. Ainsworth, J. Levesley, M. Marietta, and W. Light: Wavelets, multilevel methods and elliptic PDEs W. Freeden, T. Gervens, and M. Schreiner: Constructive approximation on the sphere: theory and applications to geomathematics Ch. Schwab: p- and hp- finite element methods: theory and applications to solid andfluid mechanics J.W. Jerome: Modelling and computation for applications in mathematics, science, and engineering A. Quarteroni and A. Valli: Domain decomposition methods for partial differential equations Monographs marked with an asterix (*) appeared in the series 'Monographs in Numerical Analysis' which has been folded into, and is continued by, the current series.

Domain Decomposition Methods for

Partial Differential Equations ALFIO QUARTERONI Professor of Numerical Analysis, Politecnico di Milano, Italy, and Ecole Polytechnique Federate de Lausanne, Switzerland ALBERTO VALLI Professor of Functional Analysis, University ofTrento, Italy

C L A R E N D O N PRESS

1999

• OXFORD

CONTENTS

1 1.1 1.2 1.3 1.4

1.5

1.6 1.7 2 2.1

2-2

2-3

2-4

The Mathematical Foundation of Domain Decomposition M e t h o d s Multi-domain formulation and the Steklov-Poincare interface equation Variational formulation of the multi-domain problem Iterative substructuring methods based on transmission conditions at the interface Generalisations 1.4.1 The Steklov-Poincare equation for the Neumann boundary value problem 1.4.2 Iterations on many subdomains The Schwarz method for overlapping subdomains 1.5.1 The multiplicative and additive forms of the Schwarz method 1.5.2 Variational interpretation of the Schwarz method 1.5.3 The Schwarz method as a projection method 1.5.4 The Schwarz method as a Richardson method 1.5.5 A characterisation of the projection operators 1.5.6 The Schwarz method for many subdomains The fictitious domain method The three-field method

1 2 5 10 18 22 24 26 26 28 29 31 32 33 34 38

Discretised Equations and Domain Decomposition Methods Finite element approximation of elliptic equations 2.1.1 The multi-domain formulation for finite elements 2.1.2 Algebraic formulation of the discrete problem Finite element approximation of the Steklov-Poincare operator 2.2.1 Eigenvalue analysis for the finite element SteklovPoincare operator Algebraic formulation of the discrete Steklov-Poincare operator: the Schur complement matrix 2.3.1 Preconditioners of the stiffness matrix derived from preconditioners of the Schur complement matrix The case of many subdomains

41 41 43 45 46 48 49 51 55

xii 2.5

3 3.1 3.2 3.3

3.4 3.5 3.6

3.7

4

CONTENTS Non-conforming domain decomposition methods 2.5.1 The mortar method 2.5.2 The three-field method at the finite dimensional level Iterative Domain Decomposition Methods at the Discrete Level Iterative substructuring methods at the finite element level The link between the Schur complement system and iterative substructuring methods Schur complement preconditioners 3.3.1 Decomposition with two subdomains 3.3.2 Decomposition with many subdomains The Schwarz method for finite elements Acceleration of the Schwarz method 3.5.1 Inexact solvers Two-level methods 3.6.1 Abstract setting of two-level methods 3.6.2 Multiplicative and additive two-level preconditioners 3.6.3 The case of the Schwarz method 3.6.4 Convergence of two-level methods Direct Galerkin approximation of the Steklov-Poincare equation

4.4 4.5 4.6

Convergence Analysis for Iterative Domain Decomposition Algorithms Extension theorems and spectrally equivalent operators 4.1.1 Extension theorems in i ? 1 ^ ) 4.1.2 Extension theorems in iJ(div; Qj) 4.1.3 Extension theorems in .ff (rot; fi;) Splitting of operators and preconditioned iterative methods 4.2.1 The finite dimensional case 4.2.2 The case of symmetric matrices 4.2.3 The case of complex matrices Convergence of the Dirichlet-Neumann iterative method 4.3.1 An alternative way to prove convergence Convergence of the Neumann-Neumann iterative method Convergence of the Robin iterative method Convergence of the alternating Schwarz method

5 5.1

Other Boundary Value Problems Non-symmetric elliptic operators

4.1

4.2

4.3

59 60 66

71 71 73 77 77 79 86 91 96 96 97 98 99 99 100

103 104 104 111 114 117 122 125 128 133 133 135 135 137 141 141

5.1.1 5.1.2 5.1.3 5.2

5.3

5.4

5.5

5.6

5-7

6 6-1 6-2 6-3

tJUlNlijJlNTS

xiii

Weak multi-domain formulation and the SteklovPoincare interface equation Substructuring iterative methods The finite dimensional approximation

142 146 147

The problem of linear elasticity 5.2.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.2.2 Substructuring iterative methods 5.2.3 The finite dimensional approximation The Stokes problem 5.3.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.3.2 Substructuring iterative methods 5.3.3 Finite dimensional approximation: the case of discontinuous pressure 5.3.4 Finite dimensional approximation: the case of continuous pressure 5.3.5 Methods based on the Uzawa pressure operator The Stokes problem for compressible flows 5.4.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.4.2 Substructuring iterative methods 5.4.3 The finite dimensional approximation The Stokes problem for inviscid compressible flows 5.5.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.5.2 Substructuring iterative methods 5.5.3 The finite dimensional approximation First-order equations 5.6.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.6.2 Substructuring iterative methods The time-harmonic Maxwell equations 5.7.1 Weak multi-domain formulation 5.7.2 The finite dimensional Steklov-Poincare interface equation 5.7.3 Substructuring iterative methods Advection—Diffusion Equations The advection-diffusion problem and its multi-domain formulations Iterative substructuring methods for one-dimensional problems Adaptive iterative substructuring methods: ADN, A R N and AR^N

147 148 151 151 153 156 170 173 185 189 190 191 194 195 197 198 201 202 203 204 208 210 211 213 214 219 220 224 227

CONTENTS

xiv 6.3.1

6.4

6.5 7 7.1

7.2

7.3

8 8.1

8.2

8.3

8.4

The damped form of the iterative algorithms: dADN, d - A R N and d-AR^N Coercive iterative substructuring methods: 7 - D R and 7 RR 6.4.1 The 7 - D R iterative method 6.4.2 Convergence of the 7 - D R iterative method 6.4.3 The 7 - D R iterative method for systems of advection-diffusion equations 6.4.4 The 7 - R R iterative method 6.4.5 Convergence of the 7 - R R iterative method The finite element realisation of the iterative algorithms Time-Dependent Problems Parabolic problems 7.1.1 Multi-domain formulation and space discretisation 7.1.2 Implicit time discretisation and subdomain iterations Hyperbolic problems 7.2.1 Multi-domain formulation 7.2.2 Implicit time discretisation and subdomain iterations Non-linear time-dependent problems 7.3.1 Navier-Stokes equations for incompressible flows 7.3.2 Navier-Stokes equations for compressible flows 7.3.3 Euler equations for compressible flows Heterogeneous Domain Decomposition Methods Heterogeneous models for advection-diffusion equations 8.1.1 The Steklov-Poincare reformulation 8.1.2 The coupling for non-linear convection-diffusion equations Heterogeneous models for incompressible flows 8.2.1 The coupling for the linearised Stokes problem 8.2.2 The coupling for the Navier-Stokes equations in exterior domains Heterogeneous models for compressible flows 8.3.1 The coupling between the Navier-Stokes and Euler equations 8.3.2 The coupling between the Euler equations and the full potential equation The coupling for the compressible Stokes equations 8.4.1 Variational formulation and finite element approximation 8.4.2 An alternative formulation: variational setting and finite element approximation

232 234 234 237 239 240 242 244 251 252 253 256 261 263 266 272 273 275 277 285 287 290 295 296 297 300 305 306 309 317 320 322

CONTENTS 8.5 9 9.1 9.2

The coupling for the time-harmonic Maxwell equations Appendix Function spaces Some properties of the Sobolev spaces

xv

328 ^33 3^ 339

References Index

357

1

THE MATHEMATICAL FOUNDATION OF DOMAIN DECOMPOSITION METHODS In this chapter the reader is encouraged to discover the mathematical foundation of domain decomposition methods, which are based on partitions of the computational domain into subdomains with or without overlap. We introduce the concept of transmission conditions at subdomain interfaces and the Steklov-Poincare problem for the interface variables. Both differential and variational formulations are addressed. Then we present substructuring iterative methods for disjoint subdomains, and the Schwarz alternating method for overlapping subdomains. The convergence analysis of these methods will be carried out in Chapter 4. We also comment on two other approaches: the fictitious domain method and the so-called three-field method. We deal mainly with symmetric linear elliptic boundary value problems, and, in particular, with the Poisson problem: f-Au = /

in fi

Iu = 0

on dfl.

(11) Here, and in the rest of this book, f2 is a d-dimensional domain (d = 2,3), with a Lipschitz boundary dCl, whose outer unit normal direction is denoted by n*, / is a given function of L 2 ( Q ) , A : = J 2 j = i D j D i is the Laplace operator and D j denotes the partial derivative with respect to Xj, j = 1,... ,d. To start

FIG. 1.1. Non-overlapping partition of the domain fi into two subdomains.

2

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

with, we assume that 0 is partitioned into two non-overlapping subdomains fii and 0 2 ) and denote by T : = fii n Q 2 (see Fig. 1.1). We also assume that T is a Lipschitz (d — l)-dimensional manifold. The generalisation to other boundary value problems will be done in later chapters, particularly Chapters 5, 6 and 8. The finite dimensional approximation is addressed in the next two chapters.

1.1

Multi-domain formulation and the Steklov-Poincare interface equation

We indicate by Ui the restriction to fij, i = 1,2, of the solution u to (1.1), and by n ' the normal direction on dVLt fi T, oriented outward. For simplicity of notation we also set n = n 1 . It is easily seen that the Poisson problem (1.1) can be reformulated in the equivalent multi-domain form:

(1.1.1)

k

-Aui = /

in f2i

ui = 0

on a o i n an

Ui

on

r

on

r

=

1l2

du2

dui

dn

dn

u2=0

on dfi 2 n dn

- AU2 = f

in fi2-

Equations (1.1.1)3 and (1.1.1)4 are the transmission conditions for iti and on r. The physical meaning of this split formulation is clear as soon as the original solution of problem (1.1) is smooth enough (say, u G C 1 ( f i ) ) . In a more general framework the equivalence between (1.1) and (1.1.1) is shown in the next section by resorting to the weak formulation of both problems. We will see in Section 1.3 that domain decomposition methods are generally amenable to iterative procedures for an interface equation that is associated with the given differential problem. This interface problem can be defined in terms of the Steklov-Poincare operator that we are going to introduce. Let us refer to the model problem (1.1) and its multi-domain formulation (1.1.1), which corresponds to the domain partition of Fig. 1.1. The same arguments apply to the other boundary value problems that will be described in Section 1.4 (for further details see also Quarteroni and Valli 1991a, b). Let A denote the unknown value of u on T. We consider the two Dirichlet problems:

tji.l

lViU-Lil

r WJrtiViUijAlIUiN AINU S f liN 1 MI.F AUB ii^UAX lUlN -AWi = f

in fij

Wi = 0

on dQi D dtl

(

. Wi = A

3

on r ,

f o r i = 1,2. We can obviously state that Wi = u? + u.

(1.1.3) where we have defined problems:

and u* to be the solutions of the following Dirichlet

-Au? = 0 (1.1.4)

in Q.i on d£li fl dfl

u° = 0

on r,

U? = A and -Au* = /

on 8Q,i n 80,

«* = 0

(1.1.5)

u:

in fij

on r.

0

For each i — 1,2, it' is the harmonic extension of A into and will be denoted HiX. We will also write Q t f instead of u*. If we proceed formally, comparing (1.1.1) with (1.1.2), it follows that (1.1.6)

Wi = Ui for i = l , 2 if and only if

^ ^ = -tt— on T. on on

The latter condition amounts to the requirement that A satisfies the Poincare interface equation (1.1.7)

5A =

on T,

X

where 9

(1.1.8)

r 2

t

dn

Qif

d

i=l a nd

S is the Steklov-Poincare

operator, which is formally defined as

Steklov-

4

THE MATHEMATICAL FOUNDATION OK DL> MhiThLuus d

d_ dn

(1.1.9) =

£

i=i In particular, S can be split as S — (1.1.10)

SiT1 : =

+ S2, with

d

i = 1

>2-

This operator, which was introduced a century ago (1896-1900), has been more recently analysed by Agoshkov and Lebedev (1985) in the framework of iterative methods. It is, however, worthwhile to point out that Agoshkov and Lebedev in fact considered the inverse operators S^1 and S2 1 , and called them Poincare-Steklov operators. Remark 1.1.1 (Generalisation) The analysis that we are going to carry out on the Poisson equation will be applied to a far more general differential problem of the form (1.1.11)

Cu = f

in 0 ,

where £ is a partial differential operator, / is a given datum, and u is the unknown solution. A broad range of problems can be considered (see Chapter 5), including non-symmetric elliptic problems, the linear elasticity problem, the Stokes problem for incompressible flows, the viscous and inviscid Stokes problem for compressible flows, and the time-harmonic Maxwell system. Should fi be partitioned into two disjoint subdomains fii and CI2 as indicated in Fig. 1.1, we can go along the same lines presented above to generate a split version of problem (1.1.11). Denoting again for i = 1,2 by m the restriction of u to Oi, it follows from (1.1.11) that

(1.1.12)

= f •£^2 = /

in H], in fl2-

To guarantee the equivalence with (1.1.11) we need to enforce transmission conditions between u\ and u^ across T. In an abstract form, such conditions can be expressed by the two relationships (1.1.13)

$ ( u i ) = $(1x2) on T tf(ui)

= ${U2) on r ,

where the functions $ and ^ will depend upon the nature of the problem. Typically, for second-order elliptic operators, (1.1.13) expresses the continuity across r of u and of the normal 'flux' (namely, the normal stress) involving firstorder derivatives of iti and ?i2. More generally, these interface conditions are most often determined noting that:

§L2

VARIATIONAL FORMULATION OF THE MULTI-DOMAIN PROBLEM

5

• The solution u belongs to a space of functions defined over the whole Q,. This requires that in fix and in 0,2 enjoy a certain regularity therein, and in addition that they satisfy a suitable matching on F. • The restrictions UjQ1 andu^2 are distributional solutions to the given equation in fli and O2> respectively. Another interface condition between them comes from the fact that u in fact satisfies the equation in the sense of distributions in the whole Q; namely, through the interface T and not only separately in Qi and Q.2 • For the Poisson problem (1.1.1) the obvious identification $(v) = v,

=

^

holds true. Keeping in mind this correspondence, all iterative substructuring methods that we are going to introduce for the Poisson problem in Sections 1.3, 1.4 can actually be extended to the more general problems (1.1.12), (1.1.13) in a straightforward manner. This is the case, in particular, for the classical methods like Dirichlet-Neumann, Neumann-Neumann or Robin, originally introduced for the Laplace operator, and here applied to a very general family of boundary value problems (see Chapter 5). • Remark 1.1.2 In transforming the Dirichlet boundary value problem (1.1) into an equation on F, we have chosen as interface unknown the physical variable that has to match on T as described in the first condition in Remark 1.1.1, while the interface equation (1.1.7) for the Steklov-Poincare operator S is based on ensuring that the second condition is satisfied. The same procedure will be constantly followed in the remainder of this book, but clearly other choices of interface equation could be devised. These are amenable to different forms of the Steklov-Poincare operator, and, correspondingly, different iterative substructuring methods. •

1.2

Variational formulation of the multi-domain problem

In this section we formulate (1.1) in a variational way. This requires us to introduce Sobolev spaces and to take into account some of their properties. We will not dwell here on this argument, and refer the interested reader to the comprehensive presentation of this theory that can be found, for example, in J.-L. Lions and Magenes (1972) (see also Chapter 9). By integrating by parts in fi, it is easily seen that the weak formulation of (1.1) reads (1-2.1) where

find

u G V : a{u, v) = (/, v)

V v G V,

6

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

{w,v)

:= / w v Jn a(w, v) : = ( V W , Vv) := {u e L2{9,)\DjV

€ L2(n),

:={v£H1(n)\vldn=

Hl0({l)

l,...,d}

0}

fi)

V

j =

and v\qq denotes the trace of v (that is, its restriction) on dft. The norm of H1(Q) will be denoted by || • ||i,n, while || • ||o,fj will indicate the norm of L2(fl). We recall that IMIo,Q =

1

M

/

2

,

while

1/2

d Mli,n =

I IMlo,n +

Sll^i«llo,n

j=i for each v e H1(fi). The Poincare inequality states that there exists a constant Cq > 0 such that (1.2.2)

f v Jn

2

< c

u

[ V p ^ ) Jq -

2

v . e

= 1

Therefore, the norm ||u||I,n is equivalent to the norm ||VU||o,n FOR EACH v € //q (fi). It is worthwhile to note that the same result is true for functions that vanish only on an open and non-empty subset S of <9fi. We also recall that the trace space of on the boundary dfl is denoted _ff1//2(<9fi). In an analogous way, the trace space on an open and non-empty subset S C ffi is indicated by H 1 ' 2 ^ ) . The trace operator from H 1 ^ ) to H ^ i d S l ) is surjective and continuous; that is, the following trace inequality holds (1.2.3)

'|en||i/2,an < C n l M k n

Vvelf1^),

where || • ||i/2,an denotes the norm in ff1//2(dQ). Moreover, it can be shown that there exist injective, linear, and continuous extension operators from Hl/2(d£l) to Let us also consider the weak multi-domain formulation equivalent to (1.2.1). First of all, let us set

a.i(wi,Vi) (1.2.4)

= / WiVi JQi = {Vwi, Vui)m

Vi r V° i

= {Vi £ i f 1 (ft*) I UiisnnaQ; = 0 }

A = {rj e H1/2(r)

I ?7 = U|r for a suitable v £ V}.

jj-^2

VAK1AUU1NAL rUJttlVi U LA i IUIN UF THJi iVlULiTl-JJUMAIiN FKUtJi^M

7

When r n = 0 we have A = H1/2(r); instead, when T n dfl ± 0 the space A, is strictly included in i J 1 / 2 ( r ) , w hich in this case is usually denoted by Hqq2(T), and is endowed with a norm || • ||A, which is larger than the norm of ff1/2(r). We refer to J.-L. Lions and Magenes (1972) for an intrinsic definition of the trace space A and for its principal properties. Note, in particular, that the trace inequality holds as well when considered on the interface T; that is, there exist q* y o, i = 1, 2, such that Ihriu

(1.2.5)

V U; e


V.

Finally, for i = 1,2 denote by TZt any possible (continuous) operator from A to Vi that satisfies (7^.^77) j r = rj. Any such operator (which can be proved to exist) will be called an extension operator from A to V;. L e m m a 1 . 2 . 1 The Poisson problem (1.2.1) can be equivalently reformulated find «i 6 li, «2 € V2 such that a1(ui,vi)

=

V

{f,vi)s

ui = u2

on T

a2{u2,v2)

= (f,v2)n2

e

Vi

as:

V:!0

(1.2.6)

a2{u2,K2fi) where IZi denotes

v « 2 e V$

= {f,Tl2pi)n2

+ (f,H

any possible extension

- ai{ui,Thn)

V/j6A,

operator from A to V;.

P r o o f Let us start by considering the solution u to (1.2.1). Setting u; : = M|q;, i = 1 , 2 , we have that m £ Vi and that ( 1 . 2 . 6 ) 1 , ( 1 . 2 . 6 ) 2 and ( 1 . 2 . 6 ) 3 are trivially satisfied. Moreover, for each /j, £ A the function IZ/j, defined as

Tc-2/U

in S22

belongs to V , therefore we have a{u,TZ/j,) =

(f,Kn),

which is equivalent to (1.2.6)4. On the other hand, let Uj, i = 1,2 be the solutions to (1.2.6). Setting _ J u\ \ u2

in f2i in

'

from (1.2.6)2 it follows that u e V. Then, taking v G V, we have that n : = U|r <5 Define Tl/j, as before; clearly ( w ^ - 11^) e Vf>, and from (1.2.6)i, (1.2.6)3, and (1.2.6)4 it follows that

8

T H E M A T H E M A T I C A L EU U JN L> ATIOJN OK D U M E T t i U I J b

un.

2

a{u, v) = ^[ai(tti, i=1

- Tli/i) + a,i(ui, TZi^)]

2

=

»=i = (/>«)•

v\Qi

~ Ki^hi

+ (/»T^itAnj

Let us point out, moreover, that (1.2.6)i and (1.2.6)3 are the weak counterparts of ( l . l . l ) i and (1.1.1) 6 , respectively, whereas (1.2.6)4 is the weak form of a the Neumann condition (1.1.1)4. We stress that the extension operator IZi : A —» V{ used in (1.2.6)4 can actually be chosen arbitrarily. Note that the weak form (1.2.6)4 of the continuity of the Neumann condition makes sense also for corner points, where the pointwise relation (1.1.1)4 would be ambiguous or even meaningless. The Steklov-Poincare operator S introduced in (1.1.9) can be characterised as follows. It acts between the space of trace functions A (introduced in (1.2.4)) and its dual A'. More precisely, applying Green's formula and recalling that Hif] is harmonic in f2; for all r? £ A, and i = 1,2 we have

(Sr), v) = Y , ( ^

H i 7 h

= ai(Hir), Tlifj.)

f")=J2jn

'

V 77, /j 6 A,

where the extension operators TZi are as in Lemma 1.2.1. Hereafter (•, •) denotes the duality pairing between A' and A. In particular, taking TZi/j, — H m , we obtain the following variational representation of S: 2

(1.2.7)

(Sv,tJ) = ^ 2 a i { H i r i , H i f i ) »=1

hence the operator S is symmetric. (1.2.2) it holds that

Moreover, from the Poincare inequality

2

(Sv,v)

= £

Vjj.^A,

2

IIVi^H^,.

> ]T

i=l

i=1

I-LH|//.r/||?ini.

'

Taking into consideration the trace inequality (1.2.5) we finally have (S^rj)

>a\\r,\\2A

V 77 G A,

for a suitable constant a > 0, therefore S is a coercive operator. Proceeding in an analogous way, from (1.1.10) we have

i

^ 2

V A K I A T I U I N A L , f ' U l t M U L A T l U l N U F T-tiii iVIUJ^Ti-JUUMAilN F R U B L K M

(1.2.8)

(SiV, ^

=

fn

'

= ai(HiV, Hin)

9

V 77, fj, € A.

Clearly, each Si is symmetric and coercive; that is, it satisfies (Stv,v)

(1.2.9)

VryeA.

> "iIMIa

Another relevant property of the Steklov-Poincare operators Si is that they are continuous; that is, there exist constants (3i > 0 such that (1.2.10)

(Siv,v) < A I M U I H I A -

In fact {SiT],ij) < H-Hi^Hi,^ llffiMlli.Oi, and from well known estimates for the solution of elliptic boundary value problems (see, for example, J.-L. Lions and Magenes 1972) it follows that (1.2.11)

\\Hiv\\i^ < C M a

y r] G a,

hence (1.2.10) holds. These properties are of great interest, because they can be exploited to obtain a numerical solution to the Steklov-Poincare problem (1.1.7). Clearly, as soon as an approximation of A is available (1.1.2) can be reduced to the solution of two independent Dirichlet problems. Finally, we can also give a variational interpretation of the right-hand side % in (1.1.7). From (1.1.8) it can be expressed through the functions / and Qif as follows: 2 (1.2.12)

i = 1

2 X

i=1

'

2

Qi

= f^{(f,Tllfi)ni-ai(gif,1liii)} i=1

V > G A.

Therefore, the Steklov-Poincare equation (1.1.7) can be written in a variational form as (1-2.13)

findAeA

: (SX,n) = (x,m)

V p e A.

Let us also note that, from a variational point of view, the functions it? = HiX and u* = Q^ introduced in (1.1.4) and (1.1.5) are the solutions to the following Problems: HiXsVi (1-2.14)

{

ai(Hi\,Vi)

H, A = A

: = 0 on F

\/VieV?

L/tl. 1

i t l t IViAinaiVIAli^Alj r UUiNJJAliUiN Ut UU MllitlWJJa

IU and (1.2.15)

: ai(gif,vi)

Gif

V v, e V°.

= (f,vi)

Remark 1.2.2 The variational form (1.2.13) of the Steklov-Poincare equation could also be obtained directly from the interface relationship (1.2.6)4. Indeed, from this relation using the splitting m = + u* we obtain 2

2

ai{u°,TZifj,) = Z—1

2

Tli^Qi ~ 1

ai(ui'nilJ)

V /u e A.

2=1

Integrating by parts on each side, and using (1.1.4) and (1.1.5), we deduce that ai{ui,

f du° T^-iH) = J

a,i(u*,1liLx) = (f,niii)Qi

+ J

(the integrals over T are formal expressions: indeed, they should be replaced by the duality pairing between A' and A). We therefore conclude with the following integral equations at the interface P: f fdu°

du°2 \

f (8u\

du*2\

or, equivalently,

I ( I * * " which coincides with (1.2.13).

" = " /r ( l

S l f

- l ^ ' ) "

V

"

e K

•

In Table 1.2.1 we summarise the relation between the differential arid the variational forms of the single domain problem, the multi-domain problem and the Steklov-Poincare equation.

1.3

Iterative substructuring methods based on transmission conditions at the interface

We face now the task of solving the multi-domain problem (1.1.1) by iterative procedures that will then be replicated at the finite dimensional level. These methods are traditionally referred to as iterative substructuring methods. Typically, they introduce a sequence of subproblems in fix and $^2 for which conditions (1.1.1)3 and (1.1.1) 4 provide, respectively, Dirichlet or Neumann data at the internal boundary P. This can be accomplished in several ways, some of which are presented below.

ITERATIVE SUBSTRUCTURING METHODS

r §1-3

11

Table 1.2.1 The differential and the variational forms of the single domain, the multi-domain and the Steklov-Poincare problems Differential vs variational form

—Au = f in fi

Single domain

u

: Vd£

— 0 on dQ,

a(u,v)

in fii

Ui€VI

: V «1 e V°

ai(u!,vi)

= (f,v i)n x

-Aui = f

Multidomain

u e ^(fi)

on <9fii n a n

=

Ui

=

0

-AU2

=

/

U2

= 0

on dfi2 n a n

a2(u2,v2)

Ui

= u2

on r

u\ = u2

dui

_

du2 dn

SX = x

on r

- (/, v2)n2 on r

J2iai(ui,TZip,)

on r

dn

(f,v)

: V v2 G V2°

u2eV2

in f?2

fl^fi)

=

HiifiKivht

V/i£ A

A £ A : V p, 6 A =

n

SteklovPoincare

{x,fA

(St},/J,) := Ei

x •=

ai(HiV>

72-jM)

- Gzf) ~Ei

ai(0if,

Hill)

In general, two sequences of functions { u j } , { u ^ } are generated starting from an initial guess uj, u2, and will converge to and u2, respectively. 1- The Dirichlet-Neumann

method

Given A 0 , solve for each k > 0: ' -

A fc+i

(1.3.1)

.u

=

= = 0 Afc

f

in f2i on dQi on r,

n dQ

THE MATHEMATICAL FOUNDATION OF DD METHODS

12

<Jh. 1

then ' -

A

=

< u\+1 = 0

(1.3.2)

/

in 0 2 on dfl2 n dO on r

with A ^ 1 : = ffu^t1 + (1 - 6)\ k

(1.3.3)

6 being a positive acceleration parameter. This method was considered, for example, by Bj0rstad and Widlund (1986); Bramble et al. (1986a); Funaro et al. (1988); and Marini and Quarteroni (1988, 1989). The same method without relaxation (that is, with 0 = 1) does not necessarily converge, unless special assumptions are made about f2j and This can be easily seen already for one-dimensional problems: if the length of Qi (the Dirichlet subdomain) is larger than that of Q 2 we have convergence, otherwise the unrelaxed method diverges. An example is furnished in Fig. 1.3.1, where the method is applied to approximate the (null) solution of the problem -u"(x)=0 in (0,1) u(0) = u ( l ) = 0. Note that for this particular problem, whatever partition is chosen, a suitable parameter 6 yields exact convergence in two iterations.

o

FIG. 1.3.1. Divergence (left) and convergence (right) for the unrelaxed Dirichlet-Neumann iterations. Let us point out that a similar iterative procedure can be obtained by applying a relaxation procedure on the Neumann boundary condition, defining v!{ + l

rrmATivt; «ubstku<jtukijng m e t h o d s

§1.3 and U2 +l

as

^

solutions

13

to

r -Auk+1 = f -.fc+1 =_ 0 du k 2 +1

on dVt2 D 5 0 k

= /

' -Auk+X < uk+1

in n 2

=0

in fii

n dQ,

on

^ uk+1 = uk+1

on T,

setting then

When we present the finite dimensional version of the Dirichlet-Neumann scheme and its algebraic formulation (see Sections 3.2 and 3.3), it will be useful to resort to the weak formulation of the iterative procedure (1.3.1), (1.3.2). It reads as follows: ' find u k + 1 e Vi : < a1(uk+1,v1) ^ uk+1

' find uk+1

=Xk

= (f,v1)Ql

VvieV?

on T

G V2 :

< Muk2+1,v2) . a2{uk2+\ll2iA

= (f,v2h2 = (/,n2i^h2

vv2ev2° + (f,niAi)0l

- ai{uk+1

V n e A,

w here

a ; (-,-) have been introduced in (1.2.4) and Hi denotes any possible extension operator from A to Vi, and the notations are the same as in Section 1.2. Let us show how this iterative scheme can be interpreted as a preconditioned Richardson procedure. For the sake of exposition, we refer to the differential formulation (1.3.1)—(1.3.3); however, the same result can be proved for the weak formulation introduced above. Since we have chosen n = n 1 = —n2, we have

14

T-tihJ M A T i l E M A T i f A L f UUiNJJ/Vl IUiN U f

L)U

' - A ( < 4 + 1 - g2f) = o

Ju+

k 1 -g2f

Noting that uk+1

oil. i

in ft2 on <9ft2 n 9ft

= o

d(4+1-g2f) mj. dn2

MfiitlULio

l duk+ — dn

_ =

dg2f dn

,

+

on r.

= HxXk + Qxf, it follows that u^1

= (uk+1

-

g2f),r dH1xk

\

J2

agj

dn

,

dn

+

<9n

= 5 2 " 1 (-5 1 A f c + x), and therefore

Xk+1 =

+

A* + x ) - A*]

#[3-1

= Xk + 9S21(—SXk

+ x).

We are thus left with a Richardson procedure for the Steklov-Poincare equation (1.1.7) with the operator S2 as a preconditioner. This interpretation plays a central role in the convergence analysis of the Dirichlet-Neumann method. The proof that the Dirichlet-Neumann iterations converge for very general problems will be given in Section 4.3, as a consequence of Theorem 4.2.2, which is concerned with Steklov-Poincare iterations. Besides, we prove the convergence of this iterative scheme in the framework of finite element discretisation, and in particular it will be shown that its rate of convergence is independent of the discretisation parameter h. 2. The Neumann-Neumann

method

This method was considered by Bourgat et al. (1989); a former version had been investigated already in Agoshkov and Lebedev (1985). In this case, for each k > 0 we have to solve ' - Auk+1 (1.3.4)

< uk+1 uk+1

= f

in Qi

=0

on dfli n d n

= Xk

on

r.

and then in ft.

' -A^f4"1 = 0 (1.3.5)

<

=

on dft;

o

d
dn

duk+1 dn

on r ,

n dft

ITERATIVE SUBSTRUCTURING METHODS

r §1-3 for

15

i = 1,2, with A ^ 1 : = \k - 6M1+1

(1.3.6)

-

a2ipk+}).

As before, 8 > 0 is an acceleration parameter, a\ and a 2 are two positive averaging coefficients, while A0 is a given datum. Using the same notations as for the Dirichlet-Neumann scheme, the weak formulation of the Neumann-Neumann method reads: ( find uk+1

G Vi :

ai(uk+1,Vi)

= (f,Vi)n.

u k + 1 = Afc

on T

V Vi G VP

for i = l , 2 , and ' find

G Vi :

ai(^+1."i)=0 a1(^+\nlti)

V i>i G V]°

= -(f,n^)Ql

-

(f,n2fj-)a2

+a1{ukl+1,1lin)

V /i G A,

+ a2{uk2+1 ,^2/i)

find V>2+1 G V2 : a2(ipk+1,v2)

0

V

6 V?

/,*+1 -a1{uk1+\ll1n)-a2{vk2+1,Tl2n) V / i G A. Also, the Neumann-Neumann scheme can be interpreted as a preconditioned Richardson scheme. In fact, we have already noted that w,fc+1 = Hi\ k + therefore

,k+1 _ Vllr

! /affiA^ V

dn

SGi/ +

dn

fl&A 8n

dn

)

and similarly, recalling that n = - n 2 ,

Therefore

A ^ 1 = A* + efaS? + &2S21)(X - S\k), which is a Richardson procedure for the Steklov-Poincare equation (1.1.7) with ^ e operator (cxiSf 1 + a a S ^ 1 ) " 1 as a preconditioner.

THE MATHEMATICAL FOUNDATION OF DD METHODS

16

<Jh. 1

The proof of the convergence of this algorithm will be presented in Section 4.4, also for the finite element discretisation. In that case, the rate of convergence is shown to be independent of the grid-size h. 3. The Robin method This time, for each k > 0, we solve (see P.-L. Lions 1990 for the presentation and the analysis of the algorithm; see also Agoshkov 1988 for a similar formulation at the algebraic level) in

An^1 = /

on 5 0 i

= 0

(1.3.7) du

k+1

+ 7i"i

dn

_

=

94 + jm2 dn

n dn

on T,

and then r-A (1.3.8)

in ft2

= /

,k+1 _ duf

on a n 2 n d a

72 « £ + 1

dn

=

dut1

k+1

on

r,

where u\ is given, and 71 and 72 are non-negative acceleration parameters satisfying 7 1 + 7 2 > 0. For the sake of parallelisation, in (1.3.8) we could also consider Uj instead of assigning in that case also The proof of the convergence of this scheme will be given in Section 4.5. 4. The method by Agoshkov and Lebedev Along the same lines, we mention the following iteration-by-subdomain algorithm that has been proposed by Agoshkov and Lebedev (1985). Given u j and Uj, for each k > 0 we have to solve in Cti

r - A ui+i'2 = / (1.3.9)

k+1/2

auj+1/2 — ~

(1.3.10)

uf+1

on dQ 1

= 0 —

k+1/2

'

dull k = ^-+pkuK2

=uk1+ak+1(uk1+1/2-ui)

on T,

infii,

n

dfl

r §1-3

(1.3.11)

ITERATIVE SUBSTRUCTURING METHODS -Au2+1/2 = /

in n 2

u2+1/2 = 0

on 9 0 2 n dtt

d4+1/2 ~Qk—^—+U2

(1.3.12)

uk+1

k+1/2

= -qk

du\+1 ^

k+1

+ uf

17

on r ,

in02.

=uk+(ik+l(uk2+1/2-uk)

In the above procedure, pk > 0, qk > 0, a f c + 1 , and f3k+1 are free parameters. In fact, this algorithm encompasses and generalises many other methods: the Dirichlet-Neumann method (1.3.1)-(1.3.3) is a particular case of the Agoshkov-Lebedev one when pk = qk — 0 and ak+\ = 1 (note, however, that in the Agoshkov-Lebedev method the roles of fii and are reversed, and that the relaxation is carried out on the Neumann boundary datum). Similarly, the Robin method (1.3.7), (1.3.8) can be obtained from (1.3.9)-(1.3.12) by taking ak=Pk = 1 and 71 = pk, 72 = l/qk {qk > 0). Remark 1.3.1 (Subdomain iterations and parallelism) With the exception of the Neumann-Neumann method, the different iterative procedures introduced thus far share the feature of generating at each step two boundary value problems, the former set in fi1; the latter in $l2, to be solved sequentially. A simple modification of this procedure making it more interesting in view of parallel implementation is in order. As a matter of fact, when solving the boundary value problem in 0,2 at the new step k + 1 it is enough to use as data on F those generated by u\ (rather than w f + 1 ) . For instance, following this approach, the Dirichlet-Neumann method (1.3.1)(1.3.3) should be modified by simply replacing in (1.3.2) the Neumann condition on T by the new one duk+1

_ du\

dn

dn

on

Similarly, for the Robin method the interface condition in (1.3.8) should become du2+1

k+1

=

duk

k

onF.

A similar modification arises in problem (1.3.11) of the Agoshkov-Lebedev algorithm. However, this 'parallel' procedure in the case of two subdomains generates sequence that has as a subsequence the 'sequential' procedure. For instance, if for i — it2 and k > 1 we denote by { u f s } the sequence generated by the sequential' Dirichlet-Neumann method and by { u \ p } the sequence generated by the 'parallel' Dirichlet-Neumann method, we have that u k s = u\k~x and U2,s = ulkp, k> 1. a

18

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

On the other hand, the issue of parallelism is relevant in the case of partitions of ft using many (more than two) subdomains, and will be addressed in a more general framework in Section 1.4.2. Q

1.4

Generalisations

The whole set of considerations that we have developed so far for the Poisson equation can be straightforwardly extended to the homogeneous Dirichlet boundary value problem

{

Lu = /

in ft

u = 0

on <9ft,

associated with the more general symmetric elliptic operator d Lw := — ^ Di(aijDjw)

(1.4.2)

+ a0w,

1,3 = 1

where the coefficients a 0 and aij are assumed to belong to The operator L is elliptic if the coefficients aij satisfy d Jl

> a0|f|2

L°°(Q).

V £ £ R d , for almost all x £ ft

i,j=i for a suitable constant «o > 0. The operator L is symmetric if its coefficients satisfy aij(x) = ajt(x) V l,j -l,...,d, for almost all x £ ft. The associated bilinear form is (1.4.3)

a*(w,v):=

f

( Y] \i,j=i

aijDjwDiv

do wv

We assume that ao(x) > 0 for almost all x £ ft, hence a*(-,-) turns out to be symmetric, continuous, and coercive in Hq(Q), because from the Poincare inequality (1-2.2) it follows that a*(v,v)

>ao|!V«||2!n > a 0 ( l + C y ^ l M l l a

V v £ if^ft).

In particular, the form a*(-, •) induces a scalar product in Hq (ft). The week formulation of (1.4.1) reads: (1.4.4)

find u £ i ^ f t )

: a*(u,v) = (f,v)

VuGJ^fi),

and, as a consequence of the Lax-Milgram lemma, there exists a unique solution'

19

ULILN EK.AL1S A'L'LUJN S

Denoting by Ui the restriction of u to fij, i = 1,2, u being the solution to (1.4.1), the interface conditions satisfied on T = dfii fl <9f22 are

0 blem

pr«

on r

u i = u2 (1.4.5)

dui

du2

driL

driL

where the conormal derivative

on r,

is defined as

dw

(1.4.6)

jwni

driL

and, as usual, we have set n = n 1 on T. Note that the conormal derivative coincides with the normal derivative ^ when L = — A . Having defined for i = 1, 2 the bilinear forms (1.4.7)

a*(wi,Vi)

/ Jn<

aijDjWj DjVj + a0Wj Vj ), \i,j=i /

the multi-domain weak formulation of (1.4.1) can be written in the following way: find ui £ V\, u2 £ V2 such that al(ui,v x ) = (f,vi)n l Uy = u2

on r

a2{u2,v2)

= {f,v2)n2

VuieVi0

(1.4.8) Vv2e

V2°

a2(u2,TZ2fJ.) = (f,TZ2fi)n2 + (f,Tl x^n,

- a^u^fti/i)

V n £ A,

where 72.,; denotes any extension operator from A to V*, and the notations are the same as in Section 1.2. As we have already observed, the last equation is the weak form of the interface condition (1.4.5) 2 . The more general mixed non-homogeneous boundary value problem in Q,

' Lu = f (1-4.9)

on r ^

U = ifiD du . 0nT =

^

on f V

be considered as well. Here, f , i f D and ipnf are given functions, r ^ u r j v = <9fI, nr^r = 0, and n* is the unit outward normal vector on dQ.

20

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

In this case, the variational formulation reads: (1.4.10)

find W e i J p D ( f i ) : a*(W,v)

= {f,v)

+

{tpN,v\rN)rNN

-a*(ipD,v)

VvtHlJ

H),

where (1.4.11)

fl^(fi)

(1.4.12)

:= {u £ H 1 ^ ) | v\rD - 0}

{4>,tp)rN ••= / 0,

and (po G H 1 (fi) denotes any extension in f2 of the non-homogeneous Dirichlet datum (po- The solution u is obtained by adding W and (poAgain, we can go through the formalism needed in the two-domain formulation, with the simple warning that, since the right-hand side (f,v) of the onedomain weak formulation (1-4.4) has been substituted in (1.4.10) by (1.4.13)

l / » + {
-

a*{
similarly the right-hand side (f,Vi)n i in (1-4.7) must be substituted by the restriction of (1.4.13) to the domain fi;, i = 1,2. However, it is worthwhile to note that, in the case a 0 (x) = 0, to avoid the compatibility conditions on the data that have to be satisfied when solving a pure Neumann problem, the subdomain decomposition has to be chosen in such a way that ]?£> fl <9fi2 0Finally, the pure Neumann case (that is, when F/} = 0) deserves a more careful analysis. No change occurs if the coefficient ao satisfies a0(x)

> fi0 > 0

for almost all x G O

because the bilinear form a*(-, •) would be coercive in iJ 1 (f2) in this case. In the case where a0 = 0, it is well known that the boundary value problem

(1.4.14)

is solvable if and only if the compatibility

condition

(1.4.15) is satisfied. In this case, sl possible variational formulation reads:

GENERALISATIONS

§1.4 (1.4.16)

findueff^fl)

: a*(u, v) = (/, v) + (yN,v{im)dn

where ^ ( f i ) : = (1.4.17)

21 V u e ^(fi),

n Lg(fi) and L

2

(0):=|

W €

L

2

(N)|

^

=

OJ.

The compatibility condition has to be satisfied also in every subdomain where a pure Neumann problem has to be solved. For instance, in the present situation the Dirichlet-Neumann scheme reads f Luk+1

=f

in fix

= ipN

on <9fii n 80,

i-i

duk+1

(1.4.18)

dn*L

on r,

xk+1 = xk then Lu^=f

in f?2

duk2+1 (1.4.19)

dn*. duk+1 dnL

on dQ 2 n dQ

=

-

8uk+1 1

dni

on r ,

with Xk+1 — Ou^1 +

(1.4.20)

(1-8)\k.

Therefore, the compatibility condition has to be assumed for the pure Neumann Problem (1.4.19), and is given by (1.4.21)

/f) z

Jdn2nan

Jr

°nL

(Recall that we have chosen n = n 1 on F.) On the other hand, we can write duk+1 dn

=

f duk+1

Jr dnL f~ fi,

r

duk+1

Jdni nan dn* JdQindn

/



therefore the local compatibility condition (1.4.21) is a consequence of the global c °ndition (1.4.15).

22

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh.

1

We can verify that the same result holds for the Neumann-Neumann iterative scheme as well. 1.4.1

The Steklov-Poincare equation for the Neumann boundary value problem

The construction of the Steklov-Poincare operator associated with problems (1.4.9) and (1.4.14) mimics that for problem (1.1), which has been presented in Sections 1.1 and 1.2. In the case of the mixed boundary value problem (1.4.9), the Steklov-Poincare operator is symmetric, continuous, and coercive in the trace space A. The same happens for the Neumann problem (1.4.14), provided that a 0 (x) > > 0. Instead, if ao = 0 the Steklov-Poincare operator turns out to be singular. In fact, let us introduce for i — 1,2 and rj G H1/2(r) the function E*r], solution to 'EtrieH^Qi) < a*(E*r],vi) . E*rj = r}

: = 0

V V i G H^Sli)

on F.

The Steklov-Poincare operator associated with the Neumann boundary value problem (1.4.14) is defined as S = Si + S2, with (Si^ri-aXEfrEtri

Vrj^eff1/2^),

i = 1,2.

Clearly, E* 1 = 1, consequently Si 1 = 0 and (5,;r?, 1) = 0 for each tj € ff1/2(r). The operator Si is therefore symmetric and continuous, but not coercive in H1'2^). To derive the associated Steklov-Poincare equation, let us start by noting that the solutions to (1.4.14) are defined up to an additive constant. However, since 51 = 0, the Steklov-Poincare equation for the trace A = it|p of any such solution reads as usual (1.4.22)

findA e H^2{T)

where

: <SA,M) = (x,ju>

V /i G

Hl'2(T),

2

(X.M) ^^[{LEi^)^

+{VN,E*ii)dnindQ

-a*(u*,E*v)}

V / i G f f 1 / 2 ( r),

and u* is the solution of u* G I?R(FTI) : a*(u*,Vi) = ( f , V i ) Q i + (
V

G #R(FT;)-

Since the operator 5 is singular in i J 1 / / 2 ( r ) , it is convenient to consider a different space of functions on T, defined as

§1.4

GENERALISATIONS

(1.4.23)

^ ( n ^ e i f ^ r ) !

j %

23

=

0

J,

endowed with the norm of i ? 1 / 2 ( r ) . In this space, the Steklov-Poincare operators 5, are coercive. In fact, taking rj 6 JT 1/ ' 2 (r) we have {Sir,,,7) = a* (E*j], E*tj) > a0\\V

E*^2^

> a 0 ( l + C f 2 , ; )- 1 ||^|| 2 n i , because the Poincare inequality (1.2.2) still holds in H^Sli) := j u i eH1^)]

J^vtlT = 0 j .

Finally, from the trace inequality (1.2.3) we find that (SlV,v)>Ml

+

Cn,y1(Cnir2\\v\\21/2,r-

Hence, we can find the unique solution A of the reduced Steklov-Poincare equation (1.4.24)

findA

e H^2(r)

: {SX,fi) = (x, £)

V/i€

H1/2(T).

The solution A indeed satisfies (1.4.25)

V^eif1/2^).

{S\,fM) = {X,n)

In fact, denoting by /ir :=

me^s r

fr fi, it holds that

(SA,/u) = (SX,fi - Mr) + Vr(SA, 1) = (X, H - Mr) = (x, V) - Mr (X, 1) = (X,M), because (x, 1) =

£[(/, i=1 = (/, !)n + {
due

to the compatibility condition (1.4.15).

The solution A does not coincide with the trace «|r of the solution to (1.4.16), a n d the difference is a constant, because A is in fact equal to the trace on P of t h e solution to ^•4.26)

find U G F 1 ^ )

: a*(u,v)

= (f,v)

+

|an)an

VveH1^),

24

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

where -

H 1.4.2

jfieff1^)!

J t)|r = oJ

Iterations on many subdomains

In view of a parallel implementation, is partitioned into many subdomains 1.4.1). In this case, denoting by Ty two adjoining subdomains Qj and tlj,

a desirable situation is that in which fi fij, i = 1 ,...,M (see, for example, Fig. dO-i (Idflj the common interface between the split form of

Lu = f

in n,

for L introduced in (1.4.2), is given by Lui = f (1.4.27)

{

in flj, V i =

l,...,M

= ${uj)

on Fy , V j such that F^ ± 0

= V(uj)

on r i j ,

V j such that Fy ^ 0,

where (1.4.28)

{v)=v,

V(v)

dv dnL

As already suggested in Remark 1.1.1, we prefer to keep this general notation, since it is convenient to treat also a larger family of boundary value problems that will be considered in Chapter 5. The extension of substructuring iterative techniques to the case of many subdomains will be addressed in Section 3.3.2. However, the reader may find it interesting to have a preliminary idea of how the previous Dirichlet-Neumann method can be adapted to the multi-domain case. This iteration-by-subdomain procedure generalises accordingly as follows: after using a black and white colouring, set IB := {1 < i < M | fi; is black} and Iw '•= I \ IB, a n d solve

(1.4.29)

f Luk+1 =f ^ {$(<4+1) =

in + (1 - 0 ) $ ( u * )

V i e JB

on r ij,

V j e i w : v l J ^ 0.

This yields a family of $-type subproblems that, when $ is given as in (1.4.28), are independent of one another and can be solved simultaneously, allowing an effective treatment within a multi-processor environment. Then solve Lu)+1=f

in Clj,

VjGAy

on Tij,

Vi G

(1.4.30) 1

= tf ( u * + 1 )

: Tij ^

§1.4

GENERALISATIONS

Q

Q

25

3

2

Q7 Q

6

FIG. 1.4.1. Black and white subdomain decomposition of the domain Q. On each white subdomain, we have, therefore, a ^-type subproblem; when is given as in (1.4.28), all these subproblems are coupled (although mildly) at the cross-points. We point out that the algorithm (1.4.29), (1.4.30) is block-sequential: each problem of the second block (1.4.30) (on the white subdomains) can be faced only after having solved those from the first block (1.4.29) (on the black subdomains). A block-parallel variant is easily achieved by replacing (1.4.30) by

(1.4.31)

However, as already pointed out in Remark 1.3.1 for the two-domain case, the block-parallel sequence generated by (1.4.29) and (1.4.31) contains as a subsequence the block-sequential sequence generated by (1.4.29) and (1.4.30). A third, fully parallel algorithm would consist of solving both a 3>-type problem and a 'i'-type problem in all subdomains (investing at each step twice as much computational work as in the previous cases). In this case, however, since at the end of each step the values of $ and/or "J are not necessarily single-valued at subdomain interfaces, we need to adopt an averaging procedure at interfaces among the values of $ from both sides (and similarly for The new algorithm can be defined as follows. Given the values {uf \ i = 1> • • •, M } , define for each i = 1 , . . . , M (1.4.32)

:= a $ ( u f ) + (1 - a ) $ ( u £ )

on T;

where a is an averaging parameter, and Ty is the part of dfti separating Ui from Obviously, = ) on d^ n <9f2. Then solve the M subproblems of the $-type:

(1-4.33)

26

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

Ql

r2

^1,2

FIG. 1.5.1. Two examples of overlapping partitions, for i = 1 , . . . , M . Now compute for each i the value (1.4.34)

and average them as follows

: = /3tf(u* + 1 / 2 ) + (1 - / 3 ) ¥ ( u J + 1 / 2 )

on

where /3 is another averaging parameter. Again, we set v?f v = $ ( u * + 1 / 2 ) on

dtii n on.

Finally, solve the M subproblems of the \£-type: ( L u : = / { [ =

(1.4.35)

in Hi ondfti

for i = 1 , . . . , M.

1.5

The Schwarz method for overlapping subdomains

The Schwarz method is undoubtedly the earliest example of a domain decomposition approach for partial differential equations. It was introduced by Schwarz (1869); however, among others, let us also mention the early contributions of Sobolev (1936); Mikhlin (1951); and Matsokin and Nepomnyaschikh (1985). This time we decompose fi into two overlapping subdomains fii and 0 2 such that fi = and denote by F x : = <9fiinn 2 , T 2 := <9fi 2 nni, := fiinfl2 (see Fig. 1.5.1). 1.5.1

The multiplicative and additive forms of the Schwarz method

The original form of the Schwarz iterative procedure, known as the alternating Schwarz method, consists of solving successively the following problems. Let u° be an initialisation function defined in CI and vanishing on <90, and set u 2 := For k > 0 we define two sequences and u 2 + 1 by solving respectively:

§1.5

THE SCHWARZ METHOD FOR OVERLAPPING SUBDOMAINS -A^

(1.5.1)

= /

1

=

u\

27

in f i j on F 1

rfc+i _

on dfti n dn

and

(1.5.2)

—A«2+1 = /

in 0 2

=

on r 2

<

on dn2 n

r^+i =_ 0

on.

On the other hand, having set := wj30i and U2 : = wj^.,, we could make the two steps independent of each other by solving -

A

(1.5.3)

Uk+l

=

f

in fix

=

on Ti

= 0

on <9fii n an

and -AUk+1 (1.5.4)

= f

in n2

k+l _ U. = Uf

on r 2

l/2fc+1 = 0

on dn2

n

an.

The two approaches (1.5.1), (1.5.2) and (1.5.3), (1.5.4) form the basis of modern formulations of the Schwarz method, named respectively multiplicative and additive, which will be addressed in the finite dimensional case in Section 3.4. The alternating Schwarz method (1.5.1), (1.5.2) converges to the solution of (1.1), provided some mild assumptions on the subdomains Hi and n 2 are satisfied. Precisely, there exist C x , C2 e (0,1) such that for all k > 0 11

(1.5.5)

(1.5.6)

|"|n2 - « 2 + 1 I U ~ ( n 2 ) <

Ck+1C*\\u-u0\\L~(r2)

The error reduction constants C\ and C2 can be quite close to one if the overlapping region is thin. In Fig. 1.5.2 we present an example showing how the

THE MATHEMATICAL FOUNDATION OF DD METHODS

28

<Jh. 1

rate of convergence depends on the dimension of the overlapping region (denoted by a shaded interval). The Schwarz method is applied to approximate the (null) solution of the problem -u"(x) = 0 in (0,1) u(0) = u( 1) = 0.

FIG. 1.5.2. Error behaviour for the Schwarz method. The proof of estimates (1.5.5), (1.5.6) can be obtained via the maximum principle (see, for example, Kantorovich and Krylov 1964, and P.-L. Lions 1989, where some convergence results for rather general geometrical configurations are also obtained). We will not adopt this point of view in our presentation, resorting instead to a variational interpretation due to P.-L. Lions (1988). For an introduction and analysis of the additive Schwarz method we refer to Matsokin and Nepomnyaschikh (1985) and Dryja and Widlund (1990). 1.5.2

Variational interpretation of the Schwarz m e t h o d

First of all, note that the alternating Schwarz method (1.5.1), (1.5.2) can obviously be generalised to the case of the symmetric elliptic operator (1.4.2). The variational formulation of the Schwarz method for the homogeneous Dirichlet boundary value problem associated with it can be stated as follows: set, as usual, V := Hq(£1), V° := H^tti), i = 1,2, and moreover (1.5.7)

V* := {v G V | v = 0 in fi \ ft;}-

Method (1.5.1), (1.5.2) reads: given u° G V, solve for each k > 0

§1.5

in£j D^nvvArLZj i v i t i n u u wf G V°

(1.5.8)

= (f,vx)Ql

: a*1(wk,v1)

uk +1/2 __

uvii±tL,Arjr"iiNU SUJjJJUMAiiNs

+1 _

V V l G Vi°

^

+

wk G K>° : O 5 K , T ; 2 ) = (f,v2) uk

- a j V , ^ )

29

ufc + l/2

wk

a2 - a*2{uk+1/2, v2)

V u2 G

^

where wk denotes the extension of wk by 0 in ft \ fij. Similarly, method (1.5.3), (1.5.4) is obtained by solving Wk (1.5.9)

G V°

Wk G

: a*1(Wk,v1)

= ( / , Vl)0l

:

= (f,v2)n2-a*2(Uk,v2)

j/fc+i = uk + Wf +

- oj(tf*, ^ )

V

Vl

V

G V^0 G

W£,

having set U° :—u°. The proof that these variational formulations are equivalent to the original ones (1.5.1), (1.5.2) and (1.5.3), (1.5.4) is obtained via the verification of the following relations: uk+1/2

=

in

f

(1.5.10) ,k+1

_ J «2+1

^

and

(1.5.11)

1.5.3

C / ^ 1 = I U k +\

+ U ^

2

- U

k

^

2

in fix,2

T h e Schwarz m e t h o d as a projection m e t h o d

Concerning the alternating Schwarz method (1.5.1), (1-5.2), using (1.5.8)i and (1-5.8)2 we see that for each v G V* a*(uk+1/2

_

— al(wk,v\ni)

_

= U,v\n1)n1 =

-

al(uk,v|nJ

(f,v)-a*(u",v)

— a*(u — uk, v). Similarly, from (1.5.8)3, (1.5.8)4 it follows that a*{uk+1

- uk+1/2,v)

= a*(u — uk+1/2,v)

Vv G

.

30

THE MATHEMATICAL FOUNDATION OF DD METHODS

Therefore, the sequences u f c + 1 / 2 and uk+l uk

(1 5 12^

satisfy

+ l/2_uk

uk+1

<Jh. 1

=V*iu_uk')

-uk+1'2

=rz{u-u>

where V*, i = 1, 2 is the orthogonal projection of V onto V* with respect to the scalar product induced by the bilinear form a*(-,-); that is, for any w 6 V it holds that V*w e V* : a*(V*w -w,v) = 0 V v £ V?. Let us denote by I the identity operator, by Ji, i = 1,2, the (non-dense) immersion of V* into V (that is, J^v — v for each v
(J?F,

v) = (F, J%v)

VFeV',ve

V?.

If we set (1.5.14)

Vi : = JiV* : V ->• V,

from (1.5.12) it follows at once that uk

(1.5.15)

+1/2

_

=

p

^uk

,

= uk +

+

V

Qf

VlG(f-Luk)

and

[

••

=uk+1/2

}

+ V2QU

~Luk+1'2),

where Q is the resolvent operator associated with problem (1.4.1); that is, Q = and, in particular, Qf = u. Therefore, the alternating Schwarz method (1.5.8) reads u k+1 =

(I.5.I7)

(J

_ p2)[(/ _

= ( i - r2)(i = uk + Qm(Qf

Vl)uk +

- v ^

V l

g f j + -p 2 g f

+ (I - W I Q f

- uk) = uk + Qmg(f

+ -

r2Qf Luk),

where (1.5.18)

Qm^n+Vi-VzVi.

In a similar way, the alternating Schwarz method (1.5.9) can be written as

THE SCHWARZ METHOD FOR OVERLAPPING SUBDOMAINS

§1.5

wk = v{{u-uk),

wk =

31

v;{u-uk),

therefore jjk+1 (1.5.19)

_

=

Vi

_

+

(Vi

p^gf

+

= uk + Qa(Qf - Uk) = Uk + Qag(f

-

LUk),

where (1.5.20)

Qa--r1+r2.

Concerning the error equations for the Schwarz method (1.5.1), (1.5.2), from (1.5.15), (1.5.16) we have u_uk+1/2 {1'5'21)

=

(I_-p1)(u_uk)

u-uk+1

=(I-V2)(u-uk+1^2).

Introducing the error ek := u — uk, the previous relations yield the error recursion formula: (1.5.22)

ek+1

V k > 0.

= {I-V2)(I-Vi)ek

These equations are the basis of the proof of the convergence of uk and uk+1/2 to u in i? 1 (f2) (see Section 4.6). Similarly, setting Ek := u — Uk and using (1.5.19), for the Schwarz method (1.5.3), (1.5.4) it holds that (1.5.23) 1.5.4

Ek+1

= (I - Vi - V2)Ek

V k > 0.

The Schwarz method as a Richardson method

On the basis of (1.5.17), the alternating Schwarz method (1.5.1), (1.5.2) can also be regarded as a Richardson iterative procedure for the solution of the new problem (1.5.24)

Qmu = g-.=

QmQf.

The multiplicative term V2V\ is responsible for Qm being a second-degree polynomial operator, and prevents the parallelisation of method (1.5.1), (1.5.2). As a matter of fact (1.5.1), (1.5.2) is a sequential algorithm. The presence of the term ^Ti justifies the adjective multiplicative, which is attributed to the alternating Schwarz method. On the other hand, the algorithm (1.5.3), (1.5.4) is called the additive Schwarz method since it can be regarded as a Richardson iterative procedure for solving a nother problem that reads

32

THE MATHEMATICAL FOUNDATION OF DD METHODS QaU = g* :=

(1.5.25)

It is easy to see that g* = g{ + a*{g%v,)

QaQf.

where g{ £ V]*,

= {f,Vi)

<Jh. 1

£ V2* and

VviGV*,

i = 1,2,

hence g* can be computed by two local solves. The projection operators T% = J{P* are symmetric with respect to the scalar product induced by the form a*{-, •), because a*(ViV,w)

=

a*(V*v,w)

= a*(V*v,T*w)

(1.5.26)

=a*(v,V*w)

= a*(v, Viw)

V

v,u> £ V.

Consequently, the operator Q a is symmetric and positive definite with respect to the bilinear form a*{-, •). This property suggests basing on Q~ 1 the construction, at the finite dimensional level, of a preconditioner of the original problem, see Section 3.5. 1.5.5

A characterisation of the projection operators

In view of the discretisation of the alternating Schwarz method that will be discussed in Section 3.4, it is useful to rewrite the projection operators and, consequently, Qm in a different form. This involves the original differential operator L, and its restrictions to the subspaces V L e t us introduce the operators Li : —» (V®)' associated with the bilinear forms a*(-, •): (1-5.27)

{ L i w l , v i ) : = a*(Wi,Vi)

V

wuVi £

as well as the extension operators pj :

—>• V*

(1.5.28)

V F ^ V ,

PJVI:=VI

r0 V°, i'

0

and the transpose restriction operators pi : (V*)' —» (Vf)1 (1.5.29)

(PiG,vt)

:= (G,pjvi)

V

G £ (V*y,Vl

£

It is easily seen that (1.5.30)

U =

piJ^LJipf.

We claim that the following identities hold: (1.5.31)

V*=pjL~lPlJ?L,

i =

l,2.

In fact, define and j*, the identification operators between V° and its dual (Vf)' and V* and its dual {V*)', respectively. In particular, we have that

§1.5

THE SCHWARZ METHOD FOR OVERLAPPING SUBDOMAINS

(1.5.32) and

=

pij*p[,

u*)-1

=

33

pJ(j°r1pt,

that, from (1.5.30),

(1.5.33)

J? LJi =

jtplirfr'L.iXj^po;.

Then, using (1.5.32) and (1.5.33) for each v £ V*,w G V it holds that a'ipjL^piJTLw^)

=

a'iJipjL^piJ^Lw^iv)

= (LJipfL-'piJ^Liu, =

Jtv)

{J^LJipfL-'p^Lw^)

= (JTLw, v) = (.Lw, Jiv) = a*(w, Jiv) — a*(w, v), and the proof of (1.5.31) is complete. As a consequence, we also obtain (1.5.34) 1.5.6

Vi = JipTLT1

PiJ?L,

V%g = ;7:p! L,

.

The Schwarz method for many subdomains

The generalisation of the Schwarz method to the case of when Q is partitioned into M > 2 subdomains (see Fig. 1.5.3 for an example) is straightforward.

FIG. 1.5.3. Overlapping decomposition into 16 subdomains. Precisely, the multiplicative Schwarz method is given by (1.5.35)

uk+^ = ( I - V i ) u k + ^ + J i p T L - l P i J ? f ,

i = l,...,M

(here we have used (1.5.34)), and the additive Schwarz method reads

THE MATHEMATICAL FOUNDATION OF DD METHODS

34

<Jh. 1

FIG. 1.6.1. Domain embedding. (

M

1=1

\

M

/

4=1

The corresponding error equations read (1.5.37)

u - uk+1

= {I — Pm) •••(/ — Vi){u -

uk)

for the multiplicative case, and

(1.5.38)

u-Uk+1

=

{u-Uk)

for the additive case.

1.6

The fictitious domain method

This is one of the earliest ideas closely related to domain decomposition. The main motivation is that whenever a problem needs to be solved on a domain 0 having a complex boundary, it may be useful to embed it into a larger domain f2 of a simpler shape; say, for instance, a rectangle as in Fig. 1.6.1, and then solve a problem of similar type in 0 (see, for example, Buzbee et al. 1971; Matsokin 1972; Proskurowski and Widlund 1976; Kuznetsov 1989; and Glowinski et al 1994). Suppose, for instance, that the given problem in Cl is the following elliptic problem with a non-homogeneous Dirichlet boundary condition { L i t = /

in 0

u = (pr> on dCt. Here, we assume that / G L2(Cl), (pD G H1^2(dCl). The operator L is given by

THE FICTITIOUS DOMAIN METHOD

§1.6

35

d Lw := -

(1.6-2)

^

Di(aijDjw)

+

a0ii,

1,3 = 1 where the coefficients a^, a 0 belong to L°°(ft) and satisfy the ellipticity condition d ^

a i j ( x >

a0|£|2

V £ G R d , for almost all x £ A

1,3 = 1 for a suitable constant ao > 0. Moreover, we assume that aij(x)

= aji(x)

V l,j =

l,...,d

for almost all x £ ft, and that ao(x) > 0 for almost all x £ ft. As a consequence, the associated bilinear form

i * ( w , v ) •= JCl

\ \i,j=i

ai

i D •wD[V + a0 w v

turns out to be symmetric, continuous, and coercive in iJg(ft). Therefore, there exists a unique solution u of the corresponding weak formulation find u £ i f 1 (ft) : (1.6.3)

Vi) Gilo 1 (ft)

a* (u, v) = f fv Jn MiSq = ipjy

on <9ft.

Let / , a^, ao be suitable extensions in ft of / , aij and a 0 , respectively, and consider the extended bilinear form a*(w,v)

/ j Q

y

y

aijDjwDiv

+ a0wv

,

\ 1,3 = 1

which is now defined in i / 1 ( f t ) . Then consider the extended problem in ft: find u £ i ^ ( f t ) : (1.6.4)

a*(u,v)

> u\dn =

= [ fv Jn

WvG on <9ft,

where we have introduced the Hilbert space

^(ft)

36

T H E M A T H E M A T I U A L i f'UUiNUATlUiN Ui< L)D i V i l i i n u J J S

Hl(fl)

(1.6.5)

1^11. i

:={veHM\v[gCl=0}.

If the extensions / , a^, a0 satisfy in Q the same assumptions that / , dij, a0 satisfy in 0 , then the bilinear form a*(-, •) is symmetric, continuous, and coercive in Hq(Q,). Therefore, problem (1.6.4) has a unique solution it, whose restriction coincides in fi with the solution u of the original problem (1.6.3). Another point of view when addressing the fictitious domain method is based on the use of Lagrange multipliers. For instance, again referring to the Dirichlet problem (1.6.3), we can instead consider the following problem in the larger domain Cl: ' find u G H£{Sl), f G H~^2(dn) '

(1.6.6)

=

TV

on

fv

: Vve

Jn

Hi (ft)

V A £ #~ 1/2 (<9fi),

/ ufi.= Jdii JdQ

where H^1^2(dtl) is the dual space of the trace space H1^2(dfl). Clearly, the solution u to (1.6.6) satisfies it^ = u, and f is a Lagrange multiplier associated with the constraint =
du\a

dn% '

where n* is the unit normal vector on dCl directed towards 0 \ 0,. For this approach we refer, for example, to Glowinski et al. (1994). The fictitious domain method can also be applied to the Neumann problem: Lu = f (1.6.7)

du

- = tpN

dnt

in Cl

on an,

where du dn*£

u n, i,i=i

is the conormal derivative of u. The data aij, ao and / satisfy the same assumptions as before, with the extra requirement that do(x) > yUo > 0 for almost all x 6 H, to guarantee the existence and uniqueness of the solution. Furthermore, we suppose that tp^ G L2(dQ). It is convenient to formulate problem (1.6.7) in a different form. Setting Pi := £ . aijDjU, we have

THE FICTITIOUS DOMAIN METHOD

§1-6

divp + aou = f

37

in fi

and aljpj - D[U = 0

in fi, V 1 = 1,...,

d,

j=l where are the entries of the inverse matrix of {<% }• Eliminating the unknown u we finally find d (1.6.8)

J2a'3Pi

- ^ ( a o ' d i v p ) = Diia^f)

in 0 ,

V/ = l,...,d.

3= 1

Let us introduce the spaces tf(div;fi)

:= { q € (L2(Cl))d

| divq G L 2 ( 0 ) }

#o(div; Cl) := { q G # ( d i v ; ft) | (q • n*) j a f t = 0}, and the bilinear form B{w,q)

P d -=152 (aljWjqi + aq1 d i v w divq). Jfli,i=i

The weak formulation of (1.6.8) reads ' find p e H(div; Cl) : (1.6.9)

q) = -

/ « o ~ 7 divq Jn

V q G ff0(div; Cl)

on dCl.

. p • n* = ifN Introducing the space

H(div; fi) := { q G H{div; SI) | (q •

= 0},

the fictitious domain formulation of (1.6.9) reads find p G f f (div; fl) : (1.6.10)

B{p, q) = -

f flo V d i v q Jn

„ p • n* = tppf

V q G H(div; Cl)

on dCl,

where B is defined through the extensions of the coefficients aij and a0 to Cl.

38

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

All problems (1.6.4), (1.6.6) and (1.6.10) can be solved iteratively. For instance, the Lagrange multiplier f in (1.6.6) can be shown to solve a variational problem for (the inverse of) a suitable Steklov-Poincare operator, symmetric and coercive in H~1^2(dCl). Therefore, it can be determined by means of preconditioned conjugate gradient iterations, which lead to the solution of a sequence of boundary value problems in the simple-shaped domain fi (see Glowinski et al. 1994). We will not discuss the fictitious domain approach further in what follows. Nonetheless, it is worthwhile to note that this method has received considerable attention in the past few years. For the early developments we refer to Marchuk et al. (1986). More recently, the analysis is carried out in several papers; in particular, in Borgers and Widlund (1990); Glowinski and Pan (1992); Glowinski et al. (1994); and Girault and Glowinski (1995). See also Girault et al. (1997) for a coupling of fictitious domain and domain decomposition techniques. Furthermore, see Kuznetsov (1997) for a discussion on the efficient preconditioned solution of systems arising from the Lagrange multiplier approach (1.6.6), and Rieder (1997) for a multi-scale Galerkin approximation to the variational approach (1.6.4) (or (1.6.10)).

1.7

The three-field method

The formulation that we present here is inspired by the so-called hybrid finite element formulation for elasticity problems (see, for example, Tong 1970), and has been introduced in the domain decomposition context by Brezzi and Marini (1994). The original problem is reformulated by relaxing the continuity requirements on both u and -j^- (see (1.4.5)), at the expense of introducing two Lagrange multipliers for each subdomain. The new weak formulation allows independent approximations within the subdomains, including the possibility of using different methods and different meshes from one subdomain to another. Still using the notations of Sections 1.2 and 1.4 we introduce the following formulation that generalises (1.4.8): for i = l , 2 find Ui G Vi, Oi G A', A G A such that

aj(ui,z;i) -

{o-1,wi|r)r = ( / ^ i ) ^

(pi,A-ui|r)r = 0 (1.7.1)

((7i + (72, \S)Y = 0 (p2, A - u 2 |r)r = 0 a2(u2,v2)

V v i G Vi

V P l G A' V/i e A V p2 G A'

- ( c 2 , f 2 | r ) r = {f,v2)n2

V v2 G V2.

THE THREE-FIELD METHOD

§1.7

39

Here, we have denoted by (p,p)r the duality pairing between A' and A; when p £ L 2 ( r ) , then (p,/j,)r = fT pfi. In the formulation above, pi is the Lagrange multiplier that is used to 'glue' the value of U; and A on T, i = 1, 2. As a matter of fact, the following equivalence result holds between the weak form of the unsplit problem (1.4.1) and the problem with Lagrange multipliers. Proposition 1.7.1 For every f £ L2(Q), problem (1.7.1) admits a unique solution (it,;, a;, A), which is related to the solution of the problem (1.7.2)

u £ V

through the following

: a*(u,v)

= (f,v)

\/v £ V

relations: Ui = u |q.,

(1.7.3)

« = ( l | )

i = 1,2 | r

-

• =

1

'2

A = U|p, where (^r)

denotes

the conormal

derivative

of u with respect to the unit

normal vector nz (issuing from Ui) on T. Proof It is straightforward to prove that, if u is the solution to (1.7.2), then (Ui,Oi, A) is a solution to (1.7.1), Uniqueness follows by inspecting the homogeneous problem. Therefore, let us take f = 0. Prom (1.7.1)2 and (1.7.1)4 we have that A = u ^ r = u2\r> thus the function u\ in Oi w := > . u2 m U2 belongs to Hq(Q) and satisfies = A. Taking p, = A in ( 1 . 7 . 1 ) 3 and summing up (1.7.1)i (with vi = ui) and (1.7.1)5 (with v2 = u2) we obtain that ai(ui,ui)

+ a2(u2,u2)

= a*(w,w)

= 0,

whence w — 0 (by the coerciveness of the bilinear form a*(•,•)), and therefore Ui = 0 and A = 0. From (1.7.l)i we have now (o"i,Wi|r)r = 0 for each £ V±, which implies 01 = 0 on F. The last equality, a 2 = 0 on F, follows similarly.

• There is a clear equivalence between the three-field method and the classical way of relaxing Dirichlet boundary conditions through Lagrange multipliers. Indeed, let us consider, for instance, equations (1.7.1)i, (1.7.1)2 for known values °f / and A. The problem reads: find ui £ Vi, <j\ £ A' such that al(u\,vi) (1.7.4)

- (
;


V pi £ A',

V vi £ Vi

40

T H E M A T H E M A T I C A L EUUJNDATJ.UJN U E U U

METHODS

<Jh. 1

and we see at once that this is the variational formulation of the boundary value problem ( Lui = f < ui = 0 „ ui = A

in fix on d f h \ T on T,

where the last equation has been imposed via the Lagrange multiplier Ai, which turns out to coincide with the conormal derivative of u\ on F (see Babuska 1973).

50

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION METHODS In this chapter the general principles previously introduced will be accommodated to treat the finite dimensional approximation of differential problems. We confine our analysis to the Poisson problem. However, its extension to the case of the more general symmetric elliptic boundary value problem (1.4.1) is straightforward and will be left to the interested reader. On the other hand, other kind of equations, either stationary or time-dependent, will be addressed from Chapter 5 onwards. After presenting the Galerkin finite element approximation, we introduce the discrete Steklov-Poincare operator and show that its algebraic counterpart is the Schur complement of the finite element stiffness matrix. Our analysis will be restricted to the so-called conforming finite element spaces (see, for example, Ciarlet 1978). In the rest of the book, an exception will be made for the boundary value problems addressed in Section 5.4, where the approximation method can also be regarded as a mixed finite element method for the Laplace operator. We also address the case of domain decompositions that are geometrically non-conforming, that is, where the subdomain grids do not match at the interfaces, and consider both the mortar method and the three-field method.

2.1

Finite element approximation of elliptic equations

In this section we give a brief presentation of the finite element approximation theory. For more details, we refer the interested reader, for example, to Ciarlet (1978) and Quarteroni and Valli (1994). To start with, assume that the set fi C R d , d = 2,3, is a polygonal domain, that is, fi is a bounded open connected subset such that ft is the union of a finite number of polygons (for d = 2) or polyhedra (for d — 3). The finite element approximation is based on a finite decomposition [J Ken

K,

where o

• each K is a polygon or a polyhedron with a non-empty internal part K • K\ H K2 = 0 for each distinct K X , K 2

iioj^jj rj^juAiiuiNS AINU JJUMA1N DECOMPOSITION

Ch. 2

• if F = Ki fl K2 0 (K\ and K2 being distinct elements of Th) then F is a common face, side, or vertex of K\ and K2 • diam(A') < h for each K e %• Th is called a triangulation of fi (see Fig. 2.1.1). In what follows we assume further that each element K of Th can be obtained as K = T K { K ) , where K is a reference polygon or polyhedron and TK is a suitable invertible affine map, i.e. TW(x) = 5 / c x + hjc, Bk being a non-singular matrix. Moreover, we will confine ourselves to considering two different cases: • The reference element K is the unit d-simplex; that is, the triangle with vertices (0,0), (1,0), (0,1) (when d = 2), or the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1) (when d = 3). As a consequence, each K = TK{K) is a triangle or a tetrahedron. • The reference element K is the unit d-cube [0, l] d . As a consequence, each K = TK{K) is a parallelogram (when D = 2) or a parallelepiped (when d = 3). Let VH denote a finite dimensional subspace of Hq(Q). element approximation to (1.2.1) is defined as follows: (2.1.1)

find uh G Vh : a(uh,vh)

= (f,vh)

A Galerkin finite

V vh 6 Vh.

The most frequent example is when Vh is given by piecewise polynomials. They can be introduced as follows. When the reference element K is the unit d-simplex, let us define (2.1.2)

XRH : = {vh G C°(Q)

|v

m

G P , ^ ) V K G Th},

r>

1,

where P R ( K ) denotes the set of polynomials defined in K and of degree less than or equal to r globally with respect to all space coordinates. Then we set (2.1.3)

VH : = {vHE

XRH | vhldn

= 0}

§2.1

FINITE ELEMENT APPROXIMATION OF ELLIPTIC EQUATIONS

43

When the reference element K is the unit d-cube, the space Vh is defined in the same way, but in this case the space XJ^ is given by (2.1.4)

XRH : = {vh G C ° ( f t ) | vh]Ko

TK G Q R(K)

V i f G TH},

where Qr(K) denotes the set of polynomials defined in K and of degree less than or equal to r with respect to each variable Xi,... , Xd. The family of triangulations Th is said to be regular if there exists a constant a > 1 such that

—
V K ETH,

VH>0,

where hpc denotes the diameter of K and pk the maximum diameter of a ball contained in K. Under this assumption, denoting by 7Th,v £ Vh the interpolant of a continuous function v at the nodes of Th, the following interpolation error estimate holds: (2.1.5)

||u - 7Thu||o,n + h\\u -

n <

Chr+1\u\r+1,n.

It is well known from the Lax-Milgram lemma that problem (2.1.1) has a unique solution under the assumption that the bilinear form a(-, •) is continuous and coercive in Hq(£1). Under the further assumption that a(-, •) is symmetric, (2.1.1) is a minimisation problem, because its solution Uh minimises the discrete energy J{vh)

••= ^a(vh,vh)

-

(f,vh)

over the finite dimensional space VhBesides, from the Cea lemma it follows that

||w -

i,n < -

inf

a vhevh

ti-m

i,n,

where 7 and a are the continuity and coerciveness constants of a ( v ) > respectively. B y the interpolation error estimate (2.1.5), we finally find that (2.1.6)

||u - uh||i,fi <

provided that u G 2.1.1 As

Chr\u\r+ifi,

Hr+1{fi).

T h e multi-domain formulation for finite elements

we have done in Chapter 1, we split into two subdomains f2i and ^2, such = that TT7 U fl 2 = n , f2i fl 0! and we set r : = fii D ^2 (see, for example, Pig. 1.1). We also suppose that the interface T does not cut any finite element T . This implies that the global triangulation Th of ft induces two triangulations Tht 1 of fti and Th,2 of ft2 that are compatible on F; that is, they share the same edges on T (see Fig. 2.1.2, where another domain, different from those in Fig. 1-1, is considered).

44

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

<Jh.

2

Let us start with the variational formulation of our problem (an alternative characterisation based on a purely algebraic argument is given in Section 2.3). To this purpose let us define (2.1.7)

Ah := {vh\r \ vh G Vh},

Viyh := {t^n, | vh G Vh},

and set V°h := {vh G Vi>h \ vh]r = 0}.

(2-1.8)

By repeating the lines of the proof of Lemma 1.2.1, we see that the finite element problem (2.1.1), after identifying u i ^ with and u2,h with uh|n2, is equivalent to the multi-domain problem ' ai(ui,h,vi,h) Ui,h = u2,h (2"L9)

= {f,v\,h)Q1 on r

a2(u2,h, v2lh) = (f,v2,h)n2 2

V v2,h G V2°h

2 ai(ui,h,

i=1

V v1:h G

) = £ ( / , Ki,hHh)ni i=1

V flh G Ah,

where 7Zith, i = 1,2, is any extension operator from A^ into VU}L. In practical implementation, these extension operators will be taken equal to the finite element interpolant ni^fth, which belongs to Vi.h, equals /J,h at the nodes on the interface P, and vanishes at the internal nodes in fi;. The formulation (2.1.9) may be generalised to many subdomains, possibly including cross-points. Note, in particular, that if IZi^^h is the restriction to fii of the finite element shape function associated with a cross-point P, then (2.1.9)4 enforces the continuity of the 'normal' derivative in P in a natural form.

§2.1

tiiNiTUttLEMHilN1 ArrxvUAlMArlUlN Uf KLbiFTIO ECj U ATKJJNS

45

R e m a r k 2 . 1 . 1 A multi-domain approach based on a dual variational formulation and the use of Lagrange multipliers, called FETI (Finite Element Tearing and Interconnecting), has been proposed by Farhat and Roux (1991, 1994), and is extensively used nowadays. •

2.1.2

Algebraic formulation of the discrete problem

The unknowns of the finite dimensional problem (2.1.1) are given by the pointvalues of Uh at the finite element nodes a r In fact, denoting by TV/1 the total number of the nodes and by ipj the basis functions of V/,; that is, the unique functions in Vh satisfying
uh{x)

(2.1.10)

=

^2uh(aj)tpj(x). 3=1

Introducing the notation (2.1.11)

{uh{a.j)}j=1,...}Nh

u :=

and f~{(f,
(2-1.12)

problem (2.1.1) can be rewritten as (2.1.13)

Au = f.

The matrix A is called the finite element stiffness matrix and is given by (2.1.14)

Au := a(
The stiffness matrix A is positive definite; that is, for any v G R ^ , v / 0 , (Av, v) > 0, where (•, •) denotes the Euclidean scalar product. Indeed, let Vh £ Vh be the function defined as Nh i=i Then

Nh (Av,v) = ] T via{ipj,
= a(vh,vh)

> 0,

AND (Av, V) = 0 if and only if Vh = 0, or, equivalently, v = 0.

4b

JJISUKJI'ILSJIL) JKYUATLUINB AINU JJUMA1JN UTLUUMFUSL 1IU1N

UIl. 2

In particular, any eigenvalue of A has a positive real part. Besides, when the bilinear form a(-, •) is symmetric (and this is the case in (1.2.1)), it follows immediately that A is also symmetric. Another important remark is concerned with the condition number (2.1.15)

K2(A)

:= P|| 2 P " i | | 2 =

yAmiin^

A)

In the symmetric case, we have the simplified relation ( A ]

K 2 { A )

_

~

AMAX(^)

XntoiA)'

and it can be proved that (2.1.16)

2.2

K2(A) =

0(h~2).

Finite element approximation of the Steklov-Poincare operator

In the framework of the finite element problem (2.1.1), the finite dimensional counterpart of the Steklov-Poincare operator can be easily characterised. For each rjh e Ah let us introduce its finite element harmonic extension in ftj (i = 1,2):

{

H^hVh € Vi,h :

Oi(Hilhr]h,Vith)=0

Vviih€VPh

HithVh\r = Vh on T. Similarly, we define (2.2.2) Qi
V viih £ V°h.

In other words, Gijif is the finite element solution of the homogeneous Dirichlet boundary value problem (i = 1,2) -Agithf = f

in fli

Gi,hf = 0

on dJV

Now we can define (formally) the finite element approximation of (1.1.7) as follows: (2.2.3)

Sh^h = Xh

on F,

§2.2

DISCRETE STEKLOV-POINCARE OPERATOR

47

where Xh

•= 2

Y; i—1

ShVh •= y~]Sithrih i=1 2

Si,km •= Y, i—1

g g^iHi'hTlh-

As in (1-2.7) and (1.2.12) we can easily get for all r}h,Hh £ Ah, 2

(ShVh^h) = y^iai{Hi,hilh>T^i,hfJ'h) (2.2.4)

^ = Ylai(Hi>hr)h' i=l

2 Hi

>hlJLh}

=

i=1

Y^i^h^h),

and 2 t=l where the discrete extension operators 7?.^ were introduced in (2.1.9). Therefore, the operators Sh and S^h, i = 1,2, are symmetric. Moreover, due to the Poincare inequality (1.2.2) and the trace inequality (1.2.5) it follows that (Si,hr]h,Vh) = \\VHithr]h\\ln. > ^

+

^

II^IIA.

namely, the Steklov-Poincare operators are coercive, uniformly with respect to h. In Theorem 4.1.3 it will be shown that there exist two positive constants C2, independent of h, such that (2-2.5)

C i l M U < H f f i . / ^ l k f i ,
V Vh e Ah,

i = 1,2.

This property is known as the finite element uniform extension theorem (see, for example, Bramble et al. 1986a; Bj0rstad and Widlund (1986); or Marini and Quarteroni 1988, 1989). In view of (2.2.4) we also immediately obtain that there exists a constant C > 0, independent of h, such that, for all rjh £ Ah (2.2.6) (Si,hVh,Vh) < (Shrjh,r)h) < C{SithVh,Vh), i = 1,2. In other words, the operators and Sh are spectrally equivalent, and therefore either Si^h ° r S2th can serve a§. an optimal preconditioner of Sh (see also the related result (2.3.14) below, which deals with the corresponding matrix Problem).

48 2.2.1

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

<Jh.

2

Eigenvalue analysis for the finite element Steklov-Poincare op. erator

Let us introduce the eigenvalue problem for the discrete Steklov-Poincare operator Sh• find j ( , £ R and wj, 6 A),, w j / 0, such that (2.2.7)

{ShVh,t*h) =

J uhjj,h

Vfj.fi £

Let us indicate by Nr the dimension of the interface space Ah- For example, in the case of piecewise-linear finite elements (r = 1), Nr is given by the number of finite element nodes lying on P, excluding the nodes on the boundary dfl (see Section 2.3). We have the following result: Proposition 2.2.1 Since the bilinear form a(-, •) is symmetric, continuous, and coercive in V, the associated discrete Steklov-Poincare operator Sh has real and positive eigenvalues Sj^, j — 1, • • -, Nr, which obey the following bounds

Kx < 5j,h <

(2.2.8)

Kih'1,

where K\ and K2 are two positive constants, both independent of h. Proof Since Sh is symmetric and coercive in A, its eigenvalues are real and positive, and are given by the Rayleigh quotient (2.2.9)

djth -

r ~2 Jr

>

where ojj^ £ Ah denotes the eigenvector associated with Using (2.2.4), the first inequality in (2.2.5) and Poincare inequality (1.2.2) it follows that

fr

w

j,h

having set K\ : = C\ + C n J - 1 . On the other hand, using now the second inequality in (2.2.5) we obtain 2

i=i By following the lines of the proof presented in Ciarlet (1978) and recalling that the space A is a subspace of H 1 / 2 ( r ) , we can obtain the inverse inequality

IM|A
O

THE SCHUR COMPLEMENT MATRIX

§2-3

49

It can be proved that the bounds of 6jth are sharp, and indeed cannot be improved (see, for example, Bj0rstad and Widlund 1986; and Smith et al. 1996, pp. 119-20, for piecewise-linear finite element or five-point finite difference approximation of the Poisson equation on an uniform grid; see also Section 3.3.1 and Brenner 1998a).

2.3

Algebraic formulation of the discrete Steklov-Poincare operator: the Schur complement matrix

In order to give the algebraic interpretation of the finite dimensional operator Sh let us distinguish between the finite element nodes belonging to T and those of Hi and ^2- We denote the corresponding vector of the finite element unknowns by u r , ui and u 2 , respectively, and their lengths by JVp, Ni and N2 (see Fig. 2.1.2 for an illustrative example, where bold nodes indicate finite element nodes belonging to F). Analogously, the vector corresponding to the datum / will be denoted by f e R N » . As we have already seen, the finite element problem (2.1.1) can be written in the algebraic form (2.1.13), where A is the Nh x Nh stiffness matrix, with Nh — Nt + Ni + N2. Our system can be written in block form as (Axx 0 \Ari

(2.3.1)

0 A22 Ar2

Air \ / u i \ A2r U2 Arr J \ u r /

=

(h\ f2 , \fr /

where we have used the following notations: for i = 1, 2 (2.3.2)

(A t i )ii ••= ai(tpf

l,j =

l,...,Nit

where a.j(-, •) is the restriction of the bilinear form a(-, •) to fli defined in (1.2.4), and ip^ are the basis functions associated with the nodes lying in f^. Moreover, we have (2-3.3)

(Arr)Sr:=ai(^r).^r))+«2(^r),^r)),

s,r =

1,...,NT,

where are the basis functions associated with the nodes lying on T. For l , 2 w e have set, moreover,

! =

( 2 - 3 -4)

{AiT)lr

-OifaP,^0),

1 = l,...,Ni,r

=

l,...,Nr,

while An denotes the transpose of A i r , i = 1,2. The algebraic system (2.3.1) can be regarded as directly associated with the of equations (2.1.9), where the extension 7Z^hPh has been taken equal to the finite element interpolant iri^hUh, which takes values 0 at all the nodes internal flj. In fact, note that we can write Se t

50

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION <J Ui,h = Ui,h +

Aft,

h.

2

1=1,2,

where u^h £ V°h takes the same values of m^ at the nodes internal at f2j and value 0 at the nodes on T, and Ah £ A^ is the common value of and u2^h on T. Then, denoting by u , £ R N l the vector of grid values of ui>fl in fi,, and by up € R W r that of u i i h on F, we obtain (2.3.1) from (2.1.9). After eliminating Ui and u 2 we obtain the reduced system: (2.3.5)

S / , u r = Xr,

with (2.3.6)

Xr : = fr - ^ r i A ^ f i -

AT2A~^i2

and (2.3.7)

E h : = A r r - ATIA~^A1V

-

A^A^A^-

Once the solution ur of (2.3.5) is available, the subdomain components and u 2 can be immediately recovered from (2.3.1) at the expense of two independent solves A^i and A^ 2 • We can split the matrix of the interface contributions as follows (2.3.8)

Arr

=^rr+4rr>

where App denotes the contribution from the subdomain fl;, i = 1,2. Then we can write (2.3.9)

EFT = S J , +

E2IA,

with (2.3.10)

E,,ft : = App - AnA^Air,

i = 1,2.

The matrix Eft is the Schur complement matrix: it is precisely the algebraic counterpart of the discrete Steklov-Poincare operator Sh- In turn, (2.3.5) is called the Schur complement system. Eft is symmetric and positive definite, because it results from (2.3.7), and also from the relationship (2.3.11)

[Eftjj, (j,] = {ShVh, Vh)

V T]h, fih £ Aft,

where [-,-] is the Euclidean scalar product in R W r , and for each /ift £ Aft,A1 denotes the set of its values at the nodes on F. Similarly, we have for i = 1,2 (2.3.12)

[EJ,FTR?,/I] = {SI,HVH,^H)

V R) H ,^H S A H .

For each i = 1,2, the matrix E^ft is dense: to compute its entries is expensive because we need as many solves as the number of nodes on T. Indeed, from

2.3

i n n , a ^ n u n , owmrij.iMvi.cjiN i m a h i i - a .

(2.3.12) we see that for each node on F we should compute the action of Sith on the Lagrangian basis function associated with that node. When the Schur complements are explicitly computed, we have the direct method called substructuring. It is c o m m o n practice in the engineering community, and corresponds to having all interiors nodes eliminated by static condensation (see, for example, Przemieniecki 1985). Most often, however, especially when the number of nodes lying on the interface F is high, conjugate gradient methods are used for an iterative solution of the Schur complement system (2.3.5). Each matrix-vector multiplication with Sh involves two subdomain solves, Aj^1 and , which can be performed in parallel. However, the matrix Sfi. is ill-conditioned. Actually, it inherits a number of properties from Sh\ in particular, from (2.2.8), its spectral condition number /t(S/i) (that is, the ratio between the maximum and minimum eigenvalue) is bounded by «(Efc) < Coh-1

(2.3.13)

for a suitable constant Co independent of h. The use of preconditioners is therefore mandatory. Their construction is addressed in Section 3.3. At this stage, we can already propose the preconditioners E j ^ , i = 1,2, which, from (2.3.11), (2.3.12) and (2.2.6), satisfy (2.3.14) Hence, either S i ^ or S 2 ,h are optimal preconditioners

of E^.

Remark 2 . 3 . 1 In the special case of a scalar second-order elliptic boundary value problem with constant coefficients, using a uniform mesh and two equalsized subdomains that are mirror images of each other (about F), it follows that = S 2 ,h = Thus, either S i ^ or S2ih. are ideal preconditioners for S^.

• In Table 2.3.1 we summarise the relation between the Galerkin and the algebraic forms of the single domain problem, the multi-domain problem and the interface equation. In the forthcoming Section 2.3.1 we show how to obtain a preconditioner for the stiffness matrix A once a preconditioner for the Schur complement S/, is available. 2.3.1

Preconditioners of the stiffness matrix derived from preconditioners of the Schur complement matrix

After (2.3.1) and (2.3.8), the stiffness matrix in the two-substructure case reads

An

0

0

'j4•22 22

An

Ar2

Air A2r

A$ +

4 r

52

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION Table 2 . 3 . 1 The Galerkin and the algebraic forms the multi-domain and the interface problems

of the single

<Jh. 2 domain,

Galerkin vs algebraic Single domain (stiffness)

uh G Vh

: M vh £ Vh

a{uh,vh)

=

«i,h e Vi l h

: v vhh e =

ai(ui,h,vi,h) Multidomain (block stiffness)

v°h 'An o

(f,vi,h)ni

u2,h e V2,h

: V v2,h e V2°h

a2{u2,h,v2,h)

= (f,v 2 ,h)n 2

,h - u 2 , h

Au = f

(f,vh)

An

0 A22

Air ' A2T

Ar2

AIT

Ul u2 ur

'

on F

= Y s i U ^ ^ h U h ) f i ; v p,h e Ah Interface problem (Schur complement)

6 Ah : V {Sh*h,Vh)

=

e Aa S ^ u r = Xr

(Xh,Vh)

It can be expressed in factored form as A =

LDLT,

where, denoting by I\, I2 and Jr the identity matrix of dimension Ni, N2 and N r , respectively, 0 0

0

h 0 ATlA^

h AT2A2^

h

(2.3.16)

D :=

(A

u

0

0

A22

V 0

0

0

\

0

Sl,ft + S2,A/

Equivalently, we have the following block LU decomposition A = LU,

THE SCHUR COMPLEMENT MATRIX

§2.3

53

where (2.3.17)

U

DLt

(An = | 0 V 0

0 A22 0

Air A2T S 1>A + S 2 l h

Assume that a convenient preconditioner Ph is available for the Schur complement matrix £/,,. Then we can devise the following preconditioner Qh of A: (2.3.18)

Qh ••= LU,

where L is given in (2.3.16) and U is obtained from U in (2.3.17) by approximating with Ph; that is, (2.3.19)

U :=

(An 0 \ 0

0 A22 0

Ait\ A2F . Ph J

Note that the blocks of Qh coincide with those of A, except for the block (3, 3), which equals (Qh)s 3 = A n A ^ 1 Air + AV2A^A2T + Ph. (In the special case in which the preconditioner is the Dirichlet-Neumann preconditioner S 2 t h, see Section 3.2, we obtain (Qh)33 = AtiA^ A\r -I- Ap2p.) Let A be an eigenvalue of Q ^ A , and w € R " ' 1 a corresponding eigenvector. Writing w = (wi, w 2 , wp), with obvious meaning of notation, from A w = AQftW we have ' (1 - A ) ( A l i w 1 + A i r w r ) = 0 (1 - A)(A 2 2 W 2 + A 2 r w p ) = 0 <

(1 - A ) ( A n w 1 + A r 2 w 2 ) + A r r w r = A(P / l w r + A n A f ^ A i r W r + AV2A22

A2t^t)-

Prom (2.3.7) the last equation can be rewritten as (1 - A ) ( i r i w i + A r 2 w 2 + A n A^ 1 A i r w r + Ar 2 A^ 2 1 J 4 2 rWr) + S^wp = APftWp. Therefore, if A ^ 1 we have A n W i + AirWr = 0

{

A22 w 2 + i 2 r w r = 0,

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

54

hence, w r

<Jh.

2

0 and SftWr = APftw r .

We conclude that the matrix Q ^ A has the same eigenvalues as plus the eigenvalue 1 (which is also an eigenvalue of P ^ H h provided that the corresponding eigenvector w satisfies w r 0). If we assume that Ph is spectrally equivalent to that is, there exist two positive constants K\ and K2, independent of h, such that (2.3.20)

< [2hn,ij\ < K2[Phr,,ri\

K^PhW]

V 17 e

RNr,

then we derive from the characterisation of the eigenvalues of Q^1A

(2.3.21)

that

4Qh'A) <

where (2.3.22)

:= m i n ^ , ^ ) ,

K2 : = m a x ( l ,

K2),

and therefore we conclude that Qh is spectrally equivalent to A. From the computational viewpoint, the solution of the preconditioned system Qhx

= b

entails the solution of the following system Ly — b ,

Ux =

y,

that is, (2.3.23)

yi = bi,

y2 = b2,

y r = b r - AnA^1^

-

AT2A^b2

and (2.3.24)

PftX r = y r ,

A

n

x 1 = bi - A i r x r ,

A22x2 = b2 -

A2rxr.

Both steps (2.3.23) and (2.3.24) require the solution of two independent Dirichlet problems, one for each subdomain. Furthermore, step (2.3.24) requires the solution of a problem associated with Ph- The whole process is therefore amenable to independent solves in fti and f l 2 , provided Ph splits into independent solves as well. In conclusion, the preconditioner Qh inherits all the good properties enjoyed by Ph in terms of both parallelism and spectral equivalence. R e m a r k 2 . 3 . 2 In the case where we take Ph = (this corresponds to the Dirichlet-Neumann preconditioner, see Section 3.2), step (2.3.24) requires us to solve

THE UASE U f MAiNY SUtSJJUMAiiNS

§2.4

=

yr,

which corresponds to a Neumann problem in

2.4

55

ft2.

•

The case of many subdomains

The case of partitions with M > 2 subdomains fij can be dealt with similarly. Let ft be partioned into M non-overlapping subdomains ft; of diameter Hi with interface T separating them, F = U ^ F * , F; 5ft, \ 5ft. Let / = U^Ni denote the indices corresponding to the internal nodes (see Fig. 2.4.1).

FIG. 2.4.1. Multi-domain partition and finite element triangulation (bold lines denote subdomain interfaces). Then, with obvious meaning of notation, we can write the algebraic problem Au = f blockwise as follows:

(t; where An = Ajv. Since the interior nodes in each subdomain will be decoupled from the interior nodes in other subdomains, we will have (A (2.4.2)

An

=blockdiag(Ati) =

u

. \ 0

••• ••• •••

0 . AMM

\

J

.

The ith block An is the principal submatrix of the local stiffness matrices that associated with either Dirichlet or Neumann problems in the subdomains ft,. Indeed, the matrix a re

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

56

An An

Ai :=

(2.4.3)

Ch. 2

irr

is the finite element matrix associated with the Poisson problem in ft with the Neumann datum on I\ (and the homogeneous Dirichlet datum on dtl n d f l , ) .

T

THE CASE OF MANY SUBDOMAINS

§2.4

where Ai is the local stiffness matrix introduced in (2.4.3), Ri is the restriction matrix from the full vector in fl to local vectors in O; U P; and consequently the matrix Rf denotes prolongation by 0 on the nodes external to fl, U Tj. In a similar way, we can split the Schur complement matrix Eft. In fact, the Steklov-Poincare operator satisfies

The notations have the following meaning: (Au)u

:=ai(
(AlT)lr

(2.4.4)

.

j = l

internal n o d , ,

<J,l
~ OiWr,
(A^l)sr-.= wVlprp w

M

1

1


Ni, are the finite element basis functions associated with in

Vh =

Arr-ArIAjI1Air,

£/lUr=Xr,

where

(2.4.12)

Xr : = fr - A n A j ^ f j .

W i t h the aim of representing Eft as a sum of local contributions, we start by Hi, i = 1 , . . . , M. Since M i=i it is easily checked that A can be split as M A =

^R?AiRi, i=i

Arr

AriA

-1

E h — ^ Rfi^i.hRrj i=1

Here, Rr, is the restriction matrix from the vector of coefficient unknowns related to the nodes on T to only those associated with and i?p. is the matrix that extends by 0 a nodal vector from I1,; to P. Note that in the case of two subdomains ( M = 2) we have r = Ti = r 2 , therefore both the restriction and the extension operators i?r ; and Rf_ coincide with the identity matrix, and consequently (2.4.12) yields Eft = E i ^ + E 2 ,ft, as already stated in (2.3.9). As reported in Le Tallec (1994) (see also Brenner 1998a), the condition number of satisfies (2.4.13)

splitting the stiffness matrix A into the contributions given by each subdomain

(2.4.9)

(see (2.2.4)). Denote now by E^ft the local Schur complement; that is, the Schur complement of the local stiffness matrix Ai with respect to the nodes of r , . Prom (2.4.3), it can be written as

M

and the associated interface problem reads:

(2-4.8)

^ai(Hi
2= 1

It is easily seen that

An 0

Vi : =

of (2.4.1) is

(2.4.7)

(Shrih>Vh) = ^2(Si,hr]h, Hh) =

(2.4.11)

where fj.' K 1 is the identity matrix of order J;, is the matrix associated with the finite element discretisation of the Poisson problem in fl^ with the Dirichlet datum prescribed on Tj (and the homogeneous Dirichlet datum on d f l fl <9fl»). The Schur complement matrix Eft associated with the interface variables up

(2.4.6)

M

while W , r = 1,. • •, J „ are those associated with

the nodes y r belonging to I V Similarly, the matrix (2.4.5)

(2.4.10)

l
ai(^rM

57

«(£h) < C

H

hHl

in'

where HMSN and H denote, respectively, the minimum and maximum diameters of the subdomains. If the family of partitions of fl is quasi-uniform; that is, there exists a positive constant r , independent of H, such that HMIN > TH, then K(E/,) = OIH^H-1). The challenging point is therefore the construction of a suitable preconditioner for the Schur complement Eft such that the convergence rate of the preconditioned iterative method is independent of both h and H (see Section 3.3 for the development of this issue). When an iterative method is applied to (2.3.5) (for example, the Richardson or the conjugate gradient method), the action of Eft on a given vector has to be computed. In view of (2.4.11) and (2.4.12), this amounts to inverting M

58

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION <J

h. 2

independent matrices An, or, equivalently, to solve M independent Dirichlet problems. Following a direct approach, generalising what was carried out in Section 2.3.1 for a two-domain subdivision, we see that the stiffness matrix A can be factored as follows: (2.4.14)

A =

with

: )•*"(*." D-

or A = LU, with U.

=

LDLt,

(An V

Air

\ /

0

As noted in Section 2.3.1, should Ph be a preconditioner of the Schur complement Sh, then the matrix (2.4.16)

Qh :=

LU,

with (2.4.17)

is a preconditioner of the stiffness matrix A. The eigenvalues of Q^XA are the same of those of plus the eigenvalue 1. When an iterative method is used to solve the linear system associated with Q ^ A , at each iteration step a system with the matrix Qh has to be solved. Using the factorisation (2.4.16), this requires us to solve first the L factor by forward substitution, which involves the inversion of An (see (2.4.2)) for each i = 1 , . . . , M, requiring therefore the solution of M independent Dirichlet problems. As for the U factor, this is solved by backward substitution, starting from the block Ph, followed by the block A n , which again yields the solution of M independent Dirichlet problems in the subdomains. As for Ph, in the special case where Ph = S 2 w i t h S2,h '•— Y J

^i'k

i€lw

(see Section 1.4.2 for notation), the inversion of Ph corresponds to a Neumann problem in fijy : = U g g / ^ f V This global problem cannot be split into independent problems in the subdomains, because they are coupled through corner points (and edges in three dimensions).

NON-CONFORMING DOMAIN DECOMPOSITION METHODS

§2.5

2.5

59

Non-conforming domain decomposition methods

In some circumstances it is useful to approximate a given partial differential equation, in a domain ft split into non-overlapping domains, by means of a numerical method that makes use of different finite dimensional spaces and/or different grids in different subdomains. The distinguishing feature of this approximation is that, in principle, the approximate solution will not be continuous across subdomain interfaces. For the sake of exposition we refer to the case of a domain ft partitioned into two subdomains fti and ft2, whose interface is T (see Fig. 2.5.1). Another case happens when, although the grids match each other on T, the structures of the approximate solutions are different in fti and ft2. Referring to the situation depicted in Fig. 2.5.2, one could use, for example, linear finite elements in fti and quadratic in ft2. Any other combination of the above situations is admissible as well and yields the same kind of difficulty. We now refer to the Poisson problem (1.1) and its weak formulation (1.2.1). A

Ql

FIG. 2.5.1. Non-matching grids: unstructured (left), structured (right).

PIG. 2.5.2. Substructuring with matching grids but different finite element approximation: piecewise-linear in fti and piecewise-quadratic in ft2.

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION <J

60

h. 2

Galerkin approximation like that introduced in (2.1.1) cannot be used this time because the finite dimensional space, being made up of discontinuous functions, cannot be a subspace of i ? 1 ( 0 ) . The alternative approach consists of giving a new weak formulation of (1.1), which, being compatible with the use of spaces defined independently in f ^ and f12, is more adapted to the finite dimensional level. This can be done in different ways: we will consider two of them, the mortar method and the three-field method (the latter has been introduced already, at the continuous level, in Section 1.7). 2.5.1

The mortar method

This method has been introduced by Bernardi et al. (1994) with the aim of combining spectral methods having different polynomial degrees, or the spectral method with the finite element method. Its generality, however, goes beyond these two specific examples. The idea is to approximate (1-2.1) by the following discrete problem: (2.5.1)

find

us 6 Vs : V [ i= i

Vu5 -Vv5

= V [ fvs i=i Ju-

V vs G Vs.

Here, 5 > 0 is a parameter describing the quality of the discretisation, and Vs is a finite dimensional space that approximates Hq ( 0 ) without being contained into C ° ( f i ) . More precisely, Vg is a subspace of the following space: (2.5.2)

Ys := {vs G L2{U) \ vs{Qi

G YhS, i = 1, 2},

where, for each i = 1,2, Yits is a finite dimensional subspace of the space V; introduced in (1.2.4). The subspace Yits can be either a finite element space, or a polynomial spectral space. In any case, no requirement of compatibility is made for the restriction on T of the functions of and Y 2 J • Heuristically, the space V() will be made up of functions belonging to Y^ that satisfy some kind of matching across T. Precisely, if v$ G Vs and v^ Vg2^

G Yitg,

G Yi.fi denotes its restriction to S7i and 0 2 , respectively, for a certain fixed

index % the following integral matching conditions should be satisfied (2.5.3)

|r(wW_42>)MW=0

v ^ e A f ,

where A ^ denotes the restriction to T of the functions of Yij(5. If we take i = 2 in (2.5.3), this amounts to letting fii play the role of master and fl2 that of slave, and (2.5.3) has to be intended as the way of generating the value of

once v i s

available. The other way round, i.e. taking j = 1 in

(2.5.3), is also admissible. Depending upon the choice of index i made in (2.5.3), the method will produce different solutions. The mathematical rationale behind the choice of the matching condition (2.5.3) (rather than a more 'natural' condition of pointwise continuity at one

§2.5

NON-CONFORMING DOMAIN DECOMPOSITION METHODS

61

set of grid nodes on F) becomes clear from the convergence analysis on problem (2.5.1). With this aim we introduce (2.5.4)

IMI* := (IMIo.n + llv^|fiillo,fia + l|Vu|fi2||o,n2)1/2,

which is a norm (the 'graph' norm) on the Hilbert space (2.5.5)

R* : = { « 6 L2(n)\v]Ul

G H1^),

v]U2 G

H1(Q.2)}.

Owing to the Poincare inequality (1.2.2), we have that

(2.5.6)

£

i=1

Vg,

/ |V^|2 > JOi

whence the discrete problem (2.5.1) admits a unique solution by a straightforward application of the Lax-Milgram lemma. For any vg G Vg we now have 2

f J U;

(2.5.7)

< Y] i

/ =

V Us -V(ug -Vg) i 2

= 52/ f(u5-vs)-Y2 .•_, Jfi;

- Y ] i=i JQi

• V(us - v$)

Vvg • V(us - vg).

Replacing / by - A u and integrating by parts on each f2j we obtain: 2

(2.5.8)

52 / i=i

2

f(us-vs)=T i=1

Jn,

Vw • V(u<5 - vg)

-fj£[(u

s

-v

s

)U-(u

s

-v

s

)W

(we recall that ^ is the normal derivative on T and n is the unit normal vector °n r pointing into Denoting by r l (!) (2) M r := ^ - ^jr the jump across T of a function vg G Vg, from (2.5.7) and (2.5.8) we have that a*\\ug - ^||2 < ||U - V«5||*||«,5 - im||* + l - l u s - v g ] . a nd

also

62

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

IK

- vs\\* <

W - V6 * +

a*

sup

WJSVJ

<Jh. 2

IKII*

By the triangle inequality \\u - Usj|» < I\u - vsll* + l\ud - vsll* we then obtain the following inequality for the error u — u^:

(2.5.9)

||u-u a |U < f l + — ) inf V a*JvseVs

||u-t;i||. + — a*

sup wseVs

H^H*

The approximation error of (2.5.1) is therefore bounded (up to a multiplicative constant) by the best approximation error (that is, the distance between the exact solution u and the finite dimensional space V^) plus an extra error involving interface jumps. The latter would not appear in the framework of classical Galerkin approximation (like (2.1.1)), and is the price to pay for the violation of the cbnforming property; that is, for the fact that Vg (jL Hq(D,). The error estimate (2.5.9) is optimal if each one of the two terms on the right can be bounded by the norm of local errors arising from the approximations in fti and ft2, without the presence of terms that combine them in a multiplicative fashion. In this way, we can take advantage of the local regularity of the exact solution as well as the approximation properties enjoyed by the local subspaces

YIIS of i? 1 (ftj). Let us focus on the last term of (2.5.9) (the interface error). Owing to (2.5.3) we have

(2)

for all /j,s

£ Ag

(2)

(the space of the restrictions to F of the functions of Y 2 J ) . We

can therefore take as

(2)

any convenient approximation of

if Y~2,s is made up of finite elements of degree s2 > 1, then ,(2) jj,y the finite element interpolant of

For instance, can be chosen as

at the finite eleme:

from the mesh used in ft2. This would give the estimate du dn

(2)

V-5

S 2 - 1 / 2

<

Ch

<

Chs22~1/2

o,r

du dn

s 2-1/2,r

lh«2

S2 + l,fi25

provided u|Q2 £ i J S 2 + 1 ( f t 2 ) and h2 is the maximum grid size in ft2. On the other hand, the identity (2.5.3) states that w ^ can be regarded as the orthogonal projection of of L2(T).

Therefore,

upon A{f> with respect to the scalar product

§2.5

NON-CONFORMING DOMAIN DECOMPOSITION METHODS

63

I I M r l k r = ||*4|r - ^ijrllo.r < Ch l 2 l2 \\wf^\\ 1/2 ^
H^'IIia,

and we conclude that (2.5.10)

sup 1/r p K H ws€Vs ll^ll*

<

gft,a||u

||

Concerning the best approximation error, a first step is to find a convenient approximation I I ^ u of u in Yit$, the space of the master domain, that satisfies an optimal local estimate. For instance, we can take I I ^ u as the finite element interpolant of degree si of u in Oi, in order to find (see (2.1.5)) (2.5.11)

H u - n ^ u l l i , ! , !
where h\ is the maximum grid size in Oi. (2) Then we define a suitable function IT5 u G Y2J, a piecewise polynomial of degree s2 that satisfies the matching condition (2.5.12)

ufK

- n Vu^f

= 0

V^

e Af>,

and is such that the function

15

fn^u

in ft!

[n^u

in Q 2

satisfies (2.5.13)

II" -

<

+

lls 2 +i,n 2 )-

The construction of n ^ ' u can be done by a slight modification of Proposition 4.6 in Bernardi et al. (1990). Finally, we can bound the norm ||u —u^H* as in (2.5.13), using (2.5.9), (2.5.10) and (2.5.13). To generate a basis for the finite dimensional space VG, we can proceed as follows. For i = 1,2, let us denote by Afi the set of nodes in the interior of ftj, and by A f ^ the set of nodes on F, whose cardinality will be indicated by Ni and ^ r , respectively. Note that, in general, A/p 1 ' and

can be totally unrelated.

Now, denote by k1 = l,...,Ni, the Lagrange functions associated w ith the nodes of A/i; since they vanish on F, they can be extended by 0 in il 2 . These extended functions are denoted by and can be taken as a first set °f basis functions for Vs.

64

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

<Jh.

2

Symmetrically, we can generate as many basis functions for V$ as the number of nodes of by extending by 0 in f2i the Lagrange functions associated with these nodes. These new functions are denoted by k" = 1 , . . . , N2. Finally, always supposing that is the master domain and Q2 its slave, for every Lagrange function in fii, m = 1 , . . . , we obtain a basis function {y> m ,r} as follows J Vm r

m

{
m

fi

2

where ~(2) Vm,r

^

C

(2)

j=i

are the Lagrange functions in 0 2 associated with the nodes of , and are unknown coefficients that should be determined through the fulfilment of the matching equations (2.5.3). Precisely, they must satisfy

(2.5.14)

r

^

( J 2

\

- ¥»Sr J v g = 0

v I = 1,..., TVf.

A basis for V& is therefore provided by the set of all functions { ^ V } , k' = . 1 , . . . , N,, {($}},

k" = 1 , . . . , JV2, and { < ? m , r } , m = 1 , . . . , A ^ .

Remark 2.5.1 In the mortar method the interface matching is achieved through a L 2 -interface projection, or, equivalently, by equating first-order moments, thus involving computation of interface integrals. In particular, from equations (2.5.3) we have two different kinds of integrals to evaluate (take, for instance, i = 2): (2)„(2)

(2) The computation of I22 raises no special difficulties, because both functions vs (2) and live on the same mesh, the one inherited from fi2- On the contrary, i^-1' and /ijj2' are functions defined on different domains, and the computation of integrals like /12 requires proper quadrature rules. This process needs to be done with special care, especially for three-dimensional problems, for which subdomain interfaces are made up of faces, edges and vertices (see Ben Belgacem and Maday 1997), otherwise the overall accuracy of the mortar approximation could be compromised. This issue is addressed in Cazabeau et al. (1997). • Any substructuring iterative method that decouples the computation in fti from that in is particularly well suited to solving the coupled problem (2.5.1).

§2.5

NON-CONFORMING DOMAIN DECOMPOSITION METHODS

65

An instance is provided by the Dirichlet-Neumann method, where this time the Neumann step should be applied in the master subdomain and the Dirichlet step (enforcing the matching relations (2.5.3)) in the slave subdomain. With obvious meaning of notations, this iterative procedure will generate new values of ( w ^ ) ^ 1 and (w^ 2) ) fc+1 from those available, ( u f ) ) k and ( u f ] ) f c , through the solution of the following subproblems: • Neumann step in fti (find ( u ^ ) ^ 1 e Y l l S

(2.5.15)

:

f JQ, i

ftp™ J Qj

f JQ!

/

^ ' = 1 , . . . , ^ f
JQ.-L

(2)

J N,

• Dirichlet step in 'find ( u f ) f c + 1 (2.5.16)

eFy

f V(42))fc+1-v^= f

J n2

V

Jci_ <5=0

k" = 1,

.,N2 (2)

VJ =

1, - - - , JV,r

'

with a possible relaxation applied on the last set of interface equations. Neumann-Neumann iterations can be defined in a similar manner (see Le Tallec 1993). For the analysis, see Achdou et al. (1996), Seshaiyer and Suri (1997), and Stefanica (1998). The discrete problem (2.5.1) can also be reformulated as a saddle point problem of the following form: findita eYs,Xs

e Af

: 2

a{us,vd)

+ b(vs,Xs)

=

V vs

eYs

i=i . b(us,ps)

- 0

(2)

v ps e Ag

where a(ws,vs)

2

f V w i l ) • Vv (i) , b{vs,p,s) ••= J (v^ Joi

-

vf])ps•

66

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

<Jh.

2

In this system, \g plays the role of the Lagrange multiplier associated with the 'constraint' (2.5.3). Denoting by ipj, j = 1 , . . . , Nx + N2 + JV^,1' + (2)

a basis of Y$ and by %[iu

(2)

/ = 1 , . . . , ATp , a basis of A^ , we introduce the matrices Asj

:= a((pj,<pa),

Bu

: = &(
Defining by u and A the vectors of the nodal values of ug and A,5, respectively, and by f the vector whose components are given by X ) i = i ( / i

?s =

+

N2 + N{.1} + iVp 2) , we have the linear system

i f ) ( ; ) - ( ; The matrix A is block-diagonal (with one block per subdomain ft;), each block corresponding to a problem for the Laplace operator with a Dirichlet boundary condition on <90; n <90 and a Neumann boundary condition on <90; \ <90. After elimination of the degrees of freedom internal to the subdomains, the method leads to the reduced linear system (still of a saddle point type)

where the matrix S is block-diagonal, C is a j u m p operator, up is the set of all nodal values at subdomain interfaces, and g r is a suitable right-hand side. This system can be regarded as an extension of the Schur complement system (2.3.5) to non-conforming approximation (the Lagrange multiplier A indeed accounts for non-matching discretisation at subdomain interfaces). In fact, the zth block of S is the analogue of S ; ^ , and corresponds to a discretised SteklovPoincare operator on the subdomain 0 , . For the efficient numerical solution of the reduced system, Achdou et al. (1995) have proposed conjugate gradient iterations that make use of a preconditioner of the following form: P

=

( S \c

CF 0

S being a convenient preconditioner of S. Basing S on the local approximation of S can bring the condition number of the preconditioned system down to 0 ( / i - 1 / 2 ) for a finite element approximation of uniform mesh size h. For other approaches involving preconditioned iterations for finite element problems with Lagrange multipliers see Kuznetsov (1995, 1997). 2.5.2

T h e three-field m e t h o d at the finite dimensional level

The three-field method in the differential case has been introduced in Section 1.7, for the general symmetric elliptic operator L introduced in (1.4.2). The

tj^.o

iNUiN-i^uiNruruvniNu jjwivimiN LinwivirusiiiuiN ivitvi n u u s

oy

finite dimensional approximation of (1.7.1) can be devised by selecting suitable subspaces V^ft of V"*, Eith of A' and Ah of A, i = 1,2, then introducing the following discrete three-field problem: find Uij. G VI,H, G AH G Aft such that ' at(uiih,v-

((7I,/L,UII/L|R)R = (f,vIIH)N!

(Pi,h.Ah-ui,ft|r)r=0 (2.5.17)

< (cri,h +
V vlth £ V\,h

V/£?i,ft e Hijft V ft £ A/, V p2,h

G H2,FT

, A 2 (I<2,ft, V2,h) ~ {02,h, ^2,H|R)R = (/, U2,H)N2

V v2,h G VJJ.A-

It is worthwhile to point out that, in principle, all the spaces above can be chosen independently, as is suggested from Fig. 2.5.3.

Grid for V l.h

Grid for " l.h

Grid for Ah

Grid for

Grid for V2,h

2,h

FIG. 2.5.3. Different meshes for the three-field discrete problem (2.5.17). This arbitrarity should, however, undergo suitable compatibility conditions in order for problem (2.5.17) to be non-singular. Obviously, if all meshes in Fig. 2.5.3 are compatible (i.e. they all coincide on T), then the above problem reduces to the standard finite element approximation (2.1.1), where Vh is the space of functions whose restriction to ft; coincides with V ^ , for i = 1,2. Regarding the compatibility conditions between the finite dimensional spaces appearing in (2.5.17), we can make the following considerations. Once the space of traces A ft has been chosen, the spaces of multipliers H and H2ift must be rich enough to control the traces. This demands a discrete inf-sup condition of the following type: there exists /3o,h. > 0 such that for each ^ G A^ it is possible

68

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

to find two multipliers px
<Jh.

2

0, such that

1/2

(2.5.18)

r > Po,h

J ] WPi^Wl' 1

IKIU-

Afterwards, the finite element functions in the subdomain spaces V\,/, and V2)h should suffice to control the multipliers. This means that the following inf-sup conditions need to be satisfied: for each i = 1,2 there exist (3ith > 0 such that for each p i i h g H,,/, it is possible to find £ Vij,, ^ 0, such that (2.5.19)

(Pi,h,

> Pi,h |\Pi,h 11A' |

|i,n;

Once (2.5.18) is satisfied, a possible way to satisfy (2.5.19) is based on the use of a stabilisation procedure through enriching the subdomain spaces Vi,/, and V2jt via suitable bubble functions (Brezzi 1998). R e m a r k 2 . 5 . 2 The mortar method can be regarded as a special case of the three-field method, corresponding to the following choices of the subspaces. The trace space A/, is exactly the space of traces of V i j , hence there is no need for the space of multipliers Hi,/, (Hi,/, = 0, and (2.5.17)2 disappears). Instead, the space of Lagrangian multipliers H2,/, coincides with the space of traces of V2ts n (see (2.5.3), which corresponds to (2.5.17)4). W i t h obvious choice of notations, problem (2.5.17) can be stated in algebraic form as follows: A i u i - Bfax

= fi

- f l i u i + C[ A = 0 (2.5.20)

G\01 + C2a2 —B 2 u 2 + A2u2

- Bja2

= 0 A= 0 = f2.

Proceeding by block-Gaussian elimination we obtain the following linear system for A: (2.5.21)

SA = g,

where E : = S i + S 2 , g : = gi + g 2 and Si-CiD^Cf,

gi^CiD-'BiA;1^

Di : = BiA~1Bj,

i =

l,2.

In the current context, (2.5.21) is a generalised Schur complement system and, indeed, S is the Schur complement with respect to u and a of the whole system (2.5.20).

§2.5

NON-CONFORMING DOMAIN DECOMPOSITION METHODS

69

The compatibility conditions on the discrete spaces can be expressed through the requirement that keri?^ = kerC^ = 0 for i = 1,2, in which case £ is symmetric and positive definite. The reformulation (2.5.20), (2.5.21), whose structure resembles that investigated in Section 2.3, suggests the way to solve problem (2.5.17). As £ is symmetric and positive definite, the interface system (2.5.21) can be solved first by preconditioned conjugate gradient iterations (the issue of which kind of preconditioner should be used is still an open problem). Note that, at each step, the calculation of the residual rk = g — requires the solution of local elliptic equations in the subdomains.

ITERATIVE DOMAIN DECOMPOSITION METHODS AT THE DISCRETE LEVEL In this chapter the iterative methods introduced in Sections 1.3 and 1.5 will be reformulated at the discrete level for the solution of the finite element problem (2.1.1). Their relation with domain decomposition preconditioners will be pointed out, and several acceleration techniques will be considered for the case where many subdomains are used. The convergence analysis will be carried out in Chapter 4. We will mainly focus on the Laplace operator —A, but the extension of the results in this chapter to the symmetric elliptic operator L introduced in (1.4.2) is straightforward.

3.1

Iterative substructuring methods at the finite element level

Following what was carried out in Section 1.3, to solve the finite element problem we can introduce iterative algorithms that exploit the subdomain partition. Let a s in us begin with the partition of 0 into two disjoint subdomains fii and Sections 2.1 and 2.2. The Dirichlet-Neumann method for solving the finite element problem (2.1.1) can be defined as follows. Let G Ah be an initial guess. For each k > 0, let us define u ^ 1 and u2 /j1 to be the solution of the following problems: ' < y (3.1.1)

e

vlth

01 ( " f ^ . v i . h ) = (/.«i,fc)ni = K

(3 1 2)

G V2'h

:

V vhh e V°h

on r

= (f,v 2 ,h)n 2 + (/, 72.i,/i^2,/i|r)fii -ai(u^ 1 ,'7li,/ l V2,/ l |r)

V v2,h G V2%h>

where notation is as in Sections 1.2 and 2.2, and 7Z±^ is any extension operator from A/i into V±^ (as previously noted, in practice 1Zi,h is an interpolant operator); then update (3.1.3)

\ k h + 1 =9u k 2 + h ] r + ( l ~ e ) \ k h

onT.

72

Oh. 3

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Note that (3.1.1) is the finite element approximation of the Dirichlet problem (1.3.1), whereas (3.1.2) is the finite element approximation of the mixed problem (1.3.2). As such, (3.1.2) can be equivalently rewritten as

«2(W2X1

(3.1.4)

' V2'h)

U2,/i)fi2

=

V V2,h

£

V£h

-aiiu^ilh^fih)

V ^

£ Ah,

where 1l 2 ,h is any extension operator from A^ into V2thIf the iterative method converges, the limit solutions uith •= lim u{h, '

k—too

u2:h :=

lim u2

k—too

h

satisfy (2.1.9).

The other iterative methods introduced in Section 1.3 can be similarly reformulated at the finite element level. Let us focus, for instance, on the N e u m a n n Neumann method. Again, let A° G Ah be an initial guess and for each k > 0, define and u ^ 1 to be the solution of the following problems: ,k+1 find u^ G Vi,h (3.1.5)

a

i(

U

,k+1 iji

>Vi,h)


=

{f,Vi,hhi

v

Vi,k

e

v?h

on r

for i = l , 2 , and { find V i l 1 G V h h V vith G (3.1.6)

+ a i ( u f + 1 , K i i h ( j , h ) + a 2 {u2 + h 1

V^

G Ah,

1

§3.2

SCHUR COMPLEMENT AND ITERATIVE UU METHODS

73

' find iP'X1 £ V2,h : 2lh)

= 0

Vv2,hev2\

(3.1.7) (^2,ft1 ' ^ 2 , / i f t ) = ( Z ^ i . h f t t a i + -aiiv!^

U^2,hPh)n2

- a2(uk+h1, U2,h^h)

V f t G Ah,

and then update (3.1.8)

\hh+1 = \ k h -

r

~ ^2
011 r -

In the case of convergence of the sequence A^ to a value A^, the limit functions ipith a n d ip2would

satisfy

in fli

A 4>i,h = 0 =

on dQi fl 80

0

o'lV'i./iir = <^2^2, h|r dip2

dipi ,h dn

dn

on

r

on

r.

(Both conditions on the interface T are satisfied in a weak form.) This yields -i[)i h = 0 in fti: in fact, when ui = 02, the function ._ / i>i,h 1p2,

in fli in Q 2

is harmonic in 0 with a boundary datum equal to 0 on 80; when a1 ^ a 2, the result follows from the algebraic formulation that we are going to present in Section 3.2. Since 1p^h — 0 in Oi, it follows at once that the limit functions and u2,h satisfy (2.1.9), and therefore Uh • =

uith "2 ,h

in Oi in 02

is the solution we are looking for.

3.2

The link between the Schur complement system and iterative substructuring methods

In the previous section, we have reformulated the methods introduced in Section 1-3 within a finite element framework. This is accomplished by replacing any

74

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Oh. 3 1

boundary value problem on each subdomain by its finite element approximation. As a consequence, each one of the methods there denoted by 1 - 4 can be regarded as if we were operating on Ah = it^r, namely, the restriction of the finite element solution to the interface P. Precisely, since the kih step provides a new value that depends linearly on the previous one, A^, each of the schemes 1 - 4 can be regarded as an iterative procedure applied directly to the discrete Steklov-Poincare equation (2.2.3). Identifying Ajj with the vector Xk of its values at the finite element nodes on r , we obtain a sequence that approximates the solution ur of the Schur complement system (2.3.5). For instance, approach 1 (Dirichlet-Neumann method) introduced in Section 1.3 and approximated in Section 3.1 yields the following iteration procedure for (2.3.5): (3.2.1)

Ph(\k+1

-\k)=9(Xr-Zh\k),

k>

0,

with Ph = ,/i- Equation (3.2.1) can be recognised as a Richardson iteration on (2.3.5), using the matrix U2th a s a preconditioner for Eh. As a matter of fact, the Dirichlet-Neumann scheme, carried out at the finite element level (see Section 3.1), corresponds to the following steps. Let Xk be given, and solve (3.2.2)

Anu*+1 =fi

-AirXk,

which corresponds to the Dirichlet step in

(3-2-3)

( A"

A ! )

( x

k l

^ )

=

(see (3.1.1)). Now solve

( fr - i n u r

- A$\h ) '

which in turn corresponds to the Neumann step in Cl2 (see (3.1.2) or (3.1.4)). Here, we are making use of the split form of the matrix A r r introduced in (2.3.8). Finally, we carry out the relaxation step (3.1.3): (3.2.4)

A fc+1 = 0A f c + 1 / 2 + (1 - 0)A\

A straightforward elimination of u^"1"1 carried out in (3.2.3) yields

(3 2 5)

{A^-AT2A^A2T)Xk+l'2 = f r - A n u ^ 1 - AppAfc - A r 2 A 2 2 1 f2 1

and similarly from (3.2.2) we find that (3.2.6)

Ul fc+1

= A ^ f i - Ai11AirXk.

Therefore, we can conclude with the following recursive formula

§3.2

SCHUR COMPLEMENT AND ITERATIVE DD METHODS

(3.2.7)

75

=Xr-£i,/,Afc,

Z2,hXk+1/2

where we have used the definitions (2.3.6) and (2.3.9). Owing to (3.2.4) we have therefore (3.2.8)

Xk+1

= eH-\{Xv

~ Ei,/,A*) + (1 -

9)Xk.

Written in this form, the Dirichlet-Neumann algorithm can be regarded as a successive under-relaxation method to solve (2.3.5), with the splitting Sh = £2 ,h — ( — On the other hand, using the identity — £ 1 ^ = £2,h ~ Sh, we can easily derive (3.2.1) from (3.2.8). At this stage, if 9 is properly chosen, in view of (2.3.14) we can infer that the error reduction factor at each iteration is C - 1

Thus, the convergence rate of Xk to up is independent of the grid size h (in the case of two subdomains). Remark 3.2.1 On the grounds of the previous considerations, the same terminology (Dirichlet-Neumann) is used to identify either the substructuring iterative procedure (3.1.1)—(3.1.3) for the solution of the finite element problem (2.1.1), or the preconditioner £ 2 i h, which can be used in the framework of any iterative method to solve the Steklov-Poincare equation (2.3.5). • Remark 3.2.2 In view of (3.2.8) it follows that, under the assumptions made in Remark 2.3.1, the Dirichlet-Neumann method converges exactly in two iterations if we take 9 = 1/2. • A similar property can be established for approach 2 considered in Section 1.3, the Neumann-Neumann method. Indeed, this iteration procedure can be regarded as a preconditioned iterative process like (3.2.1) using now a preconditioner Ph given by (3.2.9)

Ph =

(<71E-;,

+

<72

S - l ) -

1

,

for <71 > 0 and cr2 > 0. In fact, in the algebraic form, the Neumann-Neumann iteration reads (3.2.10)

Au^+1

= ft - AirXk,

A22uk+1

= f2 -

A2TXk

76

(3.2.11)

Oh.3

ITERATIVE DD METHODS AT THE DISCRETE LEVEL An An

fc+i

Air Afl)

uk+1 4! A k - f r + A n U j + 1 + A ^ A * + A r 2 u * + 1 + A{2) r r\

A22 3.2.12

Ur

V

Aor\

l.k+l xl>k2

J

{uk+1

2

.

2

fr - A n u ^ 1 -

- Ar2u2fc+1 -

A^A*

with (3.2.13)

= A* - 9 { a l f j . k + 1 - a 2 f i k + 1 ) .

Eliminating x/) k+1 and ip 2 + 1 from (3.2.11) and (3.2.12), and computing uj 4 " 1 and u * + 1 from (3.2.10), we find that (3.2.14)

= -XT

+ SFCA*

and (3.2.15)

=Xr~Zh\k,

therefore

(3.2.16)

A* + 1 = Xk + d i a ^ i x r =

Afc

+

+ a2S2"J; ( x r

+ a2^h){Xv ~

-

S h A & ).

The Neumann-Neumann preconditioner (3.2.9) can still be proved to be optimal in the case of two subdomains (see De Roeck and Le Tallec 1991 and Le Tallec 1994). We will give a proof of this result in Section 4.4. R e m a r k 3 . 2 . 3 The Neumann-Neumann method can also be interpreted as a Schwarz method, see Dryja and Widlund (1990). • Also the more general Agoshkov-Lebedev method (1.3.9)-(1.3.12) can be reformulated as an interface iterative process, making use of the Steklov-Poincare operator. Since Ph is symmetric and positive definite in both the Dirichlet-Neumann and Neumann-Neumann cases, conjugate gradient (rather than Richardson) iterations could be used, based on the same kind of preconditioner. These approaches can still be interpreted as subdomain iteration methods that differ slightly from (1.3.1)—(1.3.3) and (1.3.4)-(1.3.6) (see Quarteroni and Sacchi Landriani 1989 and Bourgat et al. 1989).

1

§3.3

bUrLUrl OUMfLUMKJNT FKliUUiNJJiTIUlNJiKS

77

Similar analogies between substructuring iterations and preconditioned Richardson iterations on the Schur complement system can also be established in the case of many subdomains. This will be pointed out in Section 3.3.2 for both Dirichlet-Neumann and Neumann-Neumann iterations. However, in this case, the spectral condition number is no longer independent of the number of subdomains, demanding the development of a coarse grid mechanism and new preconditioners.

3.3

Schur complement preconditioners

The Schur complement problem (2.3.5) (for decompositions with two subdomains), or (2.4.7) (for the case of many subdomains) allows in principle the direct computation of the vector u r of interface variables. However, the explicit calculation of the matrix E^ is expensive, because its components are much denser (albeit smaller) than the original stiffness matrix. Iterative substructuring methods aim at solving the Schur complement system by Krylov-type algorithms, without forming the different components E j ^ of E/, (see (2.3.10) and (2.4.11)), and using a parallel preconditioner. From (2.4.13) we see that the condition number of E^ blows up either when the grid size h tends to 0 (for a fixed number of subdomains), or when Hm tends to 0 (that is, the number of subdomains tends to infinity, even for a fixed grid size). Finding a preconditioner for £/,. is therefore recommended to obtain a convergence rate that depends as little as possible on the geometrical parameters h and H. In the next sections we will briefly review the most popular preconditioned; we will keep the discussion for the two subdomain case separate from that for many subdomains. In the former case, it is important to investigate the dependence of the condition number on h, while in the latter on H (or M, the number of subdomains). For a more extensive overview and for a discussion of the algorithmic aspects we refer, for example, to J. X u (1992), Chan and Mathew (1994a), Le Tallec (1994), Smith et al. (1996), J. X u and Zou (1998) and the references therein. 3.3.1

D e c o m p o s i t i o n with two subdomains

In the two subdomain case (2.3.5) the condition number of E^ grows proportionally to / i - 1 (see (2.3.13)). In very particular cases, much can be said about E^. For instance, as we noted in Remark 2.3.1, if we consider the Poisson problem (1.1) and use a uniform mesh and two equal-sized subdomains that are mirror images of each other (about the interface T), then E j ^ = E 2 ,h = Thus, both Ex /j and £2,/j are ideal preconditioners for £/j. Still for the same problem (1.1) on a rectangular domain 0 , discretised with a uniform mesh using piecewise-linear finite elements (or, equivalently, the standard five-point finite difference stencil), the Schur complement E/,, can be explicitly computed. Suppose that Q = ( 0 , 1 ) x (0,6), fti = (0,1) x ( 0 , a ) , = (0,1) x (a, b), and take the grid size h =

N1+1

and 7Vr(mi + m2 + 1) grid

78

ITERATIVE DD METHODS AT THE DISCRETELEVELOh.31

points, with (mi + 1 )h = a, + 1 )h = b — a. Then the Schur complement is given by (Bj0rstad and Widlund 1986) (3.3.1)

Eh =

FAF,

where F is the orthogonal sine transform

while A is the diagonal matrix whose elements are fl+Tp+1

1 + Tp+1\

/

—

where

The eigenvalues A j j of E/, satisfy c\h < Ajj < &2, and this estimate is sharp. Since the eigenvalues 5j,h of the discrete SteklovPoincare operator Sh are bounded as follows: czh1~dAjj

< Sj,h <

cih1~dAjj,

where d is the dimension of 0 (and d — 1 the dimension of T), it follows that estimate (2.2.8) is sharp. If mi and m 2 are large enough, then two good approximations of E/t are (3.3.2)

: = FMl'2F

and P h GM := F(M +

M2/4)^2F,

where M := diag(<7j). The preconditioner P® was proposed by Dryja (1982) in the functional setting of the trace space ff1>'2(r). Its improvement P ® M was proposed by Golub and Mayers (1984). Both preconditioners are spectrally equivalent to S^; that is, K,(P^ 1 T,f l ) is bounded independently of h. Moreover, using the Fast Sine Transform algorithm, the solution of the linear system associated with P® or requires O ( n l o g n ) operations. The weakness of these preconditioners is that they depend on the specific problem (1.1) and on the geometry of ft, and besides, the condition number depends on the aspect ratios of the subdomains (namely, on a and b in our example).

§3-3

SCHUR COMPLEMENT PRECONDITIONERS

79

The Probing preconditioner is a purely algebraic technique proposed by Chan and Resasco (1985), motivated by the observation that the decay of the offdiagonal terms of S^ is faster than the decay of the Green function of the original symmetric elliptic operator. Consequently, the idea is to compute efficiently a banded approximation (symmetric and tridiagonal) of E/, by probing the action of on a few carefully selected vectors (remember that E^ is never computed explicitly). For the effective computation of P ^ B see Keyes and Gropp (1987) and Chan and Mathew (1992). The analysis shows that K ( ( P ^ B ) _ 1 E / j ) ~ 0(h~lf/2), hence the Probing preconditioner is not spectrally equivalent to E/,. However, adapts pretty well to the aspect ratios of the subdomains and to the variations of the coefficients. Finally, let us recall two other preconditioned that are spectrally equivalent to Sh, and that have been introduced in Section 3.2: the Dirichlet-Neumann and the Neumann-Neumann preconditioners (3.3.3)

P ° N : = E 2 , h and P ™ : =

faE"*

+ ^E^)"1,

ai and <72 being two positive parameters. The preconditioners introduced above, being all related to a single interface, are often referred to as interface preconditioners. They can be used as building blocks for the construction of preconditioners for S^ in the case of many subdomains. 3.3.2

Decomposition with many subdomains

Consider now the case of a partition of fi into M > 2 non-overlapping subdomains. We retain the notation of Section 2.4, and look for preconditioners for the Schur complement system (2.4.7) (now related to the general symmetric elliptic operator L introduced in (1.4.2)). These preconditioners should have good parallel properties on arbitrary elliptic operators: ideally, they should be perfectly scaleable; that is, their performance should be insensitive to the number of subdomains. This, in general, will require us to include a coarse grid problem to allow for fast global communication and enhance the convergence rate dramatically. Let us start by considering the two-dimensional case. The use of many subdomains introduces vertex points and edges. The former are isolated points on the interfaces that are shared by more than two subdomains. The edges are the lines (typically made up of several finite element sides) that separate two adjoining subdomains (see Fig. 3.3.1). An edge does not include its end points. The global interface T is partitioned as a union of edges - f a , k = 1 , . . . , m } and vertex points {vj,j = 1 , . . . , n } . We reorder the unknowns on the interface T, listing first those lying on each edge, then those at vertices, so that we write Up = ( u e i , u e 2 , . . ., u y ) . Using deliberately a duplicity of notation and denoting by EK the indices of the nodes lying on the edge e*, and using V to denote the indices of the vertices, with the above reordering we obtain the following block partition of

80

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Oh.

3

FIG. 3.3.1. Edges and vertices for a many-domain decomposition.

(3.3.4)

Eft =

em

Eeiv E e2 V

VeT V m

Eyv

/ E e i ei rTeie2

WlV •

•

Note that Eyy is a diagonal matrix, and T,ekei = 0 if e^ and ei are not part of the same subdomain. With obvious notation we can write

<"-5>

M l ;

What follows is a short review of parallel preconditioners that are based on the partition (3.3.5). We follow closely the presentation given by Chan and Mathew (1994a) and Smith et al. (1996). A simple block-Jacobi preconditioner P£ is obtained from E^ by dropping all the couplings between different edges and between edges and vertex points. The result is a block diagonal preconditioner given by f t <>•«>

0

\

> « ' = ( . - s w ) - * - 4

/seiei o

0

...

0

\

» : : : c J '

where t e k e k is either E e f c e i or one of the interface preconditioners for the edge eu

1

§3-3

SCHUR COMPLEMENT PRECONDITIONERS

81

that we have introduced in Section 3.3.1. Also, S y v may be approximated; for instance, by the diagonal matrix A y y , the principal submatrix of A corresponding to the vertices. Since P'l does not involve global coupling between the subdomains, its spectral properties deteriorate as the number of subdomains increases. Indeed, there exists a positive constant C, independent of h and H, but possibly dependent on the coefficients of the operator L, such that (3.3.7)

k{{PI)-^u)
(l + log|j2

(see Bramble et al. 19866; Widlund 1988; and Dryja and Widlund 1994). The presence of the H~2 term can be heuristically justified as follows. Since the information is exchanged only among neighbouring substructures, the number of steps required by the conjugate gradient method to converge must necessarily be at least equal to the inverse of the diameter of fi;. On the other hand, the presence of l o g ( ^ ) can be regarded as being a measure of the maximum number of refinement steps that are required when going from any substructure (i.e. subdomain) made up of big triangles of diameter H to its finest refinement level (the triangles) of diameter h, by repeatedly cutting triangles in four. Another (more involved) argument stems from the consideration that the global preconditioner is made up of local edge preconditioners S ejcefc and of the vertex contributions S y v - The latter yield pointwise values that should be controlled in terms of the energy norm, which is possible at the expense of a logarithmic factor. Before generalizing Pfi in order to remove the H~2 factor in (3.3.7), let us describe it in terms of restriction and extension matrices. For each edge ek let R e k denote the pointwise restriction map from T onto the nodes on ; then Rj h denotes the corresponding extension map. Similarly, i?y denotes the pointwise restriction map onto the vertices V, and i?y denotes extension by 0 of nodal values on V to T. All these maps are expressed by rectangular matrices. With this notation, the block-Jacobi preconditioner Pjl satisfies (3.3.8)

m (PJ)-1 = J 2 R l ^ e k=1

k

Re

k

+ R ^

V

R V

A variant of Pfi is obtained by inserting in (3.3.8) some mechanism for global coupling, through a coarse grid problem based on a coarse triangulation; for instance, the one TH induces from the vertex points (here we are assuming that the subdomains define a coarse grid, therefore they are triangles or quadrilaterals), see Fig. 3.3.2. Let AH be the associated stiffness matrix, and let R]j denote an extension (for instance, piecewise-linear interpolation) from the nodal values on V °nto all the nodes of F. Correspondingly, RH is a weighted restriction map from r onto V. Then the modified preconditioner is defined as nia -P

82

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

0

Oh.

3

1

1

FIG. 3.3.2. The coarse grid. m (3.3.9)

( P f 3 ) - 1 :=

S . - ^

+R T h A~h 1 Rh.

k=l For this preconditioner, which was proposed by Bramble et al. (19866) with a specific choice of Y>ekek, the estimate (3.3.7) improves as follows:

(3.3.10)

^ ( P f ^ S ^ C ^ l

+ l o g ^

.

This estimate is sharp (in the sense that a similar lower bound holds, at least for H/h > 1; see Brenner 19986). In three dimensions, the condition number grows faster than H/h. The constant C, in addition to being independent of h and H, is also independent of the coefficients of the differential operator L if they are constant in each subdomain flj. From (3.3.10) we deduce that the number of iterations of a Krylov subspace method to achieve a given tolerance grows like log With the aim of removing this residual (albeit mild) dependence on ^ , additional coupling between the edges and vertex points is necessary. The Schur complement £/,, introduced in (3.3.4), is not block diagonal in the permutation { e i , e2, • • •, e m , V } , since, in general, T, ekej 0 whenever et and ej are edges of the same subdomain. Having ignored this coupling in both preconditioners Ph and P ® p s the result is the logarithmic growth factor in the condition number, see (3.3.7) and (3.3.10). Some overlap in the decomposition of interface T = (li™ = 1 {ek}) U (U" = 1 {wj}) is obtained by introducing vertex regions {ri,r2, • • • ,rn}. The j t h vertex region

§3-3

SCHUR COMPLEMENT PRECONDITIONERS

41

<>-

it

<•

41

<•

83

FIG. 3.3.3. The vertex regions. is defined as the cross-shaped region centred at the j t h vertex point v3 containing 'segments' of length 8H (0 < 6 < 1) of all the edges that emanate from Vj (see Fig. 3.3.3). Now let Rrj denote the restriction map that associates with full vectors the subvectors corresponding to the indices in the regions rj. Its transpose Rj. denotes the extension by 0 of subvectors with indices rj to full vectors. The principal submatrix of S/j corresponding to the indices r?- will be denoted by E r j ; then ST-J- = Rrj'S'hRjj • The Vertex Space preconditioner obtained from (3.3.9) as follows:

proposed by Smith (1990, 1992) is

n

(3.3.11)

(pvS)-L

: =

( P

BPS)-1

+

£ 3= 1

Since E r j is dense and expensive to compute, approximations based on the interface preconditioners of the previous section can still be used to alleviate the computational complexity of (3.3.11). . The vertex space preconditioner satisfies the following estimate: (3-3.12)

K((PrrlXh)


where C is independent of both h and H, but may depend on the variation of coefficients of the differential operator. If the coefficients of the operator L are constant in each Clit but have large jumps across the subdomain interfaces, the es timate becomes

84

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

(3.3.13)

« ( ( p V S ) - i S f t ) < C ( l + log

Oh.3

,

and now C is a constant, possibly depending on <5. For the proof of these results, see Smith (1992) and Dryja et al. (1994). The preconditioner can be interpreted as if it were generated by a decomposition of the reference finite element space Vh in overlapping subspaces (edge spaces and vertex spaces). As such, it can be regarded as a Schwarz preconditioner, and it was earlier used by Nepomnyaschikh (1984, 1986). In the three-dimensional case, it is possible to develop the extension of the vertex space preconditioner (see Dryja et al. 1994), or the so-called Wire-Basket preconditioner (see Bramble et al. 1989 and Smith 1991). In the latter case, the condition number satisfies (3.3.14)

< C (1 + l o g f )

KUP™)-1^)

,

no matter how large is the coefficient variation between subdomains. W e conclude this section by generalizing to the case of many subdomains the two interface preconditioners of Neumann-Neumann and Dirichlet-Neumann type, for both the two- and three-dimensional cases. Since the Schur complement is the sum of local elements (see (2.4.12)), and every local Schur complement Y*ith is easy to invert, then it is natural to precondition E^ by a weighted sum of inverses : M (3.3.15)

( p r r

1

:= E A ^ . S - ^ r J A , i=1

where D i is a diagonal weighting matrix with positive entries, which is often chosen in order to satisfy the partition of unity property YliLi = I , and i?r, is the restriction map from T onto F; : = T Pi <90,. The matrix is called the Neumann-Neumann preconditioner and generalises t o the case of many subdomains the one considered in Section 3.2 (see (3.2.9)). Note that, in the twodomain case, the restriction operator Hp, coincides with the identity, because

r = I\, t = l,2.

The action of

(3.3.16)

can be calculated without explicitly forming £;./-,., because

Sr£q=(0

'Mr

1

( J ) q,

which corresponds to solving a discrete harmonic problem in 0 , with a Neumann boundary condition on the interior interface Tj (and the homogeneous Dirichlet boundary condition on fl 90,;). If 0^ is an interior subdomain and the coefficient a 0 of the operator L is equal to zero, it is known that E ^ is a singular matrix (because the pure Neumann

1

SCHUR COMPLEMENT PRECONDITIONERS

§3-3

85

problem requires a compatibility condition and lacks uniqueness of the solution). We have therefore to use in (3.3.15) a regularised inverse instead of This can be the Schur complement of the modified local stiffness matrix Ai + a l , or the (more expensive to compute) Moore-Penrose pseudo-inverse. Otherwise, we can solve a linear system in the complement of the null space (see Le Tallec 1994). Concerning the spectral properties of P ™ one has (in both two and three dimensions) (3.3.17)

KKPH

- 1

^ ) < O H ' 2 ( l + log

2,

which is not better than (3.3.7) (see De Roeck and Le Tallec 1991). However, this preconditioner is much less sensitive to coefficient variations. The presence of the H~2 term is the symptom of the need for a global coupling mechanism. The balancing Neumann-Neumann preconditioner proposed by Mandel (1993) is a Neumann-Neumann preconditioner with the addition of a simple coarse grid correction constructed by using a piecewise-constant coarse grid space. The corresponding operator AH involves one unknown per subdomain in the scalar case. With obvious notation the balanced preconditioner is defined as follows: (3 3 18)

(1

^N,6)"1

:=

[(/

- W W I O x (I - ShR^A^Er)

"

1

+

+ HZAjRr] Rf.A^Rr.

This new preconditioner satisfies a spectral estimate of the same form as (3.3.10) in both two and three dimensions. A nice feature of the balancing Neumann-Neumann approach is that it allows the use of completely unstructured subdomains. Indeed, using piecewise-constant solvers, the subdomains need not form a coarse triangulation and may have an arbitrary form (for example, as in Fig. 3.3.1). This method is called balancing because the corresponding iterative algorithm is a three-step method A n+l/3 =

A„ +

A n+2/3 =

a„+1/3 +

An+1

a„+2/3 +

=

r

t

A

-l

R r { x

_

EfcAn}

(pNN)-l(x _ r

t

a

- 1Rt{x _

^A"^

3

)

SfcA»+2/3)i

and the calculation of A n + 1 / 3 and A n + 1 balances the average of A on each subdomain. Concerning the Dirichlet-Neumann iterative substructuring method introduced in Section 1.3 for the differential boundary value problem, it can be formulated at the algebraic level as follows. We use a superscript B (black) and ^ (white) to denote the colour of the subdomain. Then the Schur complement

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

86

Oh. 3

matrix (2.4.12) can be split as

i6/fl

i£lw

and the Dirichlet-Neumann (3.3.19)

preconditioner is defined through

(PD-

1

:= E i£lw

( ^ Y ^ r ^ r V -

Also for this preconditioner it is possible to include a global coarse problem, see Dryja (1988). As we have seen, a critical issue is how to obtain optimality of preconditioners when partitioning the domain into many substructures. At this stage, a multilevel approach (that makes use also of a coarse domain partition) needs to be pursued, in order to ensure a fast propagation of information among subdomains, even in the case of local grid refinement. This feature is shared by iterative substructuring methods and Schwarz methods, and will be further addressed in Section 3.6.

3.4

The Schwarz method for finite elements

The Schwarz iterative methods introduced in Section 1.5 can be straightforwardly adapted to the solution of the discrete problem (2.1.1), rewritten for the general symmetric bilinear form

a*(w,v)

:=

I Jn

I aijDjwDiv W=i

+ dp w v

Using the notations of Section 2.2 and following (1.5.8), the alternating (multiplicative) Schwarz method at the discrete level reads as follows. Given u°h £ Vh, solve for each k > 0 w I,h

e

Kh

uh+1/2 w2,h

• Oi«ih,wi,h)

= (f,vlih)0l

- aKu^v^h)

V vhh

= (f,v2,h)n2

~ a*2{uk^1/2, v2)h)

£ V£h

=uh+wi,h e V°h

: a*2(wlh,v2,h)

V v2,h £ V°h

,k+1 _ k+i/2 , k +W2,h> ~Uh

ih

where h is the finite element function that extends w k h by 0 in Q \ Oj, and a* denotes the restriction of a* to flj. Similarly, we can define the additive Schwarz method by restating (1.5.9) at the finite element level.

1

§3.4

THE SCHWARZ METHOD FOR FINITE ELEMENTS

87

FIG. 3.4.1. Overlapping partitioning of the domain ft with more than one layer of overlap.

FIG. 3.4.2. Overlapping partitioning of the domain ft with only one layer of overlap. We now reformulate these methods in algebraic terms. As we have seen in the previous chapter, the finite element approximation (2.1.1) (with a*(-, •) instead of o(-, •)) of the original boundary value problem (1.4.1) yields the algebraic system (2.1.13), where Aij : = a*(ipj,tpi). Corresponding to the overlapping subregions fti and ft2 (see Fig. 3.4.1), let 7i a nd / 2 denote the indices of the nodes in the interior of fti and ft2, respectively. Clearly, if N^ denotes the number of internal nodes of ft, and I the set of all mdices from 1 to N^, then we have that and / 2 form an overlapping partition / , i.e. I\ U / 2 = / , I\ H / 2 0. If ni and n 2 indicate the number of indices l n I\ and J 2 , respectively, due to overlap ri\ + n 2 > N^. However, note that for partitions with only one layer of overlap (see Fig. 3.4.2), i i n J2 = 0 and ni+n2 = Nh. Order now the indices in such a way that those corresponding to the nodes

88

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Oh.3

1

A A=

A2

FIG. 3.4.3. Partitioning of the stiffness matrix. internal to fli but not internal to Q 2 are first, followed by those corresponding to the nodes internal to f2i n O2, and finally we take the remaining ones. Let A\ and Ai denote the principal submatrices of A formed by the first n\ rows and columns and the last n 2 rows and columns, respectively (see Fig. 3.4.3). Then A\ is the stiffness matrix for the subdomain f l i , and A2 that of fl2. These matrices are clearly symmetric and positive definite. Moreover, they are related to the global stiffness matrix A by the algebraic relations (3.4.1)

Ax = RiAR^,

A2 =

R2ARj.

Here Rf and Rt, i — 1,2, are extension and restriction matrices, respectively. More precisely, Rj is a Nh x nt rectangular matrix whose action extends by 0 a vector of nodal values in Therefore, given a subvector v1 of length n l of nodal values of a function Vith £ V°h, we have for j £ Ii for j £ l \ I i

'

In other words, R j is the matrix whose first ri\ rows and columns form the identity matrix, whereas the entries of the last N^ — m rows are all 0 (see Fig- 5 3.4.4).

The transpose Ri of Rf is a restriction matrix whose action restricts a full vector v £ TLNh to a vector of length n» by preserving the entries with indices belonging to /,;. Thus, i?;v is the subvector of nodal values of v in the interior of S The algebraic form of the alternating (multiplicative) Schwarz method follows immediately from (1.5.8) upon replacing the operators with the corresponding matrices:

§3.4

THE SCHWARZ METHOD FOR FINITE ELEMENTS

89

FIG. 3.4.4. The extension matrices. .fe+l/2 = uk + RjA^R^t

(3.4.2)

uk+1

=

uk+1/2

+

-

Auk) _ Aufc+1/2).

RTA-lR2(J

Replacing f with A u , these equations can be rewritten as

(3.4.3)

u k+1/2 _ u& + P i ( u - u * ) = { I - Pi)nk Uk+1

uk+1/2

+

p2(u _

u*+l/2) =

+ PlU

( / _ p2)ufe+l/2

+

p2U)

where we have introduced the discrete projection operators (3.4.4)

Pi : = RfA^RiA,

i =

1,2.

Note the formal analogy of (3.4.2), (3.4.3) and (3.4.4) with (1.5.15)-(1.5.16), (1.5.12) and (1.5.31), respectively. L e m m a 3 . 4 . 1 The matrices Pi are symmetric respect to the A-scalar product (3.4.5)

( w , v ) A := ( A w , v )

and non-negative

Vw,veR"

k

definite

with

,

which is induced by the symmetric and positive definite stiffness matrix A. Moreover, Pi is the orthogonal projection in the A-scalar product onto the subspace spanned by the rows of Rf. Proof

We have (PiW,v)

(3.4.6)

A

= (AP;w,v) = (RjA~xRiAw, =

Av)

(Aw.PfAr^Av)

= ( A w , Piv) = ( w , PiV)A and

V v, w e H N h

90

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Oh.

3

(PiV,v)A = (APiV.v) (3.4.7)

=(RTAr1RiAv,Av) =

(A-LRIAV,RIA-v) > 0

V v G

RNh.

Moreover, ( P l V , i ? f w ) A = (APiv,Rf =

(3.4.8)

w)

(.Av,RjA~lRiARjw)

=

Rfw)

= {v,Rj

w)A

Vv.weR"',

which completes the proof.

•

Note also that, if we set (3.4.9)

Qi := RjAr^Ri

i = 1,2,

= PiA~\

then the correction ek := Qi(f -

Auk)

is such that P ( u — = ek\ that is, e^ is the closest vector to the error u — in the subspace spanned by the rows of Rf. In a compact form the multiplicative Schwarz method reads uk+x (3'

'10)

=uk

+

=uk

+

(Q,+Q2-Q2AQ1)(f-Au.k) [I-(I-P2)(I-P1)}A~1(f-AvLk).

Similarly, the additive Schwarz method becomes (3.4.11)

U f c + 1 = U * + (Qi + Q 2 ) ( f - A U * ) .

The generalisation to M subdomains, M > 2, is straightforward. Setting Pi := RfA-'RiA,

Qi := P ^ " 1 ,

i = 1 , . . . , M,

the multiplicative Schwarz method becomes (3.4.12)

Uk+ii

= (I - Pi)uk+T + PjU K . ' = u * + V + RfA^Riif - Auk+1*r),

i =

and the additive Schwarz method reads / Ufc+1 = ( I (3.4.13)

V =

Ufe+

M P i

\

,=i M J ]TQi)

M Rj AT1 Rif <= i (f-AUk).

l,...,M,

1

ACCELERATION OF THE SCHWARZ METHOD

§3.5

91

The error equation for the multiplicative case takes the form (3.4.14)

u - u f c + 1 = (7 - PM)

• • • (I - P i ) ( u - u*),

and

(3.4.15)

u - U * * ^

( j - 5 > ) i=1

(u-U*)

for the additive case, in analogy with the differential case (see Section 1.5.6). Note that the matrix Qa := Y^^Li P* is symmetric and positive definite with respect to the same scalar product (3.4.5), because M (<2aV,

v)A =

V) =

^(A^RiAv, 2—1

RiAv)

> 0

Vv G

and the equality is achieved if and only if RiAv = 0 for each i = 1 , . . . , M; that is, if and only if v = 0. On the other hand, each matrix Qi is clearly symmetric and, by following the argument just employed, the matrix ^ Qi is symmetric and positive definite (with respect to the Euclidean scalar product). R e m a r k 3 . 4 . 2 (Schwarz iterations and parallelism) In contrast with the additive Schwarz method, the multiplicative method has very little potential for parallel implementation. However, since we may expect that many subdomains do not share any common grid point, a strategy of subdomain colouring can be adopted to allow a simultaneous and independent update of subsets of equations by (3.4.12) (see Fig. 3.4.5 for an example, and Smith et al. 1996, pp. 19-24, for a thorough discussion of this technique). • R e m a r k 3 . 4 . 3 In special situations there is a close relationship between the additive Schwarz method and the Dirichlet-Neumann method (see, for example, Smith et al. 1996, pp. 115-6). •

3.5

Acceleration of the Schwarz method

We note that the multiplicative Schwarz method (3.4.2) coincides with a generalised block-Gauss-Seidel iteration on system (2.1.13) (using overlapping blocks). To prove it, we generalise the notation of Section 2.4 to the case of overlapping subdomains. For i = 1, 2 we define the rectangular matrix (3-5-1)

V° = (Ai

Bi),

which represents the discrete form of the differential operator L restricted to ft,. The component Ai concerns the coupling between the interior nodes, while Bi

92

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Oh. 3

FIG. 3.4.5. Subdomain colouring (with four colours). represents the coupling between the interior nodes and the nodes that lie on the artificial boundary F.t. The matrix V'- has one row for each interior node and a column for every node (including those on F,;). Correspondingly, let us denote by u, and u r ; the values of the unknown solution at the interior nodes of Q, and at the nodes on I\, respectively. The multiplicative Schwarz method (3.4.2) can be written as (3.5.2)

(Ai

B1

,fc+l Ufc+1

2 7-( c+1 — u ffc+1 T )„fc U i. ri

-— II) fi

and (3.5.3)

[M

B2)

fc+i fc+1

2

=

£2,

U

fc+1

where denotes a discrete linear operator that interpolates values from the nodes in the interior of 0 j to the nodes on the interface Fj, for i = 1, j — 2 or i = 2, j = 1. Remark 3.5.1 There are cases where the discretisation of fi is non-conforming; that is, the triangulations of the two subdomains, say and 7 ^ , do not coincide in the overlapping region. In this situation, the Schwarz method can still be formulated through (3.5.2) and (3.5.3). Since this time the nodes of on Ti are not necessarily nodes of 7^ , 7p2' is now a discrete operator that interpolates values from the nodes of 7^2 to the nodes of T^ on Fi defined similarly. From (3.5.2) and (3.5.3) we obtain

The operator

is

•

1

§3-5

AUUELUKATIUJN UF THK S(J±iWAK.Z MK'i'UUU = t -

(% 5 4)

AA2 u 2

1

93

Bili2)ui

- rf 2 - £>2-
This is precisely a block-Gauss-Seidel iteration for the linear system (3.5.5)

Binr 2I£> 2

A2

W )

U

VVe also note that (3.5.4) coincides with the classical block-Gauss-Seidel method in the case of a one-layer overlap (in such a way that the internal nodes of each subdomain ft* are not internal nodes of any other subdomain, see Fig. 3.4.2). System (3.5.5) is not symmetric even if the two submatrices A\ and A2 are symmetric; in fact, in general, B i ij^J 7= ( B 2 I ^ J ) T . However, the multiplicative Schwarz method can be easily symmetrised by including a third step

(3.5.6)

uk+1/3

= n

k

uk+2'z

= u * + 1 / 3 + RTA2lR2{f

ufc+1

=

+

RTA-IRI<J

uk+2/3

+

_

RTA-lRl(f

Aukj

-

Auk+1/3)

_ ^u F E + 2 /3).

This can also be written as a one-step method: u * + 1 = u<= + ( 2 Q i +Q2

5

-

Q2AQX

+Q1AQ2AQ1){f

-

QXAQ2

-

-

QIAQI

Auk).

Similar equivalence can be stated between the additive Schwarz method and a generalised block-Jacobi iteration method (see, for example, Smith et al. 1996, P. 14). Let us now introduce the matrix (3.5.8)

P a s : = (RjA^1R1

+ RjA^R^'1

= (Q, +

Q2)~\

and note that (3.5.9)

P-1A

=

P1+P2.

Then the additive Schwarz method can be regarded as a fixed-point method for the following system (equivalent to (2.1.13)): (3.5.10)

P~1Au

=

P~1f.

Clearly, P a s plays the role of a preconditioner, and it is given the name of additive Schwarz preconditioner. When the Schwarz method is used as a preconditioner, each application of the preconditioner is one iteration of the alternating Schwarz method with a vanishing initial guess.

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

94

Oh. 3

In the case of M subdomains, the preconditioner takes the following form:

C

M

\

/ M

\

and the preconditioned matrix becomes M (3.5.12)

P^A =

Y,Pi=Q*i= 1

As suggested by (3.4.13), the additive Schwarz method is simply a Richardson method for (2.1.13) with preconditioner P a s for A. However, since P a s is symmetric and positive definite, the preconditioned system can be more effectively accelerated by the conjugate gradient ( C G ) method, which converges faster than the Richardson method does. The C G iteration for system (2.1.13) preconditioned by Pas reads: assign u°, set r° : = f — Au°, p° : = z° : = P a n d for k > 0 solve

ak uk+1 r M-l

(3.5.13)

(zk,rk)

—

(pk,Apk) = uk + akpk = rh -

akApk

=

k+1

asP^T

fc+l) z f c + l j rr.fc+1 ...fc+1 Pk+1

(zk,Tk)

PFC+1 : = z fc+i

+pk+lPK

This method involves the same matrix operations as the original Schwarz method, namely the solution of the local subproblems associated with the local submatrices Ait for i = 1,... ,M. Clearly, each subdomain solve can be carried out concurrently. The convergence rate of the new sequence is as follows (see, for example, Golub and Van Loan 1989)

fc (3.5.14)

||U*-U1U<2|V

!K(P~1 A) - 1 U(P^A)

]

||U°-U|

+ 1,

where ||v||^ = y/(v, v ) ^ is the norm associated with the scalar product (v, W)A introduced in (3.4.5). As for the condition number K(P~S1 A), denoting as usual by H the maximum diameter of the subdomains 0 ; , i = 1 , . . . , M , and by SH the linear measure

1

ACCELERATION OF THE SCHWARZ METHOD

§3.5

95

of the overlapping region between two adjacent subdomains (0 < 5 < 1), the following estimate holds: <

C ^ ,

where the constant C possibly depends on the coefficients of the symmetric elliptic operator L (see Dryja and Widlund 1989, 1992). If a coarse grid correction is added to (3.5.11) (see (3.6.11) below), the estimate on the condition number is improved to K(P-JA)

<

CS-\

This estimate is sharp, in the sense that the condition number also satisfies «(PCas A) > C 5 _ 1 for (5 = 0(h), for both two- and three-dimensional cases (see Brenner 19986). When the coefficients of the operator L are constant in each coarse grid element, but have large jumps across interfaces, the estimate above has to be substituted by K(P~J. A)

< C

K(P~^A)


+ log

(in t w o d i m e n s i o n s )

JJ

(in three dimensions),

where C does not depend on the coefficients, but may possibly depend on the parameter <5 (see Dryja et al. 1994). A similar interpretation can be given for the multiplicative Schwarz method. This time, the multiplicative Schwarz preconditioner takes the following form: (3.5.15)

PMS

-

A[I

- ( I - P

M

) . - . ( I - Px)]"1,

which can also be written as M 1P ms

— —

i=l

Qi - Y1 j>i

Qi+

l>j>i

••• + ( - 1

QiAQjAQi

)M'1QMA---Q2AQ1

Since this is no longer symmetric, rather than using CG iterations we should use GMRES or other Krylov procedures that are suitable for non-symmetric matrices (see, for example, Saad 1996). R e m a r k 3.5.2 The restriction operator Ri can be regarded as being the sum of two terms Ri = R^ + R^, where accounts only for the nodes inside the domain excluding the overlapping region, while RI for those in the overlapping region.

96

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Oh.3

A new preconditioner has been introduced recently by Cai and Sarkis (1997), which is defined as follows: ( M PL

=

\i=l

and is called the restricted additive Schwarz preconditioner. The main motivation for designing PI s is that, in a parallel implementation, substantial communication cost can be saved because computing R(-v does not involve any data exchange with the neighbouring processors. • 3.5.1

Inexact solvers

If the Schwarz method is used as a preconditioner of an acceleration scheme, it is not necessary for the local subproblems to be solved exactly. In other words, in the preconditioning step of the GG iterations (3.5.13), P , ' 1 could be replaced by the following approximation: M P^1

—

^RjA^Ri, i=1

where Ai is a convenient approximation of A.t on the domain ft,;. A practical implementation of this idea consists of replacing the bilinear form a*(-, •) in f2j by a simpler one a*(-, •) (for example, freezing to a constant the coefficients in flj if the original elliptic problem has variable coefficients, or else by approximating A{ by an inexact factorisation). In any case, if a*(-, •) and •) are spectrally equivalent-, that is, there exist two positive constants K\ and K2 such that K{a*(Vi,Vi)

< a* {vi,

< K2a* (vi,Vi)

V vt G V*,

then the corresponding preconditioned system - P ^ A retains the same spectral properties of P ^ A , and the 'inexact' Schwarz algorithm converges with the same rate of the exact one.

3.6

Two-level methods

The convergence rate of the preconditioned iterative domain decomposition methods deteriorates when the number of subdomains becomes large. As has already been pointed out, this is due to the fact that in Schwarz algorithms, as well as in substructuring iterations, information is exchanged only between neighbouring subdomains. This weakness of the method has been overcome in previous sections; for instance, by introducing a coarse global problem set over the whole domain in order to guarantee a mechanism of global communication among all subdomains.

1

TWO-LEVEL METHODS

§3.6

97

This is not the only way, though; indeed, the coarse grid solver can be regarded as a special case of a two-level method, which is based on the simultaneous use of a 'fine' problem (the original one) and an auxiliary 'coarse' problem. 3.6.1

Abstract setting of two-level methods

Two-level methods involve the smoothing of the original problem and the solution (or preconditioning) of an auxiliary problem on a related mesh that is coarser than the original one. They can be thought of as an additive version of the general two-level multigrid algorithm. On a given grid, the preconditioner for the original problem is obtained as a superposition of the solution of an auxiliary problem on a related grid and a smoother on the original grid. The auxiliary problem should be simpler to solve, which is the case if it originates from a coarser grid or a numerical method simpler than the one on the original grid (for example, it uses piecewise polynomials of lower degree, or involves averaging the coefficients of the original operator). To formulate our problem in an abstract setting, we consider two finite dimensional problems, the original one (3.6.1)

find

uh G Vh

: ah(uh,vh)

V vh G Vh,

= Th{vh)

and the auxiliary one (3.6.2)

find

uH G VH

• CIH{UH,VH) = FH(Vh)

V vH G VH,

where A/,(•, •) and O,H(•, •) denote suitable bilinear forms on VH and VH, respectively. Given a basis
,

(3.6.3)

(

A

h ) s j

ah(ipj,ips)

(•AH)ml •= a H (ipi,il>m)-

The goal is t o find a preconditioner for Ah using either AH or a g o o d preconditioner of AH • The two spaces Vh and VH are related by an operator // t : VH —> Vh. Typically, Ih is the linear interpolation from the coarse grid to the fine grid and its matrix representation is RT (see Section 3.4). In these cases, h and H are the maximum diameter of the elements of the triangulations %, and TH , respectively. T w o different adjoint operators can be associated with Ih-

(3.6.4)

and

f Ih - - V h - ^ V H : 1 I (llwh,vH)vH = (Wh,IhVH)v h

v wh& Vh,vH

€ VH

98

(3.6.5)

Oh.3

ITERATIVE DD METHODS AT THE DISCRETE LEVEL f Jl --V^VH I [ aH(Jlwh,vH)

: = ah(wh,

V wh G Vh,vH

JhvH)

G VH-

In matrix form, keeping the same notation, this last definition reads AHJ% f £ A h , yielding = A'^ljAhA preconditioner Ph for Af, can be constructed in the following way: (3.6.6)

PH1

••= Q~H + QI1,

=

Q~H-=HA-£LZ,

where Qh is any convenient symmetric positive definite matrix, much simpler than Ah itself. The preconditioned matrix becomes p-'Ah^hJl

+

Qh'Ah.

If Qh is a g o o d preconditioner for Ah, Ph remains a good preconditioner for Ah even replacing in (3.6.6) A^1 with a good preconditioner on the coarse grid. In a two-level multi-grid context, Qh is the smoothing operator on the fine grid (in general, a few Jacobi iterations to damp the high frequencies of the error), while A ^ 1 is a coarse grid solution operator. In general, it is not enough to use Qjj 1 alone as the preconditioner, because it has a large null space. As a matter of fact, the rank of Q^1 is equal to the dimension of AH, which is lower than the dimension of AH- Therefore, any component of the residual lying in the null space of Qj/ would never be corrected. For this reason, Q ^ 1 must have full rank. For an elliptic partial differential equation, QH is designed to account for the long-range effects, Qh for the local ones. At the lowest level, Qh coincides with the diagonal part of the stiffness matrix Ah- In the domain decomposition context, the application of Qf1 will involve subdomain solves. 3.6.2

Multiplicative and additive two-level preconditioners

Define the vector f as f j : = a two-step preconditioner containing both the long-range and local-range components, QH and QH, respectively, is QH(U"

Qh(un+1

+ 1

/2_U")

= f - A f t U "

- uN+1/2) = f _

Ahun+1/2.

It can be written as a one-step method: (3.6.8)

u n + 1 = u " + {QH 1 + Q ^ 1 -

AhQ-£)(f

- Au").

This is the multiplicative two-level method. Its corresponding additive form, the additive two-level method, replaces (3.6.8) with the following equation: (3.6.9)

un+1 = u " + ( g ^ 1 + Q f t 1 ) ( f - A u n ) .

1

TWO-LEVEL METHODS

§3.6

99

This is one step of the Richardson method for the fine grid problem AHu = f with preconditioner PH (see (3.6.6)). It can be made more efficient when it is used as part of a Krylov subspace accelerator; namely, if the same preconditioner PH is used within a G M R E S or a conjugate gradient iterative procedure. Parallel multi-level preconditioners have been developed by Bramble et al. (1990); see also Zhang (1992) and Griebel (1994). 3.6.3

T h e case o f the Schwarz m e t h o d

When Qh is either a multiplicative or an additive overlapping Schwarz preconditioner (QH = P m s or Qh — Pas, see Section 3.5), we say that (3.6.8) or (3.6.9) are a two-level Schwarz methods. The generation of a coarse grid for the Schwarz method requires extra care with respect to the case of non-overlapping partitions. A commonly used technique for constructing an overlapping decomposition of fi into M subdomains Oi, • • •, fiju assumes that a non-overlapping partition • • •,LOM of CI is available. One possibility is to choose each subregion Wi as an element from a coarse finite element triangulation TH of fi of size H. Next, each UJ1 is extended to a larger domain fij, consisting of all points in fi at a distance not larger than SH from Wi, with 0 < 5 < 1. The restriction and extension maps, Ri and RF, as well as the local matrices Ai, are defined accordingly. Assume now that the finite element triangulation Th is a refinement of the coarse grid partition TH- Accordingly, let us denote by RJQ the interpolation map of coarse grid functions to fine grid functions. When using piecewise-linear elements, R j j interpolates the nodal values from the coarse grid vertices (of co,J to all the vertices of the fine grid. Its transpose RH is a weighted restriction map. Correspondingly, let AH denote the stiffness matrix of our elliptic problem on t h e c o a r s e m e s h TH, i.e. AH

=

RHAR

In the additive case, we have M (3.6.10)

Q^1

= Y,RJAJ1RI,

Q-J

=

RthA„1RH,

where the index i refers to the zth subdomain, i — 1 ,...,M, and the index H refers to the coarse grid (where subdomains play the role of elements). The resulting preconditioner is therefore

(3.6.11) where we have set, for notational convenience, RQ := RH and AO := AH- The multiplicative preconditioner is obtained similarly. 3.6.4

Convergence of two-level m e t h o d s

Let us assume that the two grids underlying problems (3.6.1) and (3.6.2), say TH and TH, are comparable, so that there exist two positive constants CO and C\

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

100

such that for any KH £ TH and KH £ TH with KH H KH (3.6.12)

Oh. 3

0 it holds that

C0 diam KH < diam Kh < G\ diam KH-

If the two grids are quasi-uniform and of comparable size (i.e. Co, Ci — 1), using either a block-Jacobi or a symmetric block-Gauss-Seidel iteration as a smoother, the preconditioner Ph is uniformly optimal for the operator Ah (the spectrum of P^xAh is uniformly bounded with respect to h). If, instead, TH is genuinely coarser than Th, which means that there are triangles KH and KH, KH n KH 0, for which (3.6.12) holds only for extremely small Co, then the above smoothers are no longer sufficient to guarantee that Ph is uniformly optimal for Ah- However, if the smoothing operator is based on the overlapping Schwarz methods (Qh given by the symmetric multiplicative Schwarz preconditioner, see (3.5.7), or Qh = Pas, see (3.5.8)), then P^1 is uniformly optimal, provided the coarse grid TH is 'comparable' with the subdomain partition. This means that there should exist positive constants Co and C\ such that for each KH £ TH such that KH D fij ^ 0 it holds that Co diam KH < diam 0 , < C\ diam KHFor the proofs of these results, see Bramble et al. (1996), and the references therein. In particular, we point out that the spectrum is bounded independently of h, H and the linear measure of the overlapping region, provided the latter is kept proportional to H, say, given by 5H. The convergence of the iterative procedures is poor for very small values of but improves rapidly as the overlap increases. Finally, it is worthwhile mentioning that the number of iterations for the symmetric multiplicative Schwarz method is roughly half of that needed for the additive Schwarz method. For numerical evidence see Smith et al. (1996); we refer to the same monograph for the analysis of multi-level methods with more than two levels. Remark 3.6.1 The generation of coarse grid preconditioners on unstructured grids is a difficult task, especially for three-dimensional problems. An algebraic way, inspired by agglomeration multi-grid techniques, is often adopted. A discussion can be found in Chan and Mathew (1994a); Bank and J. Xu (1995); and Chan and Smith (1995). See also Smith et al. (1996), Section 2.6 and the s references therein. •

3.7

Direct Galerkin approximation of the Steklov-Poincare equation

All the methods described in Section 1.3 share the property of being derived by a suitable splitting of the given boundary value problem over the subdomains. Thus, they are amenable to iterative procedures that, at each step, require the solution of at least as many independent boundary value problems as the number of subdomains. On the other hand, from a merely speculative point of view,

1

§3.7

DIRECT GALERKIN APPROXIMATION OF THE SP EQUATION

101

these iterative processes on the primitive variables Ui = (u being the solution of the given boundary value problem) can be interpreted as suitable iterative schemes for the dual variable X = u\r, which is the solution of the interface equation (1.1.7). More precisely, the iterative procedure on the primitive variables U{ induces a preconditioned iterative procedure on A. The same interpretation clearly holds at the finite dimensional level, which has been dealt with in Section 3.1. A suitable approximation is introduced first for the given boundary value problem (for example, a Galerkin finite element approximation), then an appropriate domain decomposition iterative procedure is chosen for the discrete primitive variables u^/j = U^Q., and the latter can be regarded as an iterative scheme for the dual variable Ah = Uh\T, which in turn is the solution of the discrete interface equation (2.2.3). It has to be pointed out, however, that the driving mechanism of the domain decomposition procedure is the subdomain iterative method acting on the discrete primitive variables. A different approach consists of attacking directly the interface equation (1.1.7) by a Galerkin method on a suitable subspace Ah of the space of traces A. More precisely, we start by reformulating (1.1.7) in a variational way as follows: (3.7.1)

find

A € A : S(X,p)

= (x,fA

V / i £ A,

where S(r],fj,) := (Sr],p) is the bilinear form associated with S, which is symmetric and coercive in A. Then we associate with (3.7.1) the following internal Galerkin approximation (3.7.2)

find

Aft £ A h : S{Xh,ph)

= <Xh,A»*>

V ph 6 Ah,

where \h is a convenient approximation to the right-hand side x of (1.1.7). The latter problem is completely defined after having chosen the finite dimensional subspace Ah of A. Obviously, (3.7.2) yields an algebraic problem that is symmetric and positive definite. This approach has been introduced by Agoshkov and Ovtchinnikov (1994), and has been given the name of Projection Decomposition method (PDM). Its interest relies mainly on the possibility of constructing well-conditioned, piecewisepolynomial bases for the space A^, an option that has been successively pursued by Ovtchinnikov. In a series of papers (Ovtchinnikov 1993, 1995; Gervasio et al. 1997; and Xantis and Ovtchinnikov 1994) it has been proved how to obtain these bases in a fast and accurate way for both Laplace and Stokes operators (through a Gram-Schmidt orthogonalisation procedure). The good conditioning of these bases makes it possible to solve the algebraic problem by a non-preconditioned conjugate gradient method. At each step, independent boundary value problems have to be solved in each subdomain. The accuracy with which these piecewise-polynomial functions are obtained allows us to maintain the order of accuracy of the solution of local problems in the subdomains, even when high-order methods are adopted therein. In several

102

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Ch. 3

instances, the P D M has been seen to be very effective compared with other domain decomposition methods.

4 CONVERGENCE ANALYSIS FOR ITERATIVE DOMAIN DECOMPOSITION ALGORITHMS In this chapter we present some abstract theorems that are useful for proving the convergence of iteration-by-subdomain methods. The main part of our analysis will cover the case of substructuring procedures (for disjoint subdomains). The convergence analysis of the Schwarz method for overlapping partitions will be addressed in the last section of this chapter. For a complete analysis of Schwarz methods we refer to Dryja and Widlund (1990, 1995), J. Xu (1992) and Smith et al. (1996). The chapter is organised as follows. We begin in Section 4.1 by providing some extension theorems in different function spaces H(div; fi,), i?(rot; fij)), as well as in the corresponding finite element subspaces. As outlined in the previous chapters, the local extension operators represent the basic mathematical ingredients of the Steklov-Poincare operator. On the other hand, iterative substructuring methods are amenable to preconditioned iterative algorithms on the Steklov-Poincare equation. For this reason, in Section 4.2 we focus on operators in Hilbert spaces, given in split form, and consider preconditioners whose inverses are made up of suitable combinations of inverses of suboperators. Then we provide some abstract convergence theorems for iterative methods of a preconditioned Krylov type in this framework. The analysis is also particularised to the algebraic finite dimensional counterpart of our equations. As of our paradigmatic elliptic boundary value problem, we address its convergence analysis in this chapter, immediately after having carried out the abstract analysis of Section 4.2. Since we also cover the non-symmetric and complex cases, our abstract results will be applied (through the following chapters) to a wide class of boundary value problems. In particular, Theorems 4.2.2 and 4.2.5 can be applied to Dirichlet-Neumann and Neumann-Neumann iterations, respectively, for a wide variety of situations, including general elliptic problems, the elasticity operator, the Stokes problem, and its generalisations, as well as for advection-diffusion equations. The first theorem will also be useful for some of the heterogeneous domain decomposition procedures that we will describe in Chapter 8. Theorems 4.2.10 and 4.2.13 (and Corollaries 4.2.11 and 4.2.14) are concerned with iterations for complex matrices, with a non-symmetric preconditioner, and can be applied to the Dirichlet-Neumann and Neumann-Neumann iterative scheme, respectively, for situations like those arising with the time-harmonic Maxwell equations.

104

CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

The proof of the convergence of the Robin iterative substructuring method is based on an ad hoc argument and is presented in Section 4.5.

4.1

Extension theorems and spectrally equivalent operators

As anticipated in the previous chapters, typical domain decomposition preconditioners are expressed in terms of the sum of operators, each one related to a certain subdomain ft; of the (bounded) computational domain ft C R d , d = 2,3. This is the case for the Steklov-Poincare operator S for the Laplace operator (or, similarly, for the symmetric elliptic operator L introduced in (1.4.2)), which is defined as M {.Sr),n) =

Y^iSil'V)

(4.1.1) {Si-q, y) = a,i (Hn/, Hifj.) = / VHiri • VHi/j,, JQi where rj and fj. are trace functions on the interface F, and Hi denotes the harmonic extension operator from T into ft, (see Section 1.2). The preconditioners are constructed by assembling the local operators Si. To analyse the spectral properties of a preconditioner based on substructuring; namely, based on the local operators Si, it is necessary to show that the extension operators Hi induce an equivalent norm on A, the space of traces on

r.

In the next section we will focus on the equivalence of these extension operators. Then in Sections 4.1.2 and 4.1.3 we will consider extension operators in different function spaces, that will then be used in Chapter 5 for the analysis of many other boundary value problems. As we have already mentioned in the Preface, we restrict our attention to the case of a two-domain decomposition of the domain ft. 4.1.1

Extension theorems in i f 1 (ft,)

The first result we are interested in concerns the harmonic extension. Proposition 4.1.1 Let the space A be defined in (1.2.4) and the extension operators Hi in (1.2.14). Then there exist two positive constants C\ and C2 such that (4.1.2)

(AIMIa < H ^ U i , n i
V^eA,

I=1,2.

Proof The left-hand inequality follows from the trace inequality (1.2.5), while the right-hand one follows from the a priori estimate (1.2.11) for the Dirichlet boundary value problem for (coercive) elliptic equations (see, for example, J.-L. Lions and Magenes 1972). •

EXTENSION THEOREMS

§4.1

105

An easy consequence is the following result: Proposition 4.1.2 The operators Si, i = 1,2, defined in (4.1.1) are continuous and coercive in A. Moreover, they are spectrally equivalent; that is, there exist two positive constants C3 and G\ such that (4.1.3)

C 3 (Siv, V) < (S2V,V)
V r, 6 A.

Consequently, the operator S = S\ + S2 is continuous and coercive in A, and both S1 and S2 are spectrally equivalent to S. Proof The bilinear forms ai are continuous and coercive in Hx(Qi), Proposition 4.1.1 we see that for each rj, p, G A it holds that (SiT),n)

<

(Siw)

= \\VHir,\\lni > (1 +

I M I A I I ^ I I I A

hence from

ClIMIAINIA

<

and Cn)-1llffirflllni

> C i ( l + Cn)_1ll»7llAi for i = 1,2, where CQ is the constant in the Poincare inequality (1.2.2). As a consequence, (Siv,v)

< clMll

< cKi

+

c ^ c ^ ^ n ) ,

and the same is true by interchanging the roles of Si and S2.

•

Let us consider now the finite dimensional case. The finite element approximation of the homogeneous Dirichlet problem for the Laplace operator has been introduced in (2.1.1). Following (2.2.4), the discrete Steklov-Poincare operators are defined as (4.1.4)

{Si,hVh, Hh) •= a,i(Hiihr)h,Hiihij,h)

V j]h, /j,h <5 Ah

(for notation see (2.1.7) and (2.2.1)). We prove the following theorem, known as the finite element uniform extension theorem (see, for example, Bramble et al. 1986a; Bj0rstad and Widlund 1986; or Marini and Quarteroni 1988, 1989). It provides a finite element counterpart of Proposition 4.1.1, and we have already stated it as inequality (2.2.5). We point out that different constructions of uniform extension operators have also been given by Brenner (1998a) and Bernardi and Girault (1998). Theorem 4.1.3 (Uniform extension theorem) Let ft, Oi and be Lipschitz polygonal domains. Let the space Vh be defined as in (2.1.3), with X£ introduced in (2.1.2) or (2.1.4). Assume that the family of triangulations Th is regular, and that the family of triangulations M-h, induced by Th on the interface T, is quasiuniform. There exist two positive constants C\ and C2, which depend on the relative sizes of fli and fl2 but are independent of h, such that (4-1.5)

C i I M I A < H ^ a ^ l l i A < C2\\vh\\A

V^eAfc,

1=1,2.

106

CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

Proof Since rjh is the trace on the interface T of both Hi^Vh and H2thVh, the trace inequality (1.2.5) states that there exist constants C* > 0 such that IMU

< C*\\Hl!hr]h\\ifii

yvhEAh,

1 = 1,2.

Therefore, the left-hand inequality in (4.1.5) follows by choosing Ci : = min(l/C 1 *, 1/C2). On the other hand, we have WH^VkWi,^ <

- Hir]h||ljn. + H F ^ I l ! , ^ .

From (4.1.2), it follows that ll-Hi^hlkfii < C2\\Vh\\A,

i = 1,2.

Since r]h is a piecewise-polynomial continuous function on T and fi; is a Lipschitz polygonal domain, the solution H^h belongs to the Sobolev space H1+s (fii) for a suitable s* > 1/2 (see Dauge 1988, Corollary 18.15). Denoting by || • H ^ and II • ||o-,r, respectively, the norm of the Sobolev spaces iT°"(fij) and H^iT), for each a £ R , the following regularity estimate also holds: l l ^ % | | 1 + s . , o ; < C||77/l||i/2+s*,rMoreover, we have the finite element error estimate \\Hi,hr)h - Hifihlh&i < Chs*\\Hir]h\\i+s*,n,, which is a consequence of the continuity and coerciveness of the bilinear forms ai(-,-) and of the interpolation error estimate (see, for example, Ciarlet 1978). The inverse inequality (see again Ciarlet 1978) gives hs'\\Vh\\i/2+s*,r < C\\rjh\\A

V r,h £ Ah,

with a constant C independent of h. Therefore, for a suitable constant C2 independent of h we obtain

and the right-hand inequality in (4.1.5) is proved.

•

Proceeding as in Proposition 4.1.2, we easily find Theorem 4.1.4 Under the assumptions of Theorem 4-1-3, the discrete SteklovPoincare operators Si^, S2th and Sh = <Si,/i + S2ih are continuous and coercive in Ah, uniformly with respect to h. Moreover, they are uniformly spectrally equivalent; that is, the double inequality of spectral equivalence holds with constants independent of h.

EXTENSION THEOREMS

§4.1

107

This result has important consequences at the algebraic level. In fact, let us introduce the matrices E ^ associated with the Steklov-Poincare operators 5»,/i, defined as (4.1.6)

[£i,hV, !*]••= (Si,hVh,Hh)

V^^GAA,

where [•,•] is the Euclidean scalar product in RA1*, and for each ph £ A/,, (/, denotes the set of its values at the N-p nodes on T. We obtain at once that both Si r h a r e symmetric and positive definite. In particular, [ S j ^ 17,77] > 0-i\\r]h\W for i = 1,2, having denoted by a , the coerciveness constant of SithTherefore, since E/, = + E 2 ,h, we have (4.1.7)

[S2.hf7.17] < [S/,1?,!?],

moreover


( 4 1 8 )

<

+

P2,hf7,«7],

where /3j is the continuity constant of Sijt, i = 1,2. Let us recall the following result (see, for example, Quarteroni and Valli 1994, p. 5 4 ) . T h e o r e m 4 . 1 . 5 Let P and Q be two symmetric and positive definite M x M (real) matrices. Assume that there exist constants Ki > 0 and K2 > 0 such that (4.1.9)

Ki[Pt7,rj\ < [Qr/,rf\ < K2[Pt),rj\

V T) e R M ,

where [•,•] is the Euclidean scalar product in R M . Then the eigenvalues preconditioned matrix P~1Q satisfy the inequality

of the

Ki < f m hi < fmax < -^2, and the spectral condition

number K(P~1Q)

:=

satisfies

< f • As a consequence of (4.1.7) and (4.1.8) we have thus proved that the condition number of the preconditioned matrix E ^ E h is bounded, uniformly with respect to h, hence the Dirichlet-Neumann preconditioner E2i/,, is an optimal preconditioner for E^. Clearly, the same is true for E^^Eft,. We are also in a position to analyse another preconditioner, the N e u m a n n Neumann preconditioner. First, let us point out that if a linear operator C :

108

CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

A —> A' is continuous and coercive, with continuity constant given by f3 and coerciveness constant given by a, then its inverse L~x : A' —> A exists (by the Lax-Milgram lemma), it is continuous with constant c t - 1 , and coercive with constant a / / ? 2 . In fact, for any G G A' the solution A = £~1G G A of CX = G satisfies a||A|||<(£A,A) = (G,A)<||G'||A,||A||A, hence II^GIU^IIGIU,. a Moreover, ||G||2, = \\£M\l,
=

B2 a

—(G,£_1G).

Consequently, for each > 0 and <J2 > 0 the Neumann-Neumann operator J\fh : = (viS^l + criS'l)-1 is symmetric, continuous, and coercive in A. Let us denote by /?// and a^f its continuity and coerciveness constant, respectively; by a straightforward computation, we can see that they are respectively given by

A , -

<Jiai/3| + a2a2(3{ (<Tiai(3\ + a2ot20{

)a\a\ 2 /3?/3|(aiff 2 + a 2 < r i )

and are independent of h. Introducing the Neumann-Neumann preconditioner [Nht),n] :=

(AfhT]h,nh),

it can be seen that (4.1.10)

N ^ i a ^

+

a ^ ) -

1

.

We have [NhV,r)\ = (AfhVh,Vh) < M\Vh\\l a 1 + a2

[Shij,i7]

v,eRWr,

moreover [ZhV,V] = (ShVh,Vh) < if3i + f32)\\rih\\l

Altogether, the two previous inequalities allow us to conclude that the matrix Nh is an optimal preconditioner of the matrix E/ t , owing to Theorem 4.1.5.

EXTENSION THEOREMS

§4.1

109

Similar considerations are possible if we focus now on the homogeneous Dirichlet boundary value problem for the general second-order symmetric elliptic operator

Lw := — E Di(aijDjW) i,j=i

(4.1.11)

+ a0w

introduced in (1.4.2), as well as its approximation by means of the finite elements. The local bilinear forms (4.1.12)

a*(u>i,Vi) := / Jtl'

associated with for example, ao For i — 1,2 defined on T to

Y]

+ a0 w, Vi , i = 1 , 2 , J

L are symmetric and continuous; besides, they are coercive if, > 0. we introduce the operators E*, which extend in Qi a trace rj the function E*rj, which solves the following problem (E*VeVi

(4.1.13)

aijDjWiDiVi

\i,j=i

: V Vi £ V?

a\{E:r,,Vi)=0 E*ri = r]

on T.

The Steklov-Poincare operators associated with the operator L are defined as (4.1.14)

(Sit], fj) : = a*(Eir], E*

Vj/.^A.

For the finite element approximation the definition is similar: (4.1.15)

(Si,hVh,tJ-h) ••= a*(E*hr)h,Elhnh)

V ilh,]xh £ Ah,

where now E*hr]h is a finite element extension of rjh in Qi and is given by ' E*hr]h £ Vith : (4-1.16)

a*(Elhr]h,vhh) . Elhlh. = rjh

= 0

V vith £ V &

on T.

The infinite dimensional Steklov-Poincare operators Si and S 2 , introduced in (4.1.14), as well as S = Sj + S 2 , are symmetric, and the same is true for their discrete counterparts Si,/, and S 2 ,/i, defined in (4.1.15), and for Sh = Si,h 4- S 2 ,/i. Moreover, we have

110

CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

Theorem 4.1.6 The Steklov-Poincare operators Si, S2 and S = Si + S2 introduced in (4.1.14) are continuous and coercive in A; hence, they are spectrally equivalent. If the assumptions of Theorem 4-1-3 are satisfied, the same is true for the Steklov-Poincare operators Sith> $2,h and Sh introduced in (4.1.15), and the result holds uniformly with respect to h. Proof We only consider the discrete case, the proof for the infinite dimensional case being easier. Coerciveness of Si,h is a direct consequence of the coerciveness of the forms a*(-, •) and of the trace inequality (1.2.5), which yields I M A
V^eAfc.

Hence, the result holds uniformly with respect to h. To prove the uniform continuity of S^h we proceed as follows. Since we are allowed to take viih = E*hrjh - Hithr]h in (4.1.16), we find that o-KKh^KhVh)

=

aKKh^Hi.hrjh)-

Prom the continuity and coerciveness of a*(-, •) we deduce that (4.1.17)

l l ^ ^ l k f i , < KWH^VhWi,^

< KC2||%||A-

The constant K is the ratio between the continuity constant and the coerciveness constant of a*, while C2 is the constant in (4.1.5); both K and C2 are independent of h. The result then follows from the continuity of the forms a*(•,•). • In particular, we have proved that both operators E*h, i = 1,2, are uniform extension operators from A to H As in the preceding case, from Theorem 4.1.6 we obtain at once that the matrix £2,h, associated with the discrete operator S2,h, is an optimal preconditioner of the Schur complement matrix Eh, associated with the Steklov-Poincare operator Sh- The same is true for £1 h, and for the Neumann-Neumann preconditioner Nh = i? i ^ + ^ S - j J "

1

.

Remark 4.1.7 Let us analyse the main properties that are used in the proof given in Theorem 4.1.3. We can identify the following essential ingredients: 1. The bilinear forms describing the multi-domain formulation are continuous and coercive in a suitable Hilbert space X . 2. The trace theorem from X to a space of functions defined on the interface T holds. 3. The solution in of the continuous problem, having a homogeneous righthand side and a boundary datum on d£l \ T, and being equal to a discrete function on P, belongs to a space of functions that are more regular than the functions in X . 4. For a suitable choice of finite elements, the interpolation error estimate in X holds with an optimal order of convergence.

EXTENSION THEOREMS

§4.1

111

5. The inverse inequality holds with an optimal exponent for the finite elements restricted on T. More precisely, properties 1, 3, 4 and 5 are used to prove the uniform continuity of the Steklov-Poincare operators, whereas properties 1 and 2 give their uniform coerciveness. One can note that 1 and 3 are related to the specific boundary value problem at hand, 2 is a general theorem for the trace space, and 4 and 5 concern the family of finite elements that are being used in the approximation.

•

4.1.2

Extension theorems in

H(div;Qi)

The analysis that follows is concerned with function spaces that are associated with the weak formulation of fluid dynamics problems (see Section 5.5), or with the mixed-type formulation of elliptic problems (see, for example, Brezzi and Fortin 1991). The Hilbert space H(div; ft) is defined as (4.1.18)

iJ(div; ft) := { v £ (L2(n))d

| div v £ i 2 ( f t ) } ,

endowed with the norm l|v||tf(div;fi) :=-(||v||gi{1 + | | d i v v | | ^ ) 1 / 2 , and we set (4.1.19)

# 0 ( d i v ; ft) : = { v € H(div; ft) | (v • n*)| a n = 0 } ,

where n* denotes the unit outward normal vector on 3ft (see also Chapter 9). Considering, as usual, the domain ft split into two non-overlapping subdomains fti and ft2, with interface T, for i = 1, 2 we introduce the local spaces and trace space Wi : = {v« G # ( d i v ; f i < ) | (v< • n*) | a f i n a n i = 0} (4.1.20)

W° := i?o(div; fti) V

:= {tp : T -> R | ip = (v • n)|r, v G # 0 ( d i v ; ft)}-

The trace space ^ coincides with the dual space of Hl/2(T) (see Girault and Raviart 1986, p. 27-9), and therefore is the dual of A if V n 3ft = 0, as in Fig. 1-1 (right), whereas \I> is strictly included in the dual space of A if T fl 0ft ^ 0, as in Fig. 1.1 (left). The norm in 'J' will be denoted by || • For each ip G the vector function Qi%p e Wi, i = 1,2, is the solution to (Qi^.vOn, + (div(Qi^),divvi)ni = 0 (4.1.21) (Qi4> • n)jr = ijj

on r ,

V v< G W °

112

CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

where (•, -)q ; denotes the L 2 (fi;)-scalar product. In differential form, Qiift satisfies the following problem in the sense of distributions ' Qi^

—

V div Qiip = 0

(Qiip •

= 0

in Orrii on <9fi; n dQ on T.

(Qitp • n)|r = %p

In other words, Qiip is an extension of the normal trace ^ on F from

to Wi.

Proposition 4.1.8 There exist two positive constants C\ and C2 such that (4.1.22) Proof (4.1.23)

C^M*

v^etf,

<\\QMH{div.iQi)
i = l,2.

The left-hand side inequality follows from the trace inequality IKvi • n)|r||* < C ^ V i l ^ ^ n o

V v, G Wi

(see, for example, Girault and Raviart 1986, pp. 27-8). On the other hand, we remark that Q ^ = V M ^ , where Mitp G i f 1 (ft;) is the distributional solution of Mop - AMiip ~ 0

in fi

dMiip* = 0 dn

on dSli fi <9fi

dMjip dn

on r .

Now the right-hand side inequality in (4.1.22) follows from well known a priori estimates for the Neumann problem. • Let us introduce the finite dimensional analogue of the operators Q,;, defined on the div-conforming Nedelec finite elements (see Nedelec 1980; and also Raviart and Thomas 1977 for the two-dimensional case). Let us set for r > 1 (4.1.24)

Vr := ( P r _ i ) d © {p(x) x j p € P r - i } ,

where P r - i denotes the set of homogeneous polynomials of degree r — 1. The div-conforming Nedelec finite element space is given by (4.1.25)

Nrh : = { v A G H(div; fi) | vhjK

G Vr(K)

V K G %},

where Vr(K) denotes the restriction of Vr to K. Other possible choices that we might consider are the Nedelec finite element spaces of the second type, proposed in Nedelec (1986), or, for k > 2, those introduced in Brezzi et al. (1985).

EXTENSION THEOREMS

§4.1

113

Let us define now the finite element space on ft, as well as the local spaces and trace space: Wh : = :=

(4.1-26)

w°h

fl i?o(div; ft) {vh\Qi I

<5 Wh},

i = 1,2

:= W^Htfotdivifti),

Vh : = { ( v h - n ) | r \vh

i = l,2

&Wh}.

For each tph 6 &h> denote by Q i ^ h 6

the solution of

The discrete extension operator Qi t h satisfies the two-sided bound stated in the next theorem. T h e o r e m 4 . 1 . 9 (Uniform extension theorem) Let ft, fti and ft2 be Lipschitz polygonal domains. Assume that the family of triangulations 7~h is regular, and that the family of triangulations M.h, induced by Th on the interface T, is quasiuniform. There exist two positive constants C\ and (72, which depend on the relative sizes of fl 1 and ft2 but are independent of h, such that (4.1.28)

Ci\\iph\\<s> < ||Qt,/iV'/i||.ff(div;fii) < C^IIV'/ilk

i =

1,2.

Proof W e want to show that the conditions 1 - 5 presented in Remark 4.1.7 are satisfied. Condition 1 trivially holds for the space X = Wi, because the local bilinear forms are the scalar products in i ? ( d i v ; f t , ) , i = 1,2. Condition 2 is the trace inequality (4.1.23). Since iph is a piecewise-polynomial function on r and ft,; is a polygonal domain, the regularity of Qr'/Wi = VM,iph (namely, the proof of 3) is given in Dauge (1988), Corollary 23.5, where it is shown that Miiph e H1+S'(fii) for a suitable s* > 1 / 2 , and that \\Miiph\\i+s*<

—i/2,r-

The proofs of 4 and 5 are rather technical, and we omit them, referring to Quarteroni et al. (1991). The results read - ^ V i l l ^ d i v ; ^ ) < C7i.s(||v;||s,n; + || diwj||SjQ.), where Trf^Vj € W^h is the interpolant of v ; in ftj, and \\'4,h\\s-l/2,T < Ch~S for 0 <

S

< 1.

||^|k,

114

CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

Now the proof can be completed by following the same guidelines as for Theorem 4.1.3. In fact, the left-hand inequality in (4.1.28) is the trace inequality. The right-hand inequality in (4.1.28) can be proved as follows. We have ||Qi,hV , /i||ff(div;fi i ) <

\\Qi,hi>h -

Q i V , h | | H ( d i v ; f l i ) + I I Q i ^ f c I |tf(div

)•

From (4.1.22) it follows that IIQi^hllfffdiv;^)

< C2\\4>h\\<Sf •

Moreover, ||Qi,h1ph

-

Qitph\\H(dW-,Ui)

<

HQi^h

- ^ ( Q i ^ ) l k ( d i v ; n O

and (4.1.28) follows. 4.1.3

•

Extension theorems in iJ(rot; fij)

The analysis that follows is concerned with function spaces that are associated with the weak formulation of the Maxwell equations (see Section 5.7). The Hilbert space # ( r o t ; fi) is defined as (4.1.29)

JI(rot; ft) : = { v £ (Z 2 (ft)) d | r o t v £ ( L 2 ( Q ) ) d } ,

endowed with the norm l|v||H(rot;n) :=(||v||g,n + ||rotv||gin)1/2, and we set (4.1.30)

#o(rot; A) == { v £ # ( r o t ; ft) | (n* x v ) ] a n = 0 } ,

where n* denotes the unit outward normal vector on <9fI (see also Chapter 9). For i — 1,2 the local and trace spaces are defined as Zi := { v i £ # ( r o t ; f t j ) | (n* x v;)| 9n n9fi ; = 0 } (4.1.31)

Z° :=

H0(iot;Oi)

Xr := {i/> : r

R d | t/> = (n x v)| r , v £ # 0 ( r o t ; ft)}.

A characterisation of the space of tangential traces Xr has been given in Alonso and Valli (1996), under suitable geometrical conditions on the interface T. The norm in this space will be denoted by || • ||;tr.

§4.1

EXTENSION THEOREMS

For each tp £ Xr, the vector function C {Ftil>,

115

£ Ziy i = 1, 2, is the solution to

+ ( r o t ( ^ V ) , rot v ; ) ^ = 0

V v ; G Zf

(4.1-32)

In differential form,

satisfies (in the sense of distributions) the problem

( P i t y + rot rot Fiip = 0 (n* x FiWidQindn = 0 . (n x Titj)) |r = The vector function to Z{.

in fi; on d f i ; n dfi on T.

is an extension of the tangential trace ^ on T from Xr

Proposition 4.1.10 There exists a positive constant C± such that (4.1.33)

C i l M k r < II^IU(rot;[2i)

i = 1,2.

Moreover, if T is a convex portion of the boundary of the Lipschitz domain fi;, there exists a positive constant C2 such that

(4.1.34) Proof

H^ll^rotjOO


polygonal

v V e <*t, i = 1,2.

Inequality (4.1.33) follows from the trace inequality

(4-1.35)

||(n x v.Oirlkr < ^ I K I k r o t ; Q ; ),

that is proved in Alonso and Valli (1996). In the same paper, under the assumption that T is regular enough or that F is a convex portion of the boundary 3fi;, a continuous extension operator from Xr to Wi has been constructed, satisfying (n x = Therefore, the solution T ^ can be written as PITY = Z; + UIFTY,

where z; £ Z° is the solution to (z;,v;)0i

+(rotz;),rotv;)n; =

V i ) n i - (rot(W i ,rV») l rot Vi) n i

V v ; e Z°.

Taking v ; = z; and using the Cauchy-Schwarz inequality for the scalar product in # ( r o t ; fi;), it follows that

116

CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

Hence \\^i^\\H{vot-,Qi) <

||Zt||ff(rot;nO + I [R-i,rl/>\ !ff(rot ;fi,)

<2||Wi,r^||H(rot;no


owing to the fact that TLitr is continuous.

•

We construct a finite dimensional approximation of the operator F j in the so-called rot-conforming Nedelec finite element space (see Nedelec 1980). First of all we need to define, for r > 1, (4.1.36)

H r : = ( P r _ 1 ) d © { p G (Pr)d

| p ( x ) • x = 0},

where P r denotes the set of homogeneous polynomials of degree r. The rotconforming Nedelec finite element space is given by (4.1.37)

Mrh : = {v f c £ H{rot; ft) | vh\K G Kr{K)

V K G %},

where TZr(K) indicates the restriction of TZr to K. Another possible choice relies on the Nedelec finite elements of the second type, see Nedelec (1986). Let us introduce now the finite element space on ft, as well as the local and trace spaces:

Z,h

= M l fl JT0(rot; ft) = K j f t , | v A G Zh},

(4.1-38)

= Zithr\H0(rot-,Sli), ,h

i = 1,2

= { ( n x v/j)|r | vh G Zh}.

For each iph G Xrji., denote by Ti^n

(4.1.39)

i = l,2

G Z.hh the finite element solution of

{ (Fi,hi/>h,Vi,hhi + ( r o t ( ^ i , h V h ) , r o t v i ) h ) n i = 0 I {{nx Fitht(>h)lr =iph onT.

V v i i h G Z°h

The discrete extension operator Ti^h satisfies the bounds stated in the next theorem. Theorem 4.1.11 (Uniform extension theorem) Let fi, fti and ft2 be Lipschitz polygonal domains. Assume that the family of triangulations Th is regular, and that the family of triangulations M.h, induced by Th on the interface T, is quasiuniform. There exists a positive constant C\, which depends on the relative sizes of fti and ft2 but is independent of h, such that (4.1.40)

Ci\\i>h\Ur

< H^ftVftlkrot^)

i = l,2.

Moreover, if T is a convex portion of the boundary of ftj, there exists a positive constant C2, which depends on the relative sizes o / f t i and ft2 but is independent of h, such that

§4.2

SPLITTING OF OPERATORS A N D PRECONDITIONING

(4.1.41)

| | ^ V > j H ( r o t A ) h\\Xr

V ^ e ^ ,

1=

117

1,2.

P r o o f We want to show that the conditions 1 - 5 presented in Remark 4.1.7 are satisfied. Condition 1 trivially holds, because the local bilinear forms are the scalar products in i ? ( r o t ; f t i ) , i = 1,2. Condition 2 is the trace inequality (4.1.35). The proof of 3, namely, of the regularity of Jrii/)h, is given in Alonso and Valli (1999), where it is proved that G Hl/2+s (Q,i) and that r o t ^ V / . e f j i / 2 + 5 ( Q i ) j f o r (5* > 0 small enough. Moreover, the following a priori estimate holds W ^ h W i / 2 + 6 ' , u i + | | r o t ^ f c | | i / 2 + i . i n .
< Ch1/2+6(\\vi\\1/2+6tQi

+

llrotvillj/z+s.nj,

where itfh~Vi € Z,j t is the interpolant of v , in ft,;, and l l ^ l k a n , + WdWriPllk^

<

Ch-V2-s\\1>h\\Xr,

for 0 < 5 < 1/2. The proof now follows as in Theorem 4.1.3. In fact, inequality (4.1.40) is the trace inequality. Inequality (4.1.41) can be proved as follows. We have \\Fi,h,ll>h\\H(T0t-,ni) < || Fi,h1>h -•^V'/ 1 ||fl-(rot;f1,) + I \?i*l>h I |jtf(rot

•

From (4.1.34) it follows that ll^rfMff(rot : ni) < C2\\lph\\xrMoreover, \\Fi,hl/>h - FithWmrofVi)

< W^i^h

-^(^/OllffCrotifii)

< C h V W (11Fi4>h111/2+a. >n. + II rot Fixp h I| 1/2+<5 , , n < ) < C h W { \ \ p h \ \ s . i 8 0 i + ||div r ^|| 4 ., 8 n i ) h\\Xr, which completes the proof.

4.2

•

Splitting of operators and preconditioned iterative methods

Let X be a (real) Hilbert space, and X' its dual space, with the duality pairing between X' and X denoted by (•, •). We consider a linear invertible continuous operator Q : X —> X', which can be split as

118

C O N V E R G E N C E ANALYSIS F O R I T E R A T I V E DD A L G O R I T H M S

(4.2.1)

Ch. 4 ^

Q = QI + Q2

where both Q\ and Q2 are linear operators, and they will be associated, in our application, with the subdomains and ft 2 in a way that will be made precise from case to case. We will address the solution of the problem QA = G

(4.2.2)

with G E X' given and A e X to be determined. To solve problem (4.2.2) we consider iterative methods based on a preconditioner V for Q that is given by either Q2 or ( o i Q ^ 1 + 02 S ^ 1 ) - 1 , ai a2 being positive parameters. Remark 4.2.1 In our application, we will take Q — S, the Steklov-Poincare operator introduced in Chapter 1 for infinite dimensional boundary value problems, but also Q = Sh, which represents the associated finite element approximation (see Chapter 2). In fact, we have already pointed out that both S and Sh can be regarded as the sum of contributions coming from the subdomains fli and fi2. In this framework, the choice V = Q 2 is the one underlying the DirichletNeumann method, whereas V = (<Ji Q^ 1 +
V{Xk+l

- Afc) = 0{G - QA fc ), k > 0.

If both V and Q are symmetric and positive definite matrices, a more efficient method is instead the preconditioned conjugate gradient method. An estimate of the rate of convergence of these iterative methods can be derived in terms of the spectral properties of the preconditioned operator Q. Equivalently, one can estimate the norm of the iteration operator Tg : = I — 9 V ~ l Q associated with (4.2.3). The present section is therefore devoted to the analysis of the spectrum of V 1 Q. We start by presenting an abstract convergence theorem (Alonso et al. 1998; see also Quarteroni and Valli 1991 b) concerning the Richardson iterations for equation (4.2.2) preconditioned by V = Q2. s Theorem 4.2.2 Suppose that (a) Q2 is continuous and coercive; that is, (a) 1

there exists f32 > 0 such that <2277,M> <02\\ti\\xM\X

(a)2

there exists a2 > 0 such that

Vr^eX;

§4.2

S P L I T T I N G OF O P E R A T O R S A N D P R E C O N D I T I O N I N G

119

that is, there exists (3% > 0 such that

(b) Qi is continuous;

(Qm^)


V r ^ e x .

(c) there exists a constant K* > 0 such that (Q2V,Q21Qv)

+ (Qv,v)

> K*M\2X

Vvex.

Then for any given A0 in X and for any 9 satisfying 0 < 9 < 9max,

the

with

sequence

(4.2.4)

Xk+1

converges

= Ak +

8Qz1(G-Q\k) (4.2.2).

in X to the solution of problem

Proof First of all, note that the operator Q 2 is invertible as a consequence of assumption (a) and of the Lax-Milgram lemma. Let us introduce the Q 2 -scalar product •= \{{Q2V,»)

+

{0.2V,v))-

The corresponding Q 2 -norva ||r?||g2 ( V i V ) 1 ^ = (Q2V, r l} 1 ^ 2 i s equivalent to the norm ||?/||x; actually, it satisfies the two-sided inequality a2lMlx

< IMIQ 2 <

foWvWx

v ^ e i

To prove the convergence of the sequence { A ^ } it is sufficient to show that the mapping Te:X-^X, Terj — v-OQ^Qv is a contraction with respect to the Q 2 -norm. With this aim, assuming that 0 > 0, we have IM I L

= NIL q2'tw

+92<2»?>

Q^Qv) 132)2

o. 2

- 9((Q2V, Q^QV)

+

(Qv,v))

u^ MW-B^MW \\'i\\x~ U'lttx-

Setting a|

fa

we obtain The bound Kg < 1 is guaranteed if 0 < 9 < 0 m a x -

O

120

C O N V E R G E N C E ANALYSIS F O R I T E R A T I V E D D A L G O R I T H M S

Ch. 4 ^

R e m a r k 4.2.3 In Theorem 4.2.2, the upper bound 9max, as well as the con1 /2 traction constant Kg' , depends only on a2, Pi, P2 and k*; consequently, the rate of convergence of the preconditioned Richardson iterative procedure in the Q2-norm depends only on these same parameters. • R e m a r k 4.2.4 If the operator Q 2 is symmetric, then assumption (c) reduces to the coerciveness of Q; that is,

(QV,V)>Y

MX-

Since the coerciveness of Qi is not required, we shall be able to apply Theorem 4.2.2 even to the analysis of heterogeneous domain decomposition algorithms (see Chapter 8). • We present now another abstract theorem, which concerns the preconditioned Richardson iteration for equation (4.2.2) based on the preconditioner r =

(
T h e o r e m 4.2.5 Assume that both Qi are continuous and coercive; that is, for i — 1,2 we have that (a)

there exists Pi > 0 such that
(b)

PiM\xM\x

there exists oti > 0 such that (Qtr],r])>ai\\V\\2x

V r, £ X.

Assume, moreover, that, for any choice of the averaging parameters > 0 and (72 > 0, the operator Af — (criQf 1 + o ^ Q ^ 1 ) - 1 satisfies the condition (c)

there exists re* > 0 such that + (Qr],rl)>K*\\rl\\2x

(Mr},M'1Qr1)

V rj e X.

Then there exists 9° > 0 such that for each 9 £ (0,9°) and for any given the sequence (4.2.5)

A fc+1 = Xk + 9M"1 (G -

£ X

QXk)

converges in X to the solution of problem (4.2.2). P r o o f We have already pointed out in Section 4.1 that, due to assumptions (a) and (&), the operator Af : X X' is continuous and coercive. Let us denote by p\r and a.j\; its continuity and coerciveness constants. We introduce the A/"-scalar product fa^V

:=

-^{{^,11)

+ (Affi,r])),

§4.2

S P L I T T I N G OF O P E R A T O R S A N D P R E C O N D I T I O N I N G

121

and the corresponding A/"-norm H^H./^ : = ( v , v ) ] P — W r ] , v ) 1 / 2 , which is equivalent to the norm \\rj\\x, i.e. *M\2x<M\%<M\ri\\2xTo prove the convergence of \ k we show that the following map T W :X->X,

T^r]

:= r? - 9N~X Qrj

is a contraction with respect to the norm || • ||a/"- Assuming that 9 > 0, for r] g X we have WTevWlr = WvWlf + 82(Qv,1Qr]) <Mj^ The operator A

=

+

- 9((Afrj,1

Qr?) +

(Qv,v))

e2{Qri,U-1QV)-9K*\\r]\\2x.

c i Q f 1 4- o^Q^ -1 is continuous and satisfies

\ai

ex',

a2 J

therefore
(— + —) (/?i+/3 2 ) 2 || a2 J \al

Setting 2 1 ^ , c*i

M a2 J

{Pi+for ctM

we obtain WTevWlr <

KeM\%-.

Setting K Ottf \al

a2

/

+A)3'

we conclude that T W is a contraction for all 9 £ ( 0 , 9 ° ) .

•

Again, it can be noted that the rate of convergence in the A^-norm of the preconditioned Richardson iterative procedure (4.2.5) depends only on the constants cr2, a i , a2, /?!, @2 and k*. Moreover, if Qi and Q 2 are symmetric, assumption (c) is equivalent to the coerciveness of Q. On the other hand, this property follows directly from (b); hence, in the symmetric case, assumption (c) is not necessary.

122

C O N V E R G E N C E ANALYSIS FOR ITERATIVE D D A L G O R I T H M S

4.2.1

Ch. 4 ^

The finite dimensional case

In this section we analyse the case of finite dimensional operators, and point out the conditions under which the rate of convergence in Theorems 4.2.2 and 4.2.5 is independent of the discretisation parameter h. As noted in Remark 4.2.3, the upper bound # m a x , as well as the contraction constant KXJ2 in Theorem 4.2.2, depends only on a2, Pi, @2 and K*; consequently, should these constants be independent of h, the rate of convergence of the Richardson method would also be independent of h. A similar remark holds for the method analysed in Theorem 4.2.5. It is worthwhile to note that in the finite dimensional case, if conditions (a), (b) and (c) in Theorem 4.2.2 are satisfied uniformly with respect to h, other preconditioned iterative procedures different from the Richardson method (such as, for example, G C R or GMRES, see Saad 1996) are convergent with a rate independent of h. Consider, for instance, the case of GMRES iterations. Let us denote the finite dimensional Hilbert space by Xh, and the finite dimensional operators by Qh, Qi t h and 0,2,h- The dimension of Xh will be indicated by Nh, and its norm by || • ||x. Moreover, introduce the matrix Qh as [QhV,lA '•= (QhVh^h)

V rjh,lJ.h e

Xh,

where [•, •] is the Euclidean scalar product in R ^ , and for each ft £ Xh, the vector /1 denotes the components of ft with respect to a suitable basis of Xh- The matrices Qith and Q2,h are defined in a similar way, so that Qh = Qi,h + Q2,hDenote by B the symmetric part of Q2,h, i- e -

B

:=


As a consequence of condition (a), this matrix satisfies [Br),rj\ = |[(g2,h

+ Qlh)V,V] =

- {Q2,hr]h,Vh) >

[QwV,*]

a2\\r,h\\2x

and [517,1?] <

MmW'x-

Hence, B is symmetric and positive definite, and the £?-scalar product (t],/i)b : = [Brj,fi] induces a vector norm | • \g that is equivalent to the norm || • ||x; namely, (4-2.6)

a2\\r,h\\x<\r)\2B<02\\m\\x-

The matrix Q 2 \ Q h is positive definite with respect to the B-scalar product, as from condition (c) it follows that

§4.2

SPLITTING OF O P E R A T O R S AND PRECONDITIONING

= \[(Q2,H +

(Q^QhV^B

123

Qlh)Q^hQht),v}

= l\QhV,n) + l[Q2XQhV,Q2,hv] 2 j ! = Qh,r]h,Vh) + 0-2,hVh, Q2,hQhVh)

(4.2.7)

>~\\m\\2x>~-2\r,\2B, having used (4.2.6). Therefore, the reduction factor p of the GMRES methods applied to the preconditioned matrix Q^^Qh is given by

(see, for example, Quarteroni and Valli 1994, Section 2.6), where Amin is the minimum eigenvalue of the matrix [ Q ^ Q h + { Q 2 \ Q h ) T B ] l 2 and A m a x the maximum eigenvalue of the matrix {Q2hQh) T B Q2h,Qh> the symbol ETB denoting the transpose of a matrix E with respect to the S-scalar product. We have 1

.

Amin — t: mm

(Q;\QHFI,RI)B

+

j—r=

2

\V\%

. =

«Q^HQH)TBV,R))B

(Q 2 , 1 h QhV,f])B

^ K*

—toil

due to (4.2.7). Moreover Amax — n?ax V^o = max

{(Q^Qh^Q^iQhV^i , 2 \ri\2B {Q2lh! QhT}, Q^hQh^B r—r^ , MB

and from conditions (a) and (6) (QllQhV,Q^hQhti)B

= lHQifi

+

= ^[Qh^Q^hQhfi} = [QhV,QllQhti]

QZhW^QhtiiQ^Qhti] + = (Qh,


Hence, using (4.2.6) we conclude that

liQlhQ^hQh^Q^iQhV] Q^iQhVh)

124

C O N V E R G E N C E ANALYSIS F O R ITERATIVE DD A L G O R I T H M S

Ch. 4 ^

, . (/?i + (32)2 ^max _ 0 j a2 and the GMRES method converges at a rate independent of h. The argument above can also be applied to prove the convergence of the GMRES iterations based on the Neumann-Neumann preconditioner Nh : = (cr1Q~1lk+ c a s e > denoting respectively by 07/ a n d (3m the coerciveness a2Q2h)~1and continuity constants of Nh and by B its symmetric part, we still obtain (4.2.6), with a / s and (3u instead of a 2 and fa- The rest of the proof can be repeated without modification, substituting Q2,h with Nh- The final result reads ,

.

*min il. r\n

1

x

^ {Pi +

^max _

o

P2)2

5

therefore, if the assumptions of Theorem 4.2.5 are satisfied uniformly with respect to h, the rate of convergence is also independent of h. Remark 4.2.6 The convergence of the GMRES method can be proved also by resorting to a general result. In fact, let (•, •)* be a scalar product in R M , [ • the associated vector norm and || • ||* the corresponding natural matrix norm. Consider the linear system associated with the matrix P~lQ, and suppose that the Richardson iterative method applied to it is convergent with respect to the norm | • |* for a suitable 8„ > 0; namely, there exists a constant K £ (0,1) such that (4.2.8)

||I - 0*P _1 Q||* < K.

Then also the GMRES method converges with respect to the norm | • |„, and its rate of convergence only depends on K . This is proved as follows. As we have already noted, the reduction factor of the GMRES method in the norm I • L is

where A rnm is the minimum eigenvalue of the matrix [P~1Q + (P~lQ)T*](2 and5 A max the maximum eigenvalue of the matrix (P~1Q)T* P~XQ, the symbol ET" denoting the transpose of a matrix E with respect to the scalar product (-,•)*• Hence _ ((P-1Q)T-P~1Qx,x). _ iP^Qxl2 X ~ x#0X lx|2 Ixl? ' and

. 1 . ((P-1Q + (P"1g)r-)x,x), . (P^Qx.x), Am in = - mm -H———-— = mm —-—— 2 2 x^O |x| x^O

From (4.2.8) we have

§4.2

SPLITTING OF OPERATORS AND PRECONDITIONING

|x|2 -20*(P~1Qx,x)*

+ 0 2 | p - I G X | 2
V X £ R

M

125 ,

hence (1 - K2)|x|2 + 02|p-iq x |2 < 2f?*(P~ 1 Qx,x)» < 20„|P- 1 Qx|, |x|.. As a consequence, we obtain 2 < -|x|*, (7*

1 — /T2 ( P _ 1 Q x , x ) » > ———]x| 2 , ZC7*

thus

which proves the asserted result. 4.2.2

•

T h e case o f s y m m e t r i c m a t r i c e s

A different point of view can be adopted in the finite dimensional symmetric case. Let us denote by P the matrix associated with the preconditioner V and by Q the matrix associated with Q, and assume that both P and Q are symmetric and positive definite. From now on, our analysis concerns a general preconditioner P, not necessarily Q2 (Dirichlet-Neumann) or (a\ Q^1 + o ^ Q ^ 1 ) - 1 (NeumannNeumann). First, we report a general result: Theorem 4.2.7 Let P and Q be two symmetric and positive definite M x M (real) matrices. Then the preconditioned Richardson iterations (4.2.9)

A ^ 1 = A& + 6P'1 ( G -

Q\k)

converge to the solution of QA = G if, and only if, the parameter 6 satisfies 0 <6 < where fmax

2 ^m ax

is the maximum eigenvalue of

,

P~YQ.

Proof Let us proceed for a while considering two general non-singular complex matrices P and Q, as this will be useful in what follows (see Theorem 4.2.9). Setting ek -.= Xk - A, the error equation associated with (4.2.9) is ek+1

_ (j_0p-iQ)efc,

where Tg := I - 0P~xQ is the iteration matrix. Let rs, s = 1 , . . . , M be the eigenvalues of Tg and vs those of P~lQ. Since r s = 1 — 9vs, we have \Ts\2 = (1 — 0 R e ^ s ) 2 + 0 2 ( l m ^ s ) 2

126

C O N V E R G E N C E ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch.

4^

When P and Q are symmetric and positive definite real matrices, the matrix P~XQ is symmetric and positive definite with respect to the scalar product (ij,/i)p : = [Pfj, fi}. therefore its eigenvalues v., are real and positive. Hence the spectral radius pg : = max 5 |r5| of the iteration matrix Tg is less than 1 if, and only if,

and this proves our result.

•

Let us recall that the spectral radius pg of the iteration matrix Tg satisfies pe = (see, for example, Quarteroni and Valli 1994, Lemma 2.4.1), and therefore ||A f c -A||p<^||A°-A||p. The value of pg is given by 1 - 0f m in (4.2.10)

pe = \ ^max - 1

2 for 0 < e < 2 Vmax + Umin 2 ' for < 9 < ^max "i ^min ^raax

where vm-m is the minimum eigenvalue of P ^ Q (see Fig. 4.2.1).

Now we want to analyse other iterative methods for the solution of the linear system associated with the preconditioned matrix P~1Q. Assume that P and Q are symmetric and positive definite real matrices, and that the spectral condition number K(P~1Q) = satisfies the bound ' "min (4.2.11)

<

K\

for some K* > 1 independent of the dimension of P and Q (P is therefore an optimal preconditioner for Q). Then we can prove the following convergence result.

§4.2

SPLITTING OF O P E R A T O R S A N D PRECONDITIONING

127

1. For the stationary Richardson method (4.2.9) with 6 = # o p t : = k

IA*-AIP< ( f r r i j

fc>°-

The preconditioned Richardson iterative method with a dynamical choice of the parameter 0 reads: - pzk (4.2.12)

= Yk

< Xk+1 = Xk + 6kzk =rk

-8kQzk,

where rk := G - QXk. For this iteration we have 2. For the minimum-residual Richardson method, choosing as parameter 0k that minimising the P-norm of the preconditioned residual z fc ; that is, (zk,Qzk) 9 k

( K* — 1 \ k I"0!"' 3. For the steepest-descent Richardson method (or gradient method), choosing as parameter 9k that minimising the Q-novm of the error Xk — A; namely,

e, |Afc-A|0<

k>

0.

The same type of information, namely, K(P~1Q) < K*, can be effectively exploited also in the context of different iterative methods, notably for the preconditioned conjugate gradient method (see (3.5.13), with the obvious change of notation). For the latter, we can still infer an error estimate of the form

"A|q -

2

( T S t i )

|A

°"A|q'

k

-°•

Note that, in all the previous iterative schemes, the rate of convergence only depends on K*, therefore it is independent of the dimension M of P and Q. In turn, estimate (4.2.11) can be obtained when the bounds (4.2.13)

Kx < vmin < i/ max < K2

are available, with Ki and K2 independent of M (see Theorem 4.1.5). From these estimates, one can prove the convergence of Richardson iterations even

128

C O N V E R G E N C E ANALYSIS F O R ITERATIVE D D A L G O R I T H M S

Ch. 4 ^

for a non-optimal choice of the parameter 9\ namely, for 6 chosen in a suitable interval of the real line (a motivation for this choice will be made clear in Remark 4.2.8). In fact, using (4.2.13) in (4.2.10) we conclude that for each 0 e (0,1 /K2) the spectral radius pg of the iteration matrix I — 9P~1Q satisfies 0 < pe < 1 - Kx8

< 1.

R e m a r k 4 . 2 . 8 (Cost per iteration) W e have already shown in Section 1.3 that the Dirichlet-Neumann algorithm is equivalent to a preconditioned Richardson iterative scheme for the Schur complement matrix = + £2,/>. The implementation of this algorithm in the general case has been reported in (4.2.12). In the present case, we have P = S 2 j / l , and the solution of the system Y,2ihzk = rk yields the solution of a Neumann problem in fi2- Next, when zk is available, the calculation of E/jZ& requires the solution of two Dirichlet problems, one in Qi and one in fi2- Finally, in the case of the minimum-residual method, the computation of 6k requires another Neumann solve in f l 2 , while, for the steepest-descent method, 9k can be computed without additional solves. Therefore, the global cost of each iteration amounts to solving three boundary value problems for the steepest-descent method, and four boundary value problems for the minimum-residual method. Should we take 9 without using an optimal strategy, we can reformulate the fcth iteration step as = (l - 9)\k + BE-^x

-

Zi,h\k),

which only requires the solution of two boundary value problems, a Dirichlet problem in Oi and a Neumann problem in Whether the extra cost (per iteration) of (4.2.12) would be compensated by a quicker convergence is a matter that depends on the specific problem at hand. If the preconditioner employed is the Neumann-Neumann preconditioner Nh = ( ( J i S r ^ + ^ S " ^ ) - 1 , the solution of Nhik = rk requires the solution of two Neumann problems, one in Qi and one in ft2. Therefore, the global cost of each iteration amounts to solving four boundary value problems for the steepest-descent method, and six boundary value problems for the minimum-residual method. Similar considerations can be carried out for the conjugate gradient iterations, which have the same computational cost as the steepest-descent iterations. O 4.2.3

T h e case of complex matrices

As we are envisaging the application of subdomain iterations to non-symmetric boundary value problems for complex functions, we start by formulating the complex counterpart of Theorem 4.2.7; the proof is similar and will be omitted. T h e o r e m 4 . 2 . 9 Let P and Q be two M x M non-singular Then the preconditioned Richardson iterations Xk+1

=\k

+9P~1(G-QXk)

complex

matrices.

§4.2

SPLITTING OF O P E R A T O R S AND PRECONDITIONING

129

converge to the solution A of QA = G if, and only if, ,

,2

2

\vsI < - Re vs for each eigenvalue vs of P - 1 Q , s = 1 , . . . , M. As a consequence of this theorem, we can prove the following results, in which the preconditioner for the matrix Q = Qi + Q2 is given by Q2 itself or else by (•&iQi1 + v i Q 2 l ) ~ x • Corollaries 4.2.11 and 4.2.14 below will be used for the convergence analysis of the time-harmonic Maxwell problem. Theorem 4.2.10 Suppose that A/"(f?) : = R e [Qij, rj\ Re[Q2i], 1?] V rj £ CM \ 0,

+Im[Qri,r)]Im[Q2r),ri}>0

where [•,•] denotes now the Euclidean scalar product in CM. (4.2.15)

CQ '•=

min

Moreover,

set

>0

and (4.2.16)

max I g M l l > 0. r}ecM\o \{Q2Tf, »7]|

C0:=

Then each eigenvalue vs ofQ^Q (4.2.17)

C2

~

satisfies Vs=l,...,M.

\v,|2

Proof Since (4.2.14) implies that [Qr),rj\ ^ 0 and [Q2ri,rj\ ^ 0 for each rj £ CM \ 0, the ratios in (4.2.15) and (4.2.16) make sense. Let v be an eigenvalue of Qr2 1lQ- Its associated eigenvector a; 6 CMh, satisfies Qui = vQ2u>, therefore [QU3,UJ] =

v[Q2W,uj],

or, equivalently, Re[Qu;,u>] = Re^Re[(3 2 w,a>] -

lmi/lm[Q2io,(j]

lm[Qii},tjj\ = Re vlm[Q2u),(J\ + Imj/Re[<32w,a>]. If we multiply these equations by Re[<32k>,o>] and lm[Q20J,u)], respectively, and sum up the results we find

130

C O N V E R G E N C E ANALYSIS F O R ITERATIVE D D A L G O R I T H M S ( R e v) [ ( R e [Q2OJ,U>])2 +

Ch.

4^

(Im[Q2u;,H)2]

= Re{Q<jj, w] Re[Q2w, w] + Im[Qw, o>] Im[Q2u, v] = N(u>). On the other hand, ,2 _

l[<>,w]j_ \[Q2U>M\2

<

~

so (4.2.17) follows.

•

Corollary 4.2.11 Under the assumption of Theorem 4-2.10, the Richardson erations applied to the preconditioned system (4.2.18)

Q2'QX

=

Q2'X

converge for each 9 G (0,C*) ; and the spectral radius of the iteration I — 9Q21Q

(4.2.19)

matrix

is bounded by a positive constant Kg < 1, which depends on Co and

Co, but not on the dimension Proof

it-

M.

From (4.2.17) we have \vs\2 < ? Rei>s u

V s = 1,..., M,

V 6 G (0, C , ) ,

therefore convergence follows from Theorem 4.2.9. Moreover, any eigenvalue of the iteration matrix I — OQ^1 Q has the form r = 1 -9v, where v is an eigenvalue of Q^Q. From (4.2.15) and (4.2.16), for each 6 G (0,C„) we find that |r|2 = 1 -29Rev

+ e2\v\2

<\-9{2Co-9Cl). The result follows upon setting k9 := [1 - 9(2 C0 - 9Cl)\1l2.

•

Remark 4.2.12 Note that the conclusion in Corollary 4.2.11 does not have the same strength as that in Theorem 4.2.2. Actually, by Corollary 4.2.11, we have only proved that the spectral radius of the (non-symmetric) iteration matrix is smaller than 1, whereas in Theorem 4.2.2 the convergence of the Richardson scheme is achieved with respect to a specific norm; namely, the norm induced by the symmetric part of the preconditioner Q 2 . • The analysis of the convergence of the preconditioned Richardson iterations with the Neumann-Neumann preconditioner ( o i Q ^ 1 - t - ^ Q ^ 1 ) ^ 1 is based on the following result, due to Berselli (1997). Theorem 4.2.13 Suppose that (4.2.20)

[QiVM? 0

V rj G C M \ 0,

where [•,•] is the Euclidean scalar product in CM,

i = l,2, and that

s

§4.2

S P L I T T I N G OF O P E R A T O R S A N D P R E C O N D I T I O N I N G

131

Re[QxJ7 ,rj]Re[Q2ri,rj]

(4.2.21)

+lm.[Q1r),r)}lm[Q2r),rj\ > 0

Moreover

V f? €

CM.

set

IA 9 99"! (4.2.22)

c„n — MwiW Oir:= - sup - , C 2r:= T)ecM\o |[V2»7,»?J|

-

sup - 7 r-. rjecM\o\[QiV,V\\

Then, for any choice of the averaging parameters ui > 0 and a2 > 0, every eigenvalue vs of (aiQ^1 + cr2Q2l)Q satisfies (4.2.23)

C* <

V s = 1,...

,M,

where (4.2.24) Proof

& : = min ( — L _ ,

^

(UI + A2Y

\<7I+CT2

+ H*

)

, H* := ( a ^ + ^

2

)

2

.

Since (
= (
+ <J2Q2lQX + cri<5^1<52,

any eigenvalue f of (ffiQ^ 1 + c^Q^T1)*? can be written as v = a\ + a 2 + ( , where ( is an eigenvalue of cr2Q21Qi + criQ^1Q2Let us transform Q21Q1 into an upper-triangular matrix U with diagonal elements given by the eigenvalues ( s of Q21Q1; namely, write ZQ21QiZ~1

= U,

for a suitable transformation matrix Z. The matrix U'1

=

ZQi1Q2Z~1

is also upper-triangular, and features the eigenvalues C71 of Qi1Q2 onal. In conclusion, Z{a2Q21Q1

on the diag-

+ct1Qi1Q2)Z-1

is upper-triangular, with the terms a 2 ( s + c i C " 1 on the diagonal; therefore, the eigenvalues of cr2Q21Qi + <JiQ^1Q2 are given by a2Cs + f i C " 1 We have thus proved that any eigenvalue v of (01Q^1 + a2Q21)Q can be written as v = where ( is an eigenvalue of Q2lQi• Proceeding as in Theorem 4.2.10, we find Re£ > 0 and |£| < Ci, and repeating the same argument for the matrix Q2XQ\ we also obtain R e ( £ _ 1 ) > 0 and < c2. Hence

132

o

C O N V E R G E N C E ANALYSIS F O R I T E R A T I V E D D A L G O R I T H M S R e v

_ 2

Ch. 4 ^

(Ji + cr 2 -I-o- 2 ReC + o"i Re (C *) (<Ti + c r 2 ) 2 + 2{ai + < T 2 ) [ a 2 R e C + cri Re ( C " 1 ) ] + |cr2C + ^iC" 1 1 2

> 2

cri + a2 + o-2 Re ( + (ai

+cr2)2

Re (C"1)

+ 2(a 1 + c r 2 ) [ a 2 Re ( 4- <7i Re ( C - 1 ) ] + # *

Note now that the function F*(0

:= 2

CTl + <J2 + ^ (<7! +

CR2)2

+ 2(<JI + CR2)£ +

H*

is strictly increasing when H* > (exi +
lim £->co

=

1

0-J+CT2'

hence C* is the infimum of F* for £ > 0.

•

Corollary 4 . 2 . 1 4 Under the assumption of Theorem 4-2-13, the Richardson erations applied to the preconditioned system M^1

+a2Q21)m~X)

<

it-

=0

converge for each 6 G (0,(7*), and the spectral radius of the iteration matrix I — OioiQi1 + cr2Q21)Q is bounded by a positive constant kg < 1, which depends on Ci and C2, but not on the dimension M. Proof

From (4.2.23) we have |z;s|2 < ^ R e i / S v

V s = 1,..., M,

V0e(O,(7*),

therefore convergence follows from Theorem 4.2.9. Moreover, any eigenvalue of the iteration matrix I — 9(oiQ^~1 + cr2Q21)Q can be written as r = 1 — 9v, where v is an eigenvalue of (o^Q^ 1 + <j 2 Q2 1 )Q• From (4.2.23) we find that, for each 0G (0,C*), |t|2 = 1 -29Rev

+ 92\v\2

< 1 _ e{C* - 9)\v\2 < 1 - 9(C* - 9)(cji + cr2)2, because in the proof of Theorem 4.2.13 we have shown that \u\

>Rqv>0i+02-

The result follows on setting kg : = [1 - 6{C* - 0)(ai

+ o2)2]1/2-

•

§4.3

4.3

CONVERGENCE OF DIRICHLET-NEUMANN M E T H O D

133

Convergence of the Dirichlet-Neumann iterative method

The analysis that we have developed in the previous sections can be applied to proving the convergence of the iterative substructuring procedures introduced in Sections 1.3 and 3.1. We start by considering the homogeneous Dirichlet problem for the Laplace operator (1.2.1) and its finite element counterpart (2.1.1). In Chapter 5 we will generalise our results to other families of boundary value problems. In what follows, we will confine ourselves to the case of a domain partitioned in two subdomains fii and Cl2 that do not overlap, as in Fig. 1.1. We have already noted in Section 1.3 that the Dirichlet-Neumann iterative method (1.3.1)-(1.3.3) can be interpreted as a preconditioned Richardson procedure for the Steklov-Poincare equation (1.1.7). The local Steklov-Poincare operators Si, i = 1,2, defined in (4.1.1), are symmetric. Moreover, in view of Proposition 4.1.2 they are continuous and coercive. We can therefore apply Theorem 4.2.2 taking X = A, Qi = Si and Q2 = S2, and conclude that the Dirichlet-Neumann iterative procedure (1.3.1)—(1.3.3) converges. Let us consider now the finite element approximation (2.1.1) of the same problem. Owing to (4.1.4), the discrete Steklov-Poincare operators Si}h are symmetric, coercive (with constant a ; ) , and continuous (with constant Pi). The constants en and pi are independent of h, since they only depend on the coerciveness and continuity constants of the bilinear forms a;, as well as on C\ and C2 introduced in Theorem 4.1.3. If we apply Theorem 4.2.2 with X = Ah, Qi = Sith and 0,2 — S 2 ,h, the rate of convergence of the iteration-by-subdomain method, being driven only by a* and Pi (see Remark 4.2.3), is independent of h in the norm associated with the preconditioner £2,h> or, equivalently, in the norm of the trace space A. The same conclusion holds for the general second-order symmetric elliptic operator L introduced in (1.4.2), as well as for its conforming finite element approximation. 4.3.1

A n alternative way to prove convergence

The convergence analysis of Dirichlet-Neumann iterations can be carried out by a more direct argument that does not exploit the analogy with Steklov-Poincare operators. This is derived proving that (4.3.1) where ek denotes the interface error ek :=

— U|p.

Clearly, ek+1 is related to ek through the iteration equation ek+1 = Tgek, where Tg : = I—flS^S. However, the proof of (4.3.1) can be based on a functional analysis argument for partial differential equations, and does not necessarily require the construction of Tg, and not even the reinterpretation of our problem in terms of Steklov-Poincare operators. The interested reader can refer to Marini and Quarteroni (1989), where such an approach is adopted for elliptic equations.

134

C O N V E R G E N C E ANALYSIS F O R ITERATIVE D D A L G O R I T H M S

Ch.

4^

The constant Mg in (4.3.1) is a lower bound for MT II

IIT0M||A

i|TsI|£(A)

This alternative approach can be particularly helpful for convergence analysis in those cases in which a Steklov-Poincare interpretation is not easily available. It is worthwhile to point out that the same argument can be developed at the finite dimensional level; in which case a relation between the convergence rates that are inherent in either approach can be established. As a matter of fact, let us suppose that the finite dimensional Steklov-Poincare operators and S2,h are both symmetric, and, moreover, that S2,h is coercive. In the finite dimensional space At let us choose the following scalar product {Vh,Vh)Ah •= {S2,hVh,Vh)', and denote by 11 • 11\h the associated norm. Similarly, in R W r we choose the scalar product M s : =

[S2.fc/*,t?],

where [•, •] is the Euclidean scalar product in R^ 1 - . The associated vector norm is denoted by | • |s2 h - Clearly, we have =

\\T(gh)vh\\Ah =

|(/-^sft)/i|S2ih,

where T#h^ : = I — OS^Sh, and fi and JJ denote the set of the values of p,h and rjfi at the Arp nodes on F, respectively. Hence

l i r j 'l|£(Ah) = sup —if——j-j

= sup

ryj

= III where we have denoted by || • ||s2 h the matrix norm associated with the vector norm | • |s2,h- Since the iteration matrix I — t0

[->-]a 2 , h , we finally have ||/ -

E^ is symmetric with respect = p(I - 0E~^E f t ), the spectral

radius of I - #E 22 ,h} i S h - Therefore, Mt\h) : = sup

< p(I

-

Hence, the convergence of the iterative scheme is proved if we show that Mg1'* < 1, which can be easier than proving that p(I — flE^E/,) < 1.

§4.4

4.4

C O N V E R G E N C E OF N E U M A N N - N E U M A N N M E T H O D

135

Convergence of the Neumann—Neumann iterative method

As we have already noted in Section 3.2, the Neumann-Neumann algorithm (3.1.5)—(3.1.8) is a Richardson iterative scheme for the Steklov-Poincare equation (2.2.3), with the operator —1 Mh-.= {aiS~X + a2S~)1\ x) acting as a preconditioner. The same is true in the infinite dimensional case. Since the Steklov-Poincare operators Si and S2 are symmetric, continuous, and coercive, using Theorem 4.2.5 instead of Theorem 4.2.2 we can prove the convergence of the Neumann-Neumann method for the homogeneous Dirichlet boundary value problem associated with the symmetric elliptic operator L. The same result holds for the discrete approximation by means of continuous piecewise-polynomial finite elements, provided that the assumptions of Theorem 4.2.2 are satisfied. The rate of convergence is independent of h when measured in the norm associated with the preconditioner Nh (see (4.1.10)), or, equivalently, in the norm of the trace space A. Remark 4.4.1 The results of the last two sections show that, for the twosubdomain case, both Dirichlet-Neumann and Neumann-Neumann iterations converge at a rate independent of h. The Neumann-Neumann algorithm has no particular advantage over the Dirichlet-Neumann algorithm (and actually the former requires more subdomain solves per iteration than the latter). The situation can, however, be different in the case of many subdomains (see, for example, Dryja and Widlund 1995). •

4.5

Convergence of the Robin iterative method

We present the proof of the convergence of the Robin iterative method introduced in Section 1.3 in the simplified situation 7i = 72 = 7 (we refer to P.-L. Lions 1990 for the general case). For the sake of simplicity, we will write the problem in its differential form; however, one could proceed as we will do in Theorem 6.4.2 (setting there the convective field b equal to zero), obtaining the proof of convergence for the variational formulation of the Robin method, as well as for its finite element discretisation. The local errors := u\ — u|q. , i = 1, 2, satisfy the following boundary value problems /

(4.5.1)

and

Ae?+1 = 0 ef+1 = 0

in 0 on dCli n dn

136

CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS -Ae*+1 = 0

(4.5.2)

Ch. 4 ^

in O2

„k+1

on d02 n dO

9 4 + 1

=

dn

onr,

dn

having denoted, as usual, by n the unit normal vector on T, directed from fii to O2• Multiplying (4.5.2)i by e^"1"1 and integrating by parts we find that

(4.5.3)

IIVerilU =

- /

r

de2+1 dn

ek+1 62

'

Using the identity AB = ±l(A

+

1B)2-(A-1B)2}

we can write

- I

de2+1 dn

„k+i 62

_ (de2+1 4 7J/r ^ dn " 47

_

_k+1 762

47 IT Using the boundary condition on T for e ^ 1 , we are left with

1

livens

I

(

dn

(4.5.4)

1 / / d e . 47 r \ ^7 /JT Repeating the same argument for ek+1

(4.5.5)

I Ve 1

112 Ho,ni

1

f

± J 47 7r

Adding (4.5.4) and (4.5.5) we obtain

„ t+1

7ei

we find that

r \ =

^n

dn k (94 + 7E2 V dn

§4.6

C O N V E R G E N C E OF T H E A L T E R N A T I N G S C H W A R Z M E T H O D

137

2

Now, summing over k from k = 0 to k = M — 1, it follows that M (4.5.7)

2

k=i

Consequently, the series oo

E ( l l V e J H § . n i +HV^IIg,na) k=I is convergent, and ek tends to 0 in ^ ( f t j ) , i = 1,2. In contrast with the Dirichlet-Neumann and Neumann-Neumann methods, for the Robin substructuring iterative method we have no estimates of the error reduction factor at each iteration, nor do we have any information about the rate of convergence. Remark 4.5.1 The convergence of the Robin method in the multi-domain case has been proved by P.-L. Lions (1990). •

4.6

Convergence of the alternating Schwarz method

The alternating Schwarz method presented in Section 1.5 furnishes a couple of sequences uk and uk+1/2 that converge to the solution u of the Dirichlet boundary value problem (1.4.1). Following P.-L. Lions (1988), we present here the proof based on a variational approach and not on the maximum principle, like the classical one. Recalling the results of Section 1.5, the sequences uk and uk+l/2 generated by the Schwarz method satisfy (4.6.1)

u-uk+1l2

=

u-uk+1

= (I

{I-Vi){u-uk) -V2){u-uk+1/2)

where, for i = 1,2, Vi = 1 { P * , li is the immersion of V* := {v e V | v = 0 in ft \ O J into V := IIq ( 0 ) , and V* is the orthogonal projection of V onto V* with respect to the scalar product induced by the bilinear form

138

C O N V E R G E N C E ANALYSIS F O R ITERATIVE D D A L G O R I T H M S

••=[

( i t aijDjwDiv

+

Ch.

4^

apwv

Let us denote by || • ||* the norm associated with the form a*(•,•), which is equivalent to the usual norm of H 1 (fi). Moreover, for any couple of vector spaces Wi and W2, denote by Wi © W2 their direct sum; namely, the vector space of all elements w that can be written in a unique way as the sum of an element in Wi and an element in W2. T h e o r e m 4.6.1 Assume that fi is a bounded domain in d = 2,3, with a Lipschitz boundary 9fi. If V* © V2 = V> then both uk and uk+1/'2 converge to u in Hq(Q.)

as k —> 00.

P r o o f Denoting the errors by ek := u — uk and ek+1/2 := u — uk+1^2, it is sufficient to prove the convergence to 0 of the sequence vk := ek!2. Note that vk e {V2*)x for k even and vk G (Vj*)1- for k odd. We have \\vk+1\\l + \\vk+1-vk\\l = \\vk\\l because i; f c + 1 — vk is orthogonal to vk+1. Therefore, the sequence ||wfc||* is nonincreasing and convergent, and ||t>fc+1 — tends to 0. Since vk is bounded in H1 (fi), by well known properties of Hilbert spaces (see, for example, Yosida 1974, p. 126) we can extract from each subsequence another subsequence vkm, which is weakly convergent to some limit v in H l ( f i ) , and, moreover, such that also vkm+1 is weakly convergent to another limit, say w, in IJ^iQ). Furthermore, owing to the properties of weakly convergent sequences (see Section 9.1), we find that ||iu — f||*
=0,

m—>cx>

and, consequently, w = v. It follows that v E (V{)1 n (V 2 *)^, and, recalling that ± the assumption of the theorem is equivalent to (V1*) n (V2 ) = { 0 } , we have v = 0. Therefore, we conclude that the whole sequence vk converges weakly to 0 in tf^fi). Take now k = 2l,l>0. Then ||*fc|| l=a*(el,el) = a*{el-\

=

a*((I-V2)(I-V1)el~\el)

{I - 7>i)(I - V2)el)

= a* [el~2, (I - Vi){I = a*(e°, e21"1/2)

= a*(e1-1

- V2)el+ll2)

= a*(e°,ek-1/2)

,el+1/2)

= a*(el~2, = a*(e°,

e'+3/2) v2^1).

The same result holds for k = 21 + 1 as well, therefore the weak convergence of vk implies the strong convergence to 0. • It is worthwhile to note that the assumption of Theorem 4.6.1 is always satisfied, provided that fi = fii Ufi 2 (see P.-L. Lions 1988). Under the slightly more

§4.6

C O N V E R G E N C E OF T H E A L T E R N A T I N G S C H W A R Z M E T H O D

139

restrictive assumption Vt* © V2* = V, which is satisfied for a large class of subdomains Q,i and 02, it is possible to give an estimate of the rate of convergence. Let us first prove a technical result. L e m m a 4.6.2 If (4.6.2)

ffi

V2* = V, then there exists a constant Co > 1 such that

IMI*
V v £ V.

Proof An application of the Open Mapping Theorem (see, for example, Yosida 1974, p. 75) to the surjective map (vi, v2) ->• +v2 from V* x V2 to V = V^ffiFg* yields the existence of Co > 1 such that for each v £ V it is possible to choose vi £ V* and v2 £ V2 such that v = +v2 and ( I M S + lkll*)1/2 < ColMIThen by the C'auchy-Schwarz inequality we have IM|2 = a*{vuv)+a*(v2,v)

= a*(vuViv)

< C0\\v\\*{\\Viv\\l +

+

a*{v2,V2v)

H^ull2)172,

and the result follows.

•

We are now in a position to prove a more precise convergence result. T h e o r e m 4 . 6 . 3 IfVx © V2 = V, then the iteration operators (7 - Vi)(I - V2) and (I — V\)(I — V2) are contractions in Hq(0) with respect to the norm induced by the bilinear form a*(-, •). Proof

From Lemma 4.6.2 we have | | ( / - P i H I *
Therefore, since V 2 is an orthogonal projection, 11(7 - V{)v\\2. = ||(/ - V2)(I > 11(7 - V2){I

- VM\l

+ IIW

-

- Vx)v\\l + C0"2||(7 -

VM\l VM\l

hence 11(7 - r2)(i

- v1)v\i2

< (i - CQ2)\\(I - vm\1

< (i - CO2) IMI 2 .

Interchanging the roles of the indices we find the same result, and the proof is complete. • As a consequence of this theorem, the convergence of the alternating Schwarz method takes place with a geometric rate. In fact, setting Kq : = (1 — Cq2)1''"2 , we have

140

C O N V E R G E N C E ANALYSIS F O R I T E R A T I V E D D A L G O R I T H M S

\\ek+%

=

Ch. 4

\\(I-V2)(I~V1)e%

< K0\\e%

<

Kk+1\\e°\U,

and analogously \\ek+3/2\U=\\(I-P1)(I-V2)ek+l/2\U < Kk+1\\el'2\\*

= Kk+1\\(I

- Pi)e°||. < i^+1||e0||*.

Similar convergence results hold for a decomposition with M subdomains, and in the finite dimensional case (see Bramble et al. 1991; see also Matsokin and Nepomnyaschikh 1985; and Dryja and Widlund 1989). The convergence rate turns out to be independent of the mesh parameter h, but it deteriorates as H being the maximum diameter of the subdomains fij, i = 1 , . . . , M. The introduction of a coarse grid correction makes the rate of convergence independent of both h and H, provided that the linear measure of the overlapping region between two subdomains is 8H, for 0 < <5 < 1. In the latter case, the rate of convergence depends only on 5 and on the coefficients aij and ao of the operator L.

i

^

5 O T H E R

B O U N D A R Y

V A L U E

P R O B L E M S

The theory that was developed in the previous chapters for the Poisson equation and for symmetric elliptic equations can be extended to the case of a more general differential problem of the form (5.1)

Cu = f

in ft,

where ft is a bounded domain in R d , d = 2,3, with a Lipschitz boundary, C is a partial differential operator, / is a given data, and u is the unknown solution. More precisely, we will consider: non-symmetric elliptic problems; the linear elasticity problem; the Stokes and Oseen problems for incompressible flows; a Stokes-like problem that arises from the analysis of compressible flows; the advection equation; and the time-harmonic Maxwell system. Our presentation will be developed according to the guidelines we have followed in the previous chapters. After having partitioned ft into two non-over, lapping subdomains fti and ft2, and denoted by T : = fti fl ft2 the interface (see Fig. 1.1), for each problem we derive its multi-domain variational formulation and introduce the associated Steklov-Poincare problem. Next, we illustrate iterative substructuring methods, and analyse their convergence on the basis of the abstract theorems of Chapter 4. The same presentation is then repeated for the finite dimensional approximation of each problem. Other stationary problems of a special nature may require an ad hoc interface treatment, and will be addressed in the following chapters. Precisely, convectiondiffusion equations when convection is dominant are discussed in Chapter 6, while the coupling between equations of different types is considered in Chapter 8.

5.1

Non-symmetric elliptic operators

We consider a non-symmetric elliptic operator of the following form: d L*w := — ^ Di(aijDjw)

(5.1.1)

1,3=i

d + ^ Dj(bjw)

+ a^w,

j=I

where the coefficients aij are assumed to satisfy the ellipticity condition d

Y j aij(x)£j£i > ao[£|2 1,3 = 1 for a suitable constant «o > 0.

for all £ 6 R d , for almost every x £ ft

142

O T H E R BOUNDARY VALUE PROBLEMS

Ch.

5 ^

The homogeneous Dirichlet boundary value problem associated with the operator l ) reads ( Lhu = f

in ft

{ u = 0

on 9ft .

Denoting by it* the restriction of u to ftj, i = 1,2, the interface conditions on r are the same as in the symmetric case (see (1.4.5)): ui = U2 (5.1.2)

du

L

on T

_du

driL

L

Q n p

dni

Clearly, the second interface condition in (5.1.2) can be replaced by dux dni

b 7«i =

du2 an i

h 7M2

on T,

for any function 7 defined on F. 5.1.1

Weak multi-domain formulation and the Steklov-Poincare interface equation

Let us assume that aij, bj and a0 belong to L°°(ft) for each l,j = 1 , . . . , d, and that d i v b € L°°(Sl). Then we can introduce in H1 (ft) the continuous bilinear form

ab(u;,t;):= (5.1.3)

f Jn

aij DjW Div + ( ^ d i v b + a 0 ) w v [i,j=i -

2

[ (v b • Vw — w b • Vv), Jn

associated with the operator i ) . Under the further assumption that (5.1.4)

i divb(x) + a0(x) > 0

for almost every x € ft,

the bilinear form a b (-, •) is coercive in V = HQ(£1). Therefore, the homogeneous Dirichlet boundary value problem (5.1.5)

find u G

: ab(u,v)

= (f,v)

Vvefl^ft)

has a unique solution. Defining in V; (see (1.2.4) for notation) the local bilinear forms

§5.1

NON-SYMMETRIC ELLIPTIC O P E R A T O R S

a^Wj.Wi) : = f (5.1.6)

^

Jti'

lu=i

aij DjWj DiVj + ( * d i v b + ia 0 V2

143

Wi Vi

+ o / (^ib • V ^ j - Wjb • V f j ) , * JQ i = 1,2, the corresponding two-domain weak formulation is: such that ' a\(uuvl)

(5.1.7)

k

= {f,v1)Ql

u\ = U2

on r

a\{u2,v2)

= {f,v2)n2

find

£ \\, u2 6 V2

V Vl e V°

V « 2 £ V?

a\{u 2 ,1l2lj) - (f,Tl 2 fi)a 2 +

iM)ni - a ^ u i , ^ ^ )

V fi E A,

where 7?.,; denotes any extension operator from A to Vj, and notation is the same as in Section 1.2. Note that, due to the choice of the bilinear forms a-(-, •), the last equation in (5.1.7) is the weak form of the interface condition (5.1.8)

du2 dnL

_

1,

dui

1,

2

~driL ~ 2

on r.

n U l

We introduce the Steklov-Poincare operators Si defined on the trace space A as (5.1.9)

(SiV,ti):=a\(E\v,Etn),

i|,/i£A,

where E\r) is the solution of the Dirichlet boundary value problem f EtneVi (5.1.10)

:

oS(^77,t;i)=0 „ E ^ r , = 77

V Vi £ V°

on T.

Moreover, for i = 1,2, define Xi £ A' as (5.1.11)

(xi, fj,) := (f,E\n) q ; -

V /i £ A,

where (5.1.12)

uj G Vj° : a ^ , v{) = (/, i ; , ) *

V Vi £ V?.

Then, denoting with A £ A the common value of u^p and u2|r, X turns out to be the solution of the following Steklov-Poincare equation

144

O T H E R BOUNDARY VALUE PROBLEMS

(5.1.13)

Ch.

5

SX = X,

having set S Si + S2 and x'•= Xi + XiB y following the proof given in Section 4.1.1 for the symmetric elliptic equations, it can be shown that the Steklov-Poincare operators S, Si and S2 are continuous and coercive. In fact, the solution E* satisfies the well known elliptic a priori estimate LL^LKN,


and the trace inequality (1.2.5), i.e. II^Nlkfii > C I M U . Besides, we have already noted that the bilinear forms a-(-, •) are continuous and coercive in ViMoreover, if the skew-symmetric part of S2 is small enough, we can prove that condition (c) in Theorem 4.2.2 is satisfied; namely ( ^ S ^ S r ? ) +

(£77,77)

>

K

V r j G A.

*

W i t h this aim, let us introduce the symmetric and skew-symmetric parts of the bilinear form a b (-, •), which are given respectively by r I"

(w, v):=

JA

i — (aij + aji) DjW DLv

d

V

Uj=i

L

+ [ - d i v b + ao ) wv (5.1.14) (w,v)

f r d / V

:=

Jn

i -(a tj 2

-aji)DjwDn

+ - ( « b ' Vw — w b • Vv)

Clearly, ah = as + ass. In a similar way we can define the local symmetric and 1 skew-symmetric parts a-(-, •) and a®s(-, •), i = 1,2, at the subdomain level. Setting (j, := S21Sr], from (5.1.9) we have < S 2 R S ^ S r j ) + (Sri, rj) = (S277,

- (SV, 77) + 2(Srj, 77)

= (S2r1,^)-(S2^r1) =

a2(E2r],

2(Sr],r1)

E\n) + as2s(E2r),

-a\(EE2TJ) -

+

2af(Eb2Tj, E^Srj)

E2n)

- af(EB2N, E\r]) + 2(Sri, 77) + 2{Srj, r,).

^

§5.1

NON-SYMMETRIC ELLIPTIC O P E R A T O R S

145

Denoting by a the coerciveness constant of S, we have (Si^S^Sr,)

+ (Sr/,rf) > 2a\\V\\2A - 2\as2s(E'2V, E^S,1 Srj)\.

Therefore, we only have to show that (5.1.15)

< p\\V\\2A, 0 < p < a.

lafiE^ElS^S^l

Setting : = max((aij - a^j(|(sn ; ) + ||b||L~(^)> i = 1,2, we have |afiE^E'S^Sr,^ <

^KfWE'vW^WE^SvW^

^C.K.fU'nMS^SvllA
and (5.1.15) follows if C3 h'f < a, which is a smallness assumption on the skewsymmetric part of the operator l ) in fi 2 • If the skew-symmetric part of the operator L b is small in both Qi and 0 2 , we can also prove that condition (c) in Theorem 4.2.5 is satisfied; namely, that (Afrj,Af~1Sr}) 4- (Sr),rj) > k*||t?||a where Af = (01 S f 1 + cr2S21)~l. (M^M-1

V ^ A ,

In fact, setting p. ~ Af^Sr],

Sr)) + (Srj,V) = (AfV,p)

- (SV,r]) +

= {Afrt,?) Moreover, setting pi : = S^Afrj S1P1 = S2p2,

A f p , = Siii

=

and

: = S~1Afp

we have

2(SV,V) +2{Sri,r}).

for i = 1,2, we have Afrj =

S2£2,

rj = A/"1 Afr] = aipi + (T2P2 and similarly

p. = CTi^i + cr2&!-

Therefore (Afr},p)

~(Np,v) = (Aft], (Ti& + a2(,2) ~ (Afp, aipi + <J2p2) = M(SIPI^I} - (sUi,pi)) + o2((s2p2,^2) 2 = 2 ^ > iaf^puElii) i=l 2 = 2 2 aiaf(E\S-lAfi7, E^S-'Sv).

-

(s2(2,p2))

146

Ch.

OTHER BOUNDARY VALUE PROBLEMS

5^

Proceeding as before, we obtain that condition (c) in Theorem 4.2.5 is satisfied, provided that for both i = 1,2 the constant nf is small enough. 5.1.2

Substructuring iterative methods

The Dirichlet-Neumann iterative scheme for (5.1.7) reads ' find uk+1 (5.1.16)

e Vi :

a\(uk+1,vi) _ u = X

=

V Vl G V

(f,v1h1 on r

k

find UK+1 £ V2 a'2(uk+\v2)

= (f,v2)n2

Vv2£V2°

(5.1.17) a\(uk+l ,U2y)

+ {f,K

= {f,n2^)n2

-a[(uk+1,Tl1^)

(5.1.18)

i/i)0l V/xeA

on r.

A* + 1 = 9 ^ + } + (1 - 0)\ k

Again, following the proof given in Section 1.3 for the Laplace operator, it can be shown that the Dirichlet-Neumann scheme (5.1.16)-(5.1.18) is equivalent to a preconditioned Richardson method for (5.1.13) where the preconditioner is provided by S2: Xk+1

=

\ k

+

S

-

1

(

x

- S \

k

) .

The convergence in A of these iterations is a consequence of Theorem 4.2.2, because we have already verified that all the assumptions are satisfied. Moreover, it is easily checked that the rate of convergence depends only on the continuity and coerciveness constants of SI and S2, as well as on the continuity constant of El A similar convergence result can be also found in Marini and Quarteroni (1989). Following the guidelines of what was carried out in Section 1.3 for the Laplace operator, it is also easy to introduce the Neumann-Neumann iteration scheme applied to (5.1.5). It corresponds to a Richardson method for the Steklov-Poincare equation having (ctiS']~ 1 +c72S^" 1 ) _1 as a preconditioner, therefore its convergence is a straightforward consequence of Theorem 4.2.5. The rate of convergence depends on the continuity and coerciveness constants of Si and 52, as well as on the continuity constants of EL and E\.

T H E P R O B L E M OF L I N E A R E L A S T I C I T Y

§5.2

5.1.3

147

The finite dimensional approximation

The same arguments as in the previous sections can be used for the finite element approximation of the homogeneous Dirichlet boundary value problem associated with the operator i), taking finite element spaces V"^ C H1(Qi) as in Section 2.1.1.

Although the problem is of the advection-diffusion type, a standard Galerkin approximation can be successfully used (without resorting to any stabilisation approach) if n f is small, because in this case the problem is dominated by the diffusion. The discrete Steklov-Poincare operators 5',;^,, which are the finite dimensional counterparts of those introduced in (5.1.9), are coercive and their coerciveness constants are independent of h. Actually, they depend only on the coerciveness constants of the bilinear forms a\(-,-), as well as on the constant in the trace inequality (1.2.5). Moreover, their continuity constants can be estimated in terms of those of the forms a\, as well as those of the finite element extension operators E\ h , which are defined as ' &i,hr)h e Vi,h : (5-1.19)

{ a\(E\hr]h,Vith) = 0 ElhVh

= Vh

V vi>h G V°h

on r .

Setting Vith = E\ hr]h — HijiJlh, where Hi^ are the finite element harmonic extension operators introduced in (2.2.1), we have a^El^El^n)

= a^E^h,

Hi>hVh),

hence Kfc^lliA

< C\\HithVh\\ltQi,

where C > 0 is independent of h. Assuming that the family of triangulations Th of Q is regular and the family of triangulations M h of P is quasi-uniform, we can apply the uniform extension Theorem 4.1.3, and find that also E\ h , i = 1,2 are uniformly continuous operators. All the assumptions of Theorem 4.2.2 are therefore satisfied, with constants independent of h. As a consequence, we can conclude that the Dirichlet-Neumann iterations (namely, the equivalent preconditioned Richardson iterations applied to the Steklov-Poincare equation) converge, at a rate independent of h. Applying Theorem 4.2.5, a similar result holds for the Neumann-Neumann iterative scheme.

5.2

The problem of linear elasticity

For linear materials, the stress tensor
Ch.

O T H E R BOUNDARY VALUE PROBLEMS

148

atj(w)

: = p,{DiWj + DjWi) + A d i v w ^ - ,

l,j = 1

5^

,...,d,

where fi > 0, A > 0 are the Lame's constants and Sij is the Kronecker's tensor. Let us define d (Lw); := - ^ D j a i j i w ) ,

l =

l,...,d,

3=1 and note that L w = —div a(w) = —/iAw — (fi + A ) V d i v w . Then we consider the following boundary value problem: seek u : ft —» R d such that in

(5.2.1)

ft,

I = 1 , . . . ,d

on VD

u =
o n Tn,

l =

l,...,d,

3=1 where f ,
UI = U 2

d

d

3=1

(U1 )n3 = XI °l3 3=1

(UAnj

on T,

I = 1 , . . . , d,

and they express the continuity of displacements and normal stresses on T. 5.2.1

W e a k multi-domain formulation and the Steklov-Poincare interface equation

We introduce the bilinear form (5.2.3)

e(w, v ) : = ^ f 2

Y ] {D t Wj + DjW t ) (D[Vj + DjV t ) + A [ d i v w d i v v , j,l=l

THE PROBLEM OF LINEAR ELASTICITY

§5.2

149

and set, as in (1.4.11), (1.4.12), H£D(Sl):={vGHHn)\v |rD=0} {4>A)Vn~

Jr N

Assume that f G ( L 2 ( f t ) ) d , tpD G ( H 1 / 2 { T D ) ) d and
find

W GmD(n))d

: e ( W , v ) = (f,v) +

Then the

(tpN,v\r„)rN

where tpD G (H 1 (Q,)) d denotes any extension in fl of the non-homogeneous Dirichlet datum
(5.2.5)

d

+

Mli,n

V v G [Hy d (Cl)) d .

This result is proved upon combining the inequality (see, for example, Fichera (1972), p. 382)

f

\v\2>K<Mlu

Jo. . !=1

V v G ( f f W

Ja

together with the generalised Poincare inequality / M2 < C [ E (DiVj + DjVl)2 Jn Jn. i=1

V v G { H £ b (Q))d,

which can be obtained using a standard contradiction argument. In the case of the pure Neumann problem, i.e. FD = 0, compatibility conditions on the data f and ipN are necessary to prove the existence and uniqueness of the solution. We refer the interested reader to Section 1.4 (where similar considerations for general second-order elliptic equations are presented) and to Fichera (1972). The multi-domain weak formulation of (5.2.4) has the following form. Take, for simplicity, Tjv = 0 and ipD = 0, and for each subdomain i = 1,2, let us introduce the bilinear form

Ch.

O T H E R BOUNDARY VALUE PROBLEMS

150

* (WI, VJ) : = (5.2.6)

/

2 Jn<

d, Y ] {DiWij

+ DjWij)

(DiVij

+

5^

DjVij)

3,1=1

+A / divw,; d i v v j JUi Then problem (5.2.4) can be written as: find u i 6 (V\) d , u 2 G (V2)d such that f e i ( u i , v 1 ) = (f,v1)n1 on

U! = U2

(5.2.7)

VvieiV?)'1

r

e2(u2,v2) = (f,v2)n2 e2(u2,K2fM)

Vv2G

(Vg)d

= (f ,H 2 ii)Q 2 + ( f , W i / i ) n i -ei(uunlfi)

VMG(A)d,

where u,; is the restriction of u to SI;, 72., denotes any extension operator from (A)d to [Vi)d\ the spaces Vi, V° and A have been introduced in Section 1.2. Problem (5.2.7) can be transformed into an equation set on the interface T. In fact, for i = 1,2 and for each tj G (A)d introduce the solution Erf of the Dirichlet boundary value problem (SiVe(Vi)d (5-2.8)

:

e ^ v ^ O - £if] = r)

Vvie(VP)d

on T,

and the solution u* of (5.2.9)

u* G (V°)d

: EI(u?,Vi) = (f,

Vi

)Q;

V v< G ( V ? ) d .

The Steklov-Poincare operators are defined as (5.2.10)

( S ^ H ) : = e 4 ( £ f l , £i(i)

Vr?,M€(A)d.

It is easily seen that U|Q. = £jU| r + u*; therefore, setting A : = U|r, the stress continuity condition (the last equation in (5.2.7)) can be rewritten as the following interface equation on F: (5.2.11)

=

where S := Si + S2 and x •= Xi + X2 € ( A ' ) d , with (5.2.12)

( X i , i t ) : = (f , S i i t ) a i - ei(u?,£iM)

V / i 6 (A) d .

§5.2

THE PROBLEM OF LINEAR ELASTICITY

151

The Steklov-Poincare operators Si and S2 are symmetric, while their coerciveness is a consequence of the Korn inequality (5.2.5) stated in ft; and the trace inequality (1.2.5). Consequently, S is also a coercive operator. Finally, continuity of Si and S2 follows from well known a priori estimates for the solution of (5.2.8).

5.2.2

Substructuring iterative m e t h o d s

The Dirichlet-Neumann iterative procedure for solving (5.2.7) reads: given A0 G (A) d , for each k > 0 solve find U j + 1 G (Vi) d (5.2.13)

e i ( U I + 1 , V i ) = (f, VI)Q, u

k+1

V vi G (V1°)d

on r,

= A*

then f i n d u 2 + 1 G (V 2 ) d e 2 (u 2 f c + 1 ,v 2 ) = ( f , v 2 ) f

V v 2 G (V2°)d

(5.2.14) e2(uk2+1,K2li)

=

(f,K2n) >n s 2 +

(f,Hifj,)Ul V / i g ( A)d,

-el{uk+\Thti) and finally update the interface value (5.2.15)

A * + 1 = 0 u k + x + (1 - 6)\ k

on T.

By a straightforward computation, similar to that used in Section 1.3 for the Laplace operator, we can verify that this iterative scheme on Xk reduces to the preconditioned Richardson method (5.2.16)

Xk

+1

=XK+

0S-1

_

k

>

Q

Convergence is therefore proved by applying Theorem 4.2.2. Similarly, it is also easy to introduce the Neumann-Neumann iteration scheme applied to (5.2.4), and to show that it corresponds to a Richardson method for the Steklov-Poincare equation having ( c i S j - 1 + 0 2 S ^ 1 ) - 1 as a preconditioner. Its convergence is thus a consequence of Theorem 4.2.5. 5.2.3

T h e finite dimensional approximation

The finite element approximation of the problem at hand can be based upon the usual piecewise-polynomialfinite element spaces ( V h ) d , {Vith)d, {V^Y and ( A h ) d

152

Ch.

OTHER BOUNDARY VALUE PROBLEMS

5^

defined in (2.1.3), (2.1.7) and (2.1.8). If we make, for simplicity, the assumption that 1TV = 0 and (pu = 0 on <9fi, then the finite element problem corresponding to (5.2.4) is (5.2.17)

find uh e ( V h ) d : e(uh,vh)

= (f,vh)

V vh e

(Vh)d.

The finite dimensional multi-domain weak formulation can be obtained in a similar way from (5.2.7), by projecting the first equation on {V°h)d,

the third

one on (V 2 ° /[ )
:

v iifc ) = 0

V v ^ e (V?h)d

on r ,

. €i,hVh = Vh

the discrete Steklov-Poincare operators are defined through (5.2.19)

V ifo, ph G ( A h ) d ,

(Si,hVh, t*h) •= ei(£ithrih,£ithnh)

t = 1,2,

and the Steklov-Poincare equation by (5.2.20)

Sh\h=Xh,

where Xh is the finite element counterpart of x defined in (5.2.12) and Sh : = Si,h + S 2 j i - The solution A/, of (5.2.20) is given by u ^ p , where is the solution to'(5.2.17). We have already pointed out that the bilinear forms e,-(-, •) associated with the operator L are symmetric, continuous, and coercive in ( V i ) d , therefore the Steklov-Poincare operators are symmetric, and, as a consequence of the trace inequality (1.2.5), coercive, uniformly with respect to h. Their uniform continuity can be proved as follows. Taking in (5.2.18) v.;^ = £i,hVh ~ H-i,hVh, where r)h = (rjhii, • • •, ? ? M ) , (5-2.21)

Hi,hVh : = {Hi,hrih,i, •••, HiihVh,d)

and Hi^Vh is the solution to (2.2.1), we have ei(£i,hVh,£i,hVh)

=

ei(£i,hVh^i,hVh)-

Hence, if the family of triangulations Th of f2 is regular and the family of triangulations A4h, induced on T by Th, is quasi-uniform, we can use the uniform extension Theorem 4.1.3 and find LL^^LKN,. < KWn^hVhWi,^

< KC2\\VhHA

r

THE STOKES PROBLEM

§5.3

153

where K > 0 is independent of h. Using this inequality in (5.2.19) shows that the Steklov-Poincare operators are uniformly continuous. If we now consider the preconditioned Richardson method (5.2.22)

\kh+1=x

+

kh

es-l(Xh-sh\kh),

by the usual arguments we can show that it is equivalent to the Dirichlet-Neumann iteration-by-subdomain procedure: ' findujjj1 G (V\,h)d : (5.2.23)

eiKJ;

1

, Vlih) =

(f, V l i h ) n i

V vlifc G

[Vlh)d

on r

ffindu^e^)*

:

e 2 (u*+ 1 ) v 2 , h ) = (f,v 2 , A ) n a

V v 2 , h G (V 2 \) d

(5.2.24) e 2 (u*+ 1 ,ft 2 ih /i h ) = (f ,n2,hfJ.h)Q2 + (f ^ u h P h h i -ei (uf+SWi.hMh)

V»he(Ah)d,

with (5.2.25)

XKH+1=0UK2+]R

+ (I-E)XKH

onr.

As usual, 7lith denotes any extension operator from ( A h ) d to (Vi:h)d, i = 1,2. The convergence of the Dirichlet-Neumann algorithm can be proved by applying Theorem 4.2.2, and we also conclude that the rate of convergence is independent of the grid size h. A similar result holds for the Neumann-Neumann iterative scheme, making use of Theorem 4.2.5.

5.3

The Stokes problem

Another example we consider is a classical problem in the theory of incompressible fluid dynamics. It involves the linear Stokes operator, and in this case the stress tensor T(w, q) is given by (5.3.1)

Tij (w, q) := v(DtWj + Djwi) - q5u,

l,j =

l,...,d,

where v > 0 is a given constant. The Stokes problem reads: find u : ft C R d —> d = 2,3, and p : ft ->• R such that

OTHER BOUNDARY VALUE PROBLEMS

154

FIG. 5.3.1. Boundary decomposition into —divT(u,p) = f

in ft

div u = 0

in ft

u = (pD

on VD

Ch.

5^

and

(5.3.2)

T(u,p) • n* - < p N

on I V ,

where f, (fio and N are given vector-valued functions. As usual, n* denotes the unit outward normal vector on 3ft, YD UT^y = <9ft, T o n r j v = 0, and either o r V N c o u l d b e e m p t y (see F i g . 5 . 3 . 1 ) .

The unknowns u and p denote, respectively, the velocity and the pressure of a viscous incompressible fluid that is confined to the region ft, and is subjected to a body force f, to a surface normal stress tpN on P/v, and has a prescribed velocity tfio on Note that in the momentum equation —divT(u,p) = f the pressure p occurs only through its gradient. Therefore, when TJV = 0 (namely, when we are imposing the Dirichlet boundary condition for the velocity on the whole boundary 3ft) the pressure is defined up to an additive constant. To recover uniqueness, in this case it is usually assumed that the additional condition fQ p = 0 holds. This request is not necessary when Tjv ^ 0, because the pressure p is present in the boundary condition on This different behaviour motivates the development of two alternative formulations, one for the case where Tjy = 0, the other for the case where r ^ ^ 0. A more explicit form of the boundary condition on T/v reads

+

i=i

Diui)n*3

~Pn*i

=

(W)z,

1

= 1,. • •

As is well known, other boundary conditions are admissible for the Stokes

r

THE STOKES PROBLEM

§5.3

155

problem (see, for example, Pironneau 1988, pp. 123-7; Quarteroni and Valli 1994, Section 10.1.1), which, however, will not be considered here. The incompressibility equation div u = 0 yields the more familiar identity —divT(u,p) = —vAu + Vp, which gives rise to the most common form of the conservation of momentum for an incompressible viscous flow. The interface conditions for the Stokes problem are given by the continuity of the velocity field and that of the normal stress vector T(u,p) • n; precisely, denoting by u,- = u ^ . and pi = p\Qif i = 1,2, the interface conditions take the form Ui = u 2

on T

d (5.3.3)

jUi,i)rij - pini i=i

d = u'Y(DiU2,j

on r ,

+ DjU2,i)nj - p2ni

l=

l,...,d.

J'=I

We do not, therefore, have to fulfil the continuity of the pressure field. Let us note that in the two-dimensional case, if the interface T is parallel to a coordinate axis, say the x2-axis, the interface conditions on the normal stress split into duiT , du2,n „ V——

on

+ V—-—

or

= V——+V——

on

on r .

or

Note, moreover, that, by virtue of the condition ui = u 2 on T, the latter condition can be reduced to v

dui.r ^ on

—

du2r on

—

o

n

r

-

A generalisation of the Stokes problem is the so-called Oseen problem - v A u + (b • V ) u + V p = f div u = 0 I u = tpD

in fi in fi on <9fi ,

where b is a given vector field (which could be the velocity field itself at a preceding time step, when considering a time-advancing scheme for the discretisation of the non-stationary, non-linear Navier-Stokes equations). The problem is nonsymmetric, due to the presence of the first-order term (b • V ) u . If b is continuous

Ch.

O T H E R BOUNDARY VALUE PROBLEMS

156

5^

across P, then the interface conditions for the Oseen problem are the same as for the Stokes problem. However, due to the lack of symmetry, the convergence of substructuring iterative methods can be studied by means of the same technique that we are going to present in Chapter 6 for scalar advection-diffusion equations. R e m a r k 5 . 3 . 1 The incompressibility constraint d i v u = 0 has several consequences, concerning both the existence theory and the numerical approximation of the solution. In particular, it induces compatibility conditions on the data. In fact, as a consequence of the divergence theorem we have / U|QO • n = / d i v u = 0, Jao Jo for each set O C fi, having denoted by n the unit outward normal vector on d O . Therefore, in the case where P^r = 0 (namely, for the pure Dirichlet boundary value problem), the boundary datum has to satisfy the compatibility condition (5.3.4)

[
Moreover, if Tat intersects only one of the two subdomains Qi or fi 2 , say Tat C <9fi2, a compatibility condition on T : = Qi H fi2 has to be satisfied; namely /

Jr

U|R • N =

-

/

Jr

U|R

Jn 1

N +

/

Jddi\ r

U|9NI\R-N*-

divu-/
= -

/

JdQi\r

Jen i\r

(FID

N*


having chosen on T the unit vector n directed towards • This condition cannot be avoided if T ^ = 0; on the contrary, when Pyv f 0 the decomposition of ft will be chosen in such a way that this compatibility condition on T is not necessary (see Remark 5.3.6). • 5.3.1

W e a k multi-domain formulation and the Steklov-Poincare interface equation

W e warn the reader that our analysis will present some technical subtleties, due to the special nature of the problem, particularly: • The problem data should obey special compatibility conditions (see Remarks 5.3.1 and 5.3.3). • The functional spaces for admissible velocities and pressures should satisfy a suitable relation (see (5.3.7) and (5.3.8)), which in the finite dimensional case will be expressed by the well known inf-sup condition (see (5.3.43) and (5.3.45)). • The issue of the uniqueness of the pressure, which depends upon the type of boundary conditions, is reflected by a special choice of test functions for

i nc/ a l uiviia rn,wJtsij.c/ivi

10Y

the continuity equation, for both the one-domain and the multi-domain formulation of the Stokes problem. In order to derive the variational formulation of (5.3.2), and discuss its wellposedness, let us assume that f £ (L 2 (f2)) d , ipD £ (Hl/2(TD))d and
(5.3.5)

f v d s ( w , v ) := / - V (Diwm + Dmwi){Divm Jn z ,Z,m=1

+

Dmvi)

and (5.3.6)

b(w,q) := -

JQ

q divw,

for all w , v £ ( # 1 ( 0 ) ) d and q £ L 2 (ft). First of all, we remark that there exists a constant /3° > 0 (which depends on ft) such that (5.3.7)

V q£ Ll(Q)3v£(Hi(Sl))d,

where the space £o(ft) is easily proved, taking

v^O

: 6(v, q) > /J°||v||i,nlMlo,n,

been introduced in (1.4.17). Indeed, this inequality

v e {H^(n))d

: div v = -q

in ft,

which satisfies HVH^Q < C||g||o,n (see Girault and Raviart 1986, p. 24). Similarly, we have Proposition 5.3.2 Assume that Tp ^ dCl, or, equivalently, YN ^ 0. There exists a constant (3* > 0 such that (5.3.8)

V q £ i 2 ( f t ) 3 v G ( # ^ ( f t ) ) d , v ^ 0 : b(v,q) > /3*||v||i,n|H|0,n)

where ff^D(fl) is the space introduced in (1.4.11). Proof This inequality can be proved by means of (5.3.7). In fact, given a function q £ L 2 (ft), g / 0, denote by q its extension by zero in a Lipschitz domain ft' D ft, such that c dCl' (see Fig. 5.3.2). Define, moreover, qw := (meas ft')-1 Jfl, q. In correspondence to the function q - q n ' € L^(Si') there exists v ' £ (i^ 1 (ft')) d , v ' ^ 0, such that ( Q - <7fi') divv' > /3°||v'||i,Q>||g - §h<||0,n'. Setting v : = v j n 6 v|9n-

= 0), we have

(ft)) d , and taking into account that fQ, d i v v ' = 0 (as

un. b

UittEK, tSUUiNDAK* VALUE ^KUiiLEMS

b(v,q)

= - J

d i v v ' q = -J

divv'

> ^0||v'||i,n'||9- g b ' l k n ' > / ^ I M k f i l k - gh-lkn. In particular, it follows that v ^ 0. Moreover, Ikllo.n <

II? -

gn-1lo.fi +

||?fi'||o,n,

and meas

2 =

Therefore

ft

f f

m-ea, Q ' f i ( , / , , '<)

/

meas ft \

M

meas

\

..

= (meas

,.

_

ft Si') 2

/ f

\

[Jn V

.,

and taking m e a s n y

;=

\

meas

ft'/

inequality (5.3.8) follows.

•

The inequalities (5.3.7) and (5.3.8) yield the well-posedness of the Stokes problem in the case where I V = 0 (condition (5.3.7)) or where I V i 1 0 (condition (5.3.8)); see also Remark 5.3.4. Their finite element counterpart is known as the inf-sup condition, and is crucial for the analysis of the finite element approximation of the Stokes problem. Since H 1 / / 2 ( T o ) is the trace space on I V of i ? 1 ( f t ) , we can construct a vector function $ e (iJ 1 (ft)) d such that $ | p D = ipD. We also construct a vector function tf G ( ^ D ( f t ) ) d such that (5.3.9)

div $ = — div $

in ft.

r

§5.3

THE STOKES

PROBLEM

159

The existence of $ is guaranteed by the inf-sup condition (5.3.7) or (5.3.8) (see, for example, Brezzi 1974; and Brezzi and Fortin 1991). Note that, in the case where = 0, from (5.3.4) the necessary solvability condition

L

div$ = 0

is satisfied. The case where TN ^ 0 We start by presenting the variational formulation of the Stokes problem (5.3.2) in the case where Tiv 0- It reads find W 6

( f i ) ) d , p £ Z, 2 (0) :

{

a(W,v)+6(v,p)=^(v) 6 ( W , g) = 0

Vve(^

D

(0))

d

\fqGL2(n),

where W = u — $ T {— y ) $: =and (f,v) + (^,V|rjv)rjv - s ( * + *,v). The second equation in (5.3.10) clearly implies d i v W = 0 in fi, because we can take q = d i v W as a test function. Remark 5.3.3 In the case of the pure Neumann boundary value problem, i.e. when rjv = dQ or, equivalently, r ^ = 0, a different kind of compatibility condition between f and
(5.3.11)

We will not deal with this last case in what follows, and will always assume that T/) ^ 0; however, we refer to Section 1.4, where the interested reader can find how the pure Neumann boundary value problem associated with the general second-order symmetric elliptic operator is analysed. • R e m a r k 5.3.4 The existence and uniqueness of the solution to problem (5.3.10) is a consequence of well known abstract theorems; in particular, on the coerciveness of s(-, •) in (ffpD (0))d (which follows from Korn inequality (5.2.5)) and the validity of the inf-sup condition (5.3.8). • The variational formulation of problem (5.3.2) can be equivalently rewritten in a two-domain fashion. First of all, for all Wj, v , £ ( H 1 ( f l i ) ) d and qi £ L 2 (Q,i) let us define (5.3.12)

SI(WI,VJ):=

r

I

v

-

d

+ 5MWI,J)(DIUI,RA +

^ , 1 )

O T H E R BOUNDARY VALUE PROBLEMS

160

Ch.

5^

and bi{wi,qi):=~

(5.3.13)

qi div Wi. J Qi

The two-domain formulation of (5.3.10) reads: find ( W i , p i ) £ (Vi) d x L 2 ( Q i ) , ( W 2 , p a ) G (V2)d x L 2 ( f i 2 ) such that r a ^ W i . v O + biCvi.pi) = J i ( v i ) 6i(Wi,gi) = 0 Wi = W2 (5.3.14)

VftEl

2

V v i G {V?)d

^)

on r

s 2 ( W 2 , v 2 ) + 6 2 ( v 2 , p 2 ) = ^2(v2) 62(W2,92) = 0 s2( w2,n2n)

Vv2G(V2°)d

V g2 G L2(Q,2)

+ b2(n2fi,P2)

= t2(k2H)

+

^{ihn)

-s1(w1,n1^-b1(n1fM,p1)

v /i G (A) d ,

where JFi(vi) — (f,v i )n i + {tPN^i\rN)rNndni ^ := { „ G F ^ ; ) I «|rDnan, = 0 } , 1

A := {

v

G

F1/2(r)

V^0

: = {v G

IT? = W|r for a suitable v G

+ tf1^)

|^ A ^

= 0}

(fi)},

and denotes any extension operator from ( A ) d to {Vi)d, i = 1,2. For the sake of completeness, let us show the equivalence between (5.3.10) and (5.3.14). L e m m a 5.3.5 The Stokes problem (5.3.10) is equivalent to (5.3.14), in the sense that = W i and p\Q. = pi} i = 1, 2. Proof Let us start by proving that a solution ( W , p ) to (5.3.10) yields a solution to (5.3.14). Let us set W , := W| f i ., pi p ^ . We have that W j G Vi, Pi G I/ 2 (f2j) and all the equations in (5.3.14), except the last one, are trivially satisfied simply by extending by zero in the complementary domain any local test function. To verify the last equation, for each (i G (A) d let us define the vector function [Hill infti n \ n 2 n infi2 • Then the last equation in (5.3.14) can be equivalently rewritten as 3(W,kh)

+ b(H/i,P)

=

r

THE STOKES PROBLEM

§5.3

161

Since Tfyi £ this equation is a special case of (5.3.10) when we take v = Conversely, let us show that, if ( W i , P i ) is a solution to (5.3.14) and we set W then (W,p)

= / W l [ W2

in

in Q 2 '

'

= IPl {p2

in

in ft2 '

is a solution of (5.3.10). Clearly, W G (JI^D(fl))d

Then, for every v £ (H^D(Cl))d (V|Q. -fti/i)

P

we set /z := V|r £

(A) d .

and p £ L 2 (Q).

For i = 1,2 we have

£ ( V ° ) d and thus from (5.3.14) we obtain 2

s ( W , v ) + b(v,p) = 53[si(Wi,V|n. - R ^ + S i t W ^ / i ) ] i=i 2

2

+ X]Nv|fi; i=i

=

- Kit*, Pi) +

biiHi^pi)]

- KiP) + Ti{Hm)\ = .F(v). i= 1

The set of momentum equations of (5.3.10) is therefore satisfied. Now we consider the continuity equation. For any q £ L2(fl), both restrictions q\Q. belong to L2(Cti) and can be used as test functions in (5.3.14), therefore 2

6(W,9) = £ & i ( W i ) g m J t=l yielding the desired continuity equation.

=0, •

R e m a r k 5.3.6 If we split the domain fi in such a way that either 3f2i n T ^ = 0 or <902 H F N = 0 (clearly, this is always the case if TJV = 0; namely, for the pure Dirichlet boundary condition that we will consider later on), due to the divergence theorem the normal component of the solution W has to satisfy a compatibility condition on T; namely, (5.3.15)

J W|p • n = 0.

This constraint for the space of traces on P introduces some technical problems, which, when it is possible, we prefer to avoid. Therefore, in the case where Tjv 0 we choose T in such a way that dfli n TN 0 for each i = 1,2 (see Fig. 5.3.3), and consequently there is no need for (5.3.15) to be satisfied. • We transform now the multi-domain problem (5.3.14) into an equivalent problem set on the interface F, for the associated Steklov-Poincare operator. For each

162

Ch.

O T H E R BOUNDARY VALUE PROBLEMS

5^

FIG. 5.3.3. Two-domain decomposition. rj e (A) d consider now the solution ( U ^ , Pif7) € (Vi) d x L2(Qi), problem Si(UjJj, V i ) + 6i(vi, Pit)) = 0 (5.3.16)

bi(Uitl,qi)=0 Uif) = T)

i = 1,2, of the

V v< G (V°)d

yqiGL2(ni) on

r,

which is a Stokes problem in the domain Q; with a non-homogeneous Dirichlet boundary condition on T. The couple ( U ^ , P 2 ij) is called the Stokes extension in fii of the interface value r/ on T. Introduce also the solution ( U * , P * ) € [V?)4 x L 2 ( f t i ) of the problem Si(UJ, vO + bi(vi, P*) =

V vi G (K°)°

(5.3.17) 6i(UJ,gi) = 0

V ^ L

2

^ ) ,

for i = 1,2, which is a Stokes problem in with homogeneous Dirichlet boundary data on T. The local Steklov-Poincare operators Si : (A)d —>• (A') d are defined as:

(5.3.18)

(Sitj.At) : = Si(\5ir],TliH) +bi(HiH,Pin) + 0i{K>ifl, FiT))

vj7,/ie(A)d

and the continuous linear functional Xi £ (A') d as (5.3.19)

<Xi,/i> : = FiiKiH)

-

~h{nifi,P*)

SitU^nm) V

M

e ( A) d .

I HJti; bl'UKBS fKUBLKM

163

Introducing the Steklov-Poincare operator 5 : = S1+S2 and setting x ~ we are in a position to prove the following equivalence result:

Xi+X2>

LEMMA 5 . 3 . 7 If we set A : = W|P, where ( W , p ) is the solution to (5.3.10), then A satisfies the following Steklov-Poincare equation on T: (5.3.20) Conversely, setting

Ae(A)d

: (S\,p)

V p G (A)d.

= (X,p)

from the solution A of (5.3.20) we obtain the solution of (5.3.10) by

5.3.21

1 P\Qi

: = P i \

+

P * -

PROOF Let ( W , p ) be the solution to (5.3.10). By Lemma 5.3.5 W ^ i = 1,2, are the solutions to the two-domain problem (5.3.14), hence

and p ^ ,

W | 0 i = U i W | r + UJ P\Qi

= P i W

l

T

+

P * .

Therefore, the stress interface condition (5.3.14)6 can be rewritten as 2

^ [ S i ( U , W | r + U*,KiiM) + biCRiH, P , W , r + P*)) i=1

2

=

v /x e (A) d , i=1

which coincides with (5.3.20) if A = W| R . Conversely, taking the solution A to (5.3.20) and setting for i = 1,2 W i : = U,A + U* P i

: = P i \

+

P * ,

the same argument shows that ( W i , p i ) are the solutions of (5.3.14).

•

If we take the extension operator "R^fi = Uip, introduced in (5.3.16), then from (5.3.18) and (5.3.19) we obtain (5.3.22)

( $ » / , fi) = Si(Uifj, XJiH)

V r?, p, G ( A ) d

and (5.3.23)

(Xi,M> = ^ ( U i P ) - S i i V l U i p )

V M G (A) d .

Therefore, both operators Si are symmetric in ( A ) d , and S is symmetric too. Moreover, using the Korn inequality (5.2.5) in ft, and the trace inequality (1.2.5),

164

OTHER BOUNDARY

V A L U T ; FKUTILTTMS

LjXI. O

FIG. 5.3.4. Possible decompositions: both SI and S 2 are coercive (left), or only S2 is coercive, while Si is non-negative (right).

FIG. 5.3.5. Alternative decomposition: Si is coercive (hence, it could be used as a preconditioner in place of S2), while S 2 is non-negative. we can show that the operator S» is coercive in (A) d , provided that we have chosen F in such a way that dfli fl ^ 0. Having assumed that TD 0, this choice is possible for at least one subdomain: for definiteness, in what follows we will always suppose that <9512 Ct F D 0, so that S 2 is coercive. Since we have already assumed that dfli fl T^r 0 for every i = 1,2 (see Remark 5.3.6), the geometric decomposition is as described in Fig. 5.3.4 (instead, the decomposition of f2 in Fig. 5.3.5 is such that <9fi2nr£> = 0, the case that we are not going to consider in the following). The operator S 2 is coercive and therefore invertible, and can be used as a preconditioner for the global Steklov-Poincare operator S (see Section 5.3.2). Finally, the operator Si is non-negative (or coercive, if the decomposition of fl is as in Fig. 5.3.4, left), therefore the global operator S is coercive. Finally, the continuity of the Steklov-Poincare operators S» in (Aj d is a consequence of well known a priori estimates for the solution UJJJ of (5.3.16).

r

§5.3

THE STOKES PROBLEM

165

The case where Pat = 0 We shall highlight only those points of the procedure which are different than in the preceding case. As we have already noted, for the pure Dirichlet boundary value problem the pressure p is defined up to an additive constant; therefore, to have uniqueness we seek p in the space Lg(f2). The variational formulation reads find W e (ff 0 1 (ft)) d , p e L2(fi) (5-3.24)

s ( W , v ) + b(v,p) b(w,q)

= T(v)

:

VV e

(H%{Sl))d

V?e^(fl),

= 0

where W = u — $ — fl*, ^ ( v ) := ( f , v ) - « ( * + » , v). Again, the second equation in (5.3.24) implies d i v W = 0 in 0 , because we can take q = d i v W £ L ^ f i ) as a test function. Therefore, also in the case where TAT = 0 we have fn divWifi = 0 for every ip € L2(Ci), and not only for ip e LQ(Q). The existence and uniqueness of the solution to (5.3.24) follow from the coerciveness of s(-, •) in HQ(£1) and the inf-sup condition (5.3.7). The two-domain formulation is given by: find ( W i , p i ) G (Vi)d x ( W 2 , p a ) e (V2)d x L 2 ( f t 2 ) such that f M W x . v O + M v i . p i ) =^a(v!) bi(W1,q1) Wi = W2

= 0

VgieLg(ni) on r

S2(W2,v2) + 62(v2,p2) = ^ 2 ( v 2 ) (5.3.25)

V v i G (K°)d

b2(W2,q2)=0

V v 2 6 (V 2 °) d

Vq2EL2(02)

s2(W2,H2n)+b2{Tl2fi,p2)

=

+Fi{Hili)

-Sl(Wunlfi) -b^ll^Pi)

V p 6 (A) a

/ Pl = P2, J fix Jn2 where ^i(vi) := (f,Vi)n; the spaces Vj, and A are as in Section 1.2 and Tti denotes any extension operator from (A) d to ( V i ) d , i = 1,2.

O T H E R BOUNDARY VALUE PROBLEMS

166

Ch.

5^

The further condition on the pressure field P2 in (5.3.25) guarantees that fQp

= 0.

R e m a r k 5 . 3 . 8 In the two-domain formulation (5.3.25) the role of the spaces of test functions for Qi and ft 2 is not the same. Other two-domain formulations equivalent to (5.3.24) could be defined, with different choices of these spaces. Each specific formulation induces a natural way of devising substructuring iterative methods, because the latter are driven by the way the interface conditions have been split between the subdomains. For instance, (5.3.25) induces a Dirichlet-Neumann procedure in which (5.3.25)3 is used as a Dirichlet boundary condition in fii, and (5.3.25)6 as a Neumann condition in fl 2 . The condition of zero-average pressure (5.3.25)7 has to be imposed in f2i (taking p2 from the previous iteration) to ensure uniqueness of the pressure pi (see Section 5.3.2).

•

The following equivalence result is the counterpart of Lemma 5.3.5. L e m m a 5 . 3 . 9 The Stokes problem (5.3.24) is equivalent to (5.3.25), in the sense that W|fi. = Wi and p\Q. = pit i = 1,2. P r o o f The proof follows that of Lemma 5.3.5, and can be repeated with the following changes. In the first part, the only difference arises from the fact that the extension by zero of a test function q2 £ L2(Cl2) belongs to L2(fl) but not to Lg(fi), the space of pressure test functions in 0 . However, we have already noted that b(W, q) = 0 not only for any q 6 Ll(fl), but also for every q £ L2(Q), therefore we can proceed in 0,2 as in fii, extending by zero the local test functions and obtaining that b2(W2,q2) = 0 for every q2 £ L2(Q,2). In the second part, one notes that, for a test function q £ Lq(Q), in general, its restriction does not belong to L g ( f i i ) , but only to L2(Qi). Consequently, the result b(W,q) = 0 is true for all q £ Ll(Q.) such that £ L2(Qi). For a general q £ Lq(CI), setting

one has

b(W, q) = b(W, q-qni)+

HW, 1)

W , a f i • n* = 0. The last equality holds since

= 0.

•

To state the Steklov-Poincare equation corresponding to (5.3.24), we have to introduce the space

r

THE STOKES PROBLEM

§5.3

(5.3.26)

A : = j i ? £ (A)d \ j

167

rj • n =

which is the trace space on F of divergence-free functions belonging to (H^ (Q))d (see Remark 5.3.6). Moreover, for each r) £ (A) d denote by U ^ £ ( V i ) d , Pit] £ ^ ( H . ) , U* £ (V°)d and P* £ Ll(Sli) the solutions of the problems (5.3.16), (5.3.17), where L2(fli) has to be substituted by L^Qi), i = 1,2. The local Steklov-Poincare operators Si and the functionals Xi a r e defined as in (5.3.18), (5.3.19), respectively. L e m m a 5 . 3 . 1 0 If we set A : = W|r, where ( W , p ) is the solution then A satisfies the following Steklov-Poincare (5.3.27)

to (5.3.24),

equation on T

A £ A : (SX, £ ) = ( * , £ }

V /2 £ A.

Conversely, assuming that T is regular enough, from the solution X of (5.3.27) we recover the solution of (5.3.24) by setting (5.3.28

1

p](li

:=PiX

+

P*+pi,

where pi and p2 are obtained by solving the linear

system

\ p i - p 2 = ——f{S>< - X , n ) < measT

(5.3.29)

I Pi meas fti + p 2 m e a s f t 2 = 0. Proof

Let ( W , p ) be the solution to (5.3.24), and denote by Pfi; : =

I P\Vi meas ft, JQ. 1

the mean value of p on ft;. By the equivalence Lemma 5.3.9 the restrictions W|Q. and p\Qi, i = 1,2, are the solutions to the two-domain problem (5.3.25), hence = U j W | r + U* P\n> = PiW{r

+ P*

+pni-

We have added put to -PiW| r + P* in order to restore the correct mean value of p in fti (which, in general, is not zero in ft,, but only in ft). The stress interface condition (5.3.25)6 can be rewritten as

^ M l W i r i=i

+ VlKiH)

+bi(Rai,PiWlr

+ P*)}

2

=

- bi(HiH,PQi)]

Vn £

(A)d.

Ch.

OTHER BOUNDARY VALUE PROBLEMS

168

5

Moreover, 2

~Y'bi(1liti,pni)

= y2pni

/ M-n* = (pQl - Pn2)

hence taking as test function p, G A C (A) d , we have that A = W| r satisfies (5.3.27). Conversely, take the solution A to (5.3.27) and set, for i = 1,2, W i : = UjA + U* Pi

:=PiX

+

P*+pi.

To verify that these functions satisfy (5.3.25) we have to check only the last three equations; namely, the continuity equation in f22, the interface equation on F, and the equation for the integrals of pi and p 2 . The last one is given by (5.3.29) 2 . Concerning the continuity equation in denoting by (q2)a2 the mean value of a function q2 G L2(02) we obtain b2(W2,q2)

= b2(W2,q2

- ( ® ) n 2 ) + b2(W2,

(g 2 )n a )

= ( q 2 ) n M W 2 , 1 ) = - ( g 2 ) n a J A • n 2 = 0. Finally, using (5.3.18) and (5.3.19), the interface equation on T reads 2

£ [st{Wi,HiP) i=1

+

bi{iiiti,Pi)}

2

= 53[si(UiA, %iH) + bi{HiH, Pi A) i=1 +Si(U*,%i/x) + bi(Hip,, P*) +pibiCR.iP, 1)] 2

2

i=i

i=l

n* = (SX - X, M) + iz Fi&il*) i=i

Jr

- (PI - P 2 ) / M • N. Jr

Define, for each p. G (A) rf , the mean value pr as Mr :=

n meas 1 JfT M " -

The L2 (F)-orthogonal projection of p, on A is given by p — ^ r n (due to the regularity assumption on the interface T, the normal vector n belongs to (A) d ). From (5.3.27) we have

r

THE STOKES PROBLEM

§5.3

(S A - x , M > =

169

(SA-x,/*-A»rn)+/ir(5A-x,n>

(5.3.30)

I

f

f

\

meas Therefore, 2

[^(Wj.Wj/i) + t=i =

+(

^

[ fi- n) Ur /

(—^—(SX-x,n) Vmeasr

-pi

+p2

2

t=l having used (5.3.29)i.

•

The Steklov-Poincare operators enjoy the same properties as in the case where Fn ^ 9 (with fl decomposed as in Fig. 5.3.4, left): Si, i = 1,2, and 5 are symmetric, continuous, and coercive in (A)d. It is worthwhile to remark that from equation (5.3.27) we cannot deduce that SA = x , due to the fact that the space of test functions is not large enough (each function /i £ A is subjected to the constraint f r Ji • n = 0). Indeed, from (5.3.30) the equation on T is (5.3.31)

SX

— (SA, n) n = x meas T

^-=(x,n)n. meas T

Let us introduce the operator

meas 1 and the functional X ~ X

^ { X i n ) n. measl

The trace A = W|r of the solution to (5.3.24) thus satisfies (SX,fM) = (x,fi)

V/ie(A)d,

which is the Steklov-Poincare equation in the space (A) d . However, the operator S : (A)d -»• ( A ' ) d is not surjective, because it satisfies (Sf7,n)=0

V jj G (A) d .

In particular, (Sn, n) = 0, hence S is not coercive in (A) d . We have encountered a similar situation for the Neumann boundary value problem for a symmetric elliptic operator (see Section 1.4.1). Furthermore, S is not symmetric.

Ch.

OTHER BOUNDARY VALUE PROBLEMS

170

5^

On the other hand, for each r) e (A) d , the functional Sr) and Srf coincide when acting on A; that is, (Sri,® = (Sri,fi) therefore S is symmetric and coercive in A. Since the trace W|p indeed belongs to A, we have been led to write the Steklov-Poincare equation on T in terms of A and S, instead of (A) d and S. 5.3.2

Substructuring iterative methods

Let us assume that TN ^ 0. The Dirichlet-Neumann iteration-by-subdomain procedure for the multi-domain weak formulation (5.3.14) reads: given X° € (A) f / , for each k > 0 solve find ( W * + 1 , p * + 1 ) € [Vi) d x I 2 ( f i i ) : «i(W{+1)v1) + 61(vi,p{+1)=^i(vi)

V v ! G (V?) d

(5.3.32) 6i(Wf+1,gi) = 0 w

a+I

=

y

V 9i. 6 L 2 (fti)

on r ,

find (W k 2 + 1 ,p k 2 + 1 ) e [V2)d X L2{n2)

:

s 2 ( W * + 1 , v 2 ) + & 2 (v 2 ,p* + 1 ) = F 2 { V 2 ) (5.3.33)

b2(Wk+\q2)=0 s2(Wk+1

V V2 G (V2°)D

Vq2eL2(n2)

,H2ti) + b2{K2tx,pk2+1)

= F2(K2h)

+ Fy.iH.ru) VMG

-s1(wk+\n1ix)-b1(nlti,pk1+1)

(A)d,

and finally update the interface value as follows: (5.3.34)

Xk+1 = m j l

1

+ (1 - 6)Xk

on T.

A 'parallel' version of this iterative scheme is obtained by replacing by W j and pk+1 by p\ in the last set of equations (5.3.33). The Dirichlet-Neumann iterations (5.3.32)-(5.3.34) can be rewritten as a preconditioned Richardson procedure for the Steklov-Poincare equation (5.3.20). Noting that U ^ W ^ 1 - U*|r) = W f + 1 - U*, i = 1, 2, for each ji 6 (A) d we have

r

§5.3

THE STOKES

PROBLEM

171

<S 2 W*+\ m > = < s 2 ( w ^ 1 - u ; | r ) , | i ) = s2(wk+1

- u;,u2M)

= -62(U2At,p2A+1) +J-2(U2M) - S l ( W f + \ U l M ) - 6 i ( U i M , p f + 1 ) - S2(U2,U 2 /i), having used the last equation in (5.3.33). Moreover, (5.3.35)

&i(UiM,p? +1 ) = 0

for each i — 1,2, and s1(Wk1+1,Vlti)

= s1(V1(Wk1+1 =

(SiX

,n)

-V*llT),\Jlti)

+

+

s1(V*1,Vlfi)

s i ( U i , U i / i ) .

Hence (5.3.36)

(S2W^R1,M) =

(

X

,

M

)-(S1A

K

,

M

).

Therefore, we derive from (5.3.34) that the Dirichlet-Neumann iterations can be written in (A) d as A0 € (A) d Xk+1 = Xk +8S2\x-SXk),

(5.3.37)

k>

0.

The convergence of this algorithm has been proved at the finite dimensional level by Marini and Quarteroni (1989) for finite elements and by Quarteroni (1989) for spectral approximation, provided that 0 < 9 < 9* for a suitable 9*. We can now see that convergence in (A) rf is a direct consequence of Theorem 4.2.2; in fact, as we have already pointed out, both Si and S2 are symmetric and continuous in ( A ) d , S2 is coercive in (A) d and Si is non-negative (or coercive, if dfli n r c ^ 0), so that S is also coercive. The Dirichlet-Neumann iterative scheme in the case where TAT = 0 is quite similar. Given A0 G (A) d , for each k > 0 solve

si(W?+1, (5.3.38)

Vl)

+ 61(v1,pf+1) = Ji(v2)

MWf+1,gi)=0 Wk+1 =Xk [ V JQi

P

V ^ e L ^ i )

on r

r = - f P2, JQ;

and then proceed as in (5.3.33), (5.3.34).

V v , £ (7°)d

172

Ch.

O T H E R BOUNDARY VALUE PROBLEMS

5 ^

The interpretation as a Richardson iterative scheme is still valid. Actually, set for each k > 0 and i = 1,2 k PL •= Since pk+1 we have

1 meas fi^ JQ.

1

does not belong to ^ ( f t , ) , equation (5.3.35) does not hold, but instead

2

2

2

i=1

i=1

i=l

= p^^bdV^,

1) = -(p'nt1

~ Pnt1)

f n • n>

so that (5.3.36) becomes (S2W**1,

fi) = (X, Ai) - (Si A*, /i) + ( p ^ 1 - Pot 1

In other words, (5.3.39)

(S 2 W 2 fc +\/2> = ( X , t i ) - { S ^

In the case where Fjv = 0, i.e. when (5.3.33) satisfies / W^1 •n = - / W^ Jr ' Jr ' = -

1

= ^ Q 2 \ r ( f t 2 ) , the solution W £ + 1 of

• n2 = - / W ^ Jr '

/ div W * + 1 = Jn?

V £ e A.

1

fo2(W^+1,1)

• n2 -

[ 9f!2\r

k+1 rT W 2\r • 11 2|9n

= 0

having used the continuity equation in (5.3.33). Hence, taking the initial guess A0 in A, it can be shown by an induction argument that (5.3.34) yields Xk E A. for each k > 0. Therefore, the DirichletNeumann iterations can be written as (5-3.40)

\K+L = \k + es21{x-s\k),

k> o,

the equality being valid in A. All the assumptions in Theorem 4.2.2 are now satisfied in the space A, and convergence of these iterations follows as before. Note that the convergence of the sequence A/; does not follow if we take the initial guess A0 in (A)d. In both cases where TV 0 and TN = 0, we can also consider the NeumannNeumann iterative scheme, which can be defined following the guidelines of what

r

§5.3

THE STOKES PROBLEM

173

has been carried out in Section 1.3 for the Laplace operator. Since, as usual, this algorithm can be interpreted as a Richardson method for solving the SteklovPoincare equation (5.3.20) or (5.3.27), with ( o i S f 1 + < T 2 S ^ 1 ) ~ 1 as a preconditioner, its convergence is a direct consequence of Theorem 4.2.5. 5.3.3

Finite dimensional approximation: the case of discontinuous pressure

Let us consider finite element approximations of the Stokes equations that make use of discontinuous discrete pressures across inter-element boundaries. The case of continuous pressures will be considered in the next section. We therefore consider finite element spaces Vh C Hpo(Q) for each velocity component and Q h C L 2 ( f i ) for the pressure, as well as their subdomain restrictions: Vilh (5.3.41)

l'h

Ah

=

\vh£Vh}

= Vi,h n VP = W|r \vh e 14}

Qi,h = {^ifi; I
xrh n HlD (fi) c vh,

Ah = H ,

r

I vh £ xrh n H£d ( f i ) }

for some r > 1 has been introduced in (2.1.2) or (2.1.4)). We assume that the inf-sup condition is satisfied. W h e n TN = 0, we have i ? P D ( f i ) = H i (fi), and the inf-sup condition takes the form: there exists a constant /3 > 0, independent of /i, such that V

^

e

(5.3.43)

H L g ( f i ) 3 v , £ CV h ) d C [ H l { n ) ) d , VhT^O : b(vh,qh)

> j 9 | | v h | | i , n ||%||o,n-

Moreover, we assume also that the local inf-sup condition holds: for each i = 1,2, there exists a constant /3t- > 0, independent of h, such that o ^ (5.3.44)

v

^

e

Q

i

'

h

3

e

(v°h)d

c

m m * ,

bi{vi,h,qi,h) > Allvi.hHi.fii

*

o

:

||g»,A||o,n ; -

W h e n TAT ^ 0, the inf-sup condition in fi takes the following form: there exists 0 > 0, independent of h, such that

(5.3.45)

V g/j £ Qh 3

£ (Vh)d C (J/"r ( f i ) ) d , v h ^ 0 : b { v h , q h ) > / 3 | | v h | | i , n ||?/>||o,n-

Furthermore, we also require that the following local inf-sup condition is satisfied: for each i— 1,2 there exists /3,; > 0, independent of h, such that

174

O T H E R BOUNDARY VALUE PROBLEMS

Ch.

5 ^

(5.3.46)

It is worthwhile to note that conditions (5.3.44) and (5.3.46) are very often a consequence of (5.3.43) and (5.3.45), respectively. However, if the number of triangles of Th in FII is too small, it can happen that (5.3.44) does not hold, even if (5.3.43) does, and similarly for (5.3.45) and (5.3.46) (see, for example, Boffi 1994, where a counterexample is provided in the case of continuous pressures). We now introduce the finite element counterparts of the functions $ and $ defined in Section 5.3.1. Let us assume that the Dirichlet boundary datum
G ( V h ) d : b{9h,qh)

V qh G Qh,

= -b(*h,qh)

where Qh := Qh n L20(n) if I V = 0, and Qh := Qh if I V ^ 0- The solvability of the latter problem is guaranteed by the inf-sup conditions (5.3.43) or (5.3.45) (see, for example, Quarteroni and Valli 1994, p. 248). Being an overdetermined system, its effective solution can be attained either by QR factorisation, or by the singular value decomposition (see, for example, Golub and van Loan 1989). Let us start again with the case where I V 0- The finite element variational approximation of the Stokes problem (5.3.10) reads: ' find W h G (V h ) d C (H±D(n))d, (5.3.48)

s(Wh,vh)

+ b(vh,ph)

U(Wh,?h) = 0

ph G Qh

=Fh(vh)

:

Vvte

(Vh) d

v qh e Qh,

where Fh.(vh) ••= (f,vh)n

+ (
+

%h,vh).

The corresponding two-domain formulation (which is the finite element counterpart of (5.3.14)) reads: find (Whh,Pi,h) G (V1>h)d X Qi,h, (W2,h,P2,h) € (V2,h)d x Q2,h such that

r

§5.3

THE STOKES PROBLEM

' Sl(Wiift,

Vi,/j) + bi(vith,Pl,h) = 0

bi(Wi,h,qi,h) W

l i h

=W

175

VV!,1 G

= -Fl,h(vl,h)

(V°h)d

V?i,fceQi,h onT

2 i h

S2(W 2 ,h,V 2 ,h) + b2{v2,h,P2,h)

= T2,h{^2,h)

V v 2 , f t G (Vr2%)d

(5.3.49) M W ^ , ^ )

= 0

S2(W2,h,T^2,ht1h)

Vq2,heQ2,h +b2(1l2,hl*h,P2,h)

- s i C W i ^ n ^ h P h ) - 6i(»i,fc/i/l,pllfc)

v

G (Ah)'',

+

, vi,/i)>

where ••= (f, Vj^)^; + (¥>jv, v i

ft|rjv)rA,nani

-

and TJ^ft denotes any extension operator from (A^)^ to (V^)" 1 , i = 1,2. As we have already noted in Section 2.1.1, in practical implementation Hi^Hk chosen to be the finite element interpolant V i ^ h - , which belongs to ( V 1 j l ) d , coincides with fi h at the nodes on the interface T, and vanishes at the internal nodes in fii. Following the same guidelines as the proof of Lemma 5.3.5, we can prove the following result. L e m m a 5 . 3 . 1 1 The discrete Stokes problem (5.3.48) is equivalent to the twodomain formulation (5.3.49), in the usual sense that W ^ n , . = W i t h , Ph\o,i = Pi,h fori = 1,2. For each r)h G (A/,)'' we introduce now the discrete local Steklov-Poincare operators Sijt : ( A h ) d (A'h)d defined as

(5.3.50)

(Si,hVh>Ph)

:

=

Si(Vithr)h,1iithith) +bi(Hi,hnh,Pi,hrih)

V rih,nh&(Ah)d.

+0i(/ci!hfJ,h,ri,hr)h)

vrjh,

W e also introduce the continuous linear f u n c t i o n a l Xi,h (5

.

<Xi,h^h)

:

= FuhVlifiHh)

e

(Ah)d'•

- Si(Ulh,Ki,h(*h)

-bi(Hi,hf*ht P*th)

V/i, G(Ah)d.

For i = 1, 2 we have denoted by ( U ^ j j ^ , Pi^Vh) G (Fi,/i) d x Qith the solution to

O T H E R BOUNDARY VALUE PROBLEMS

176

' 8i(Uiihrih,-viih) (5.3.52)

bi(Ui,h,Vh'
that is, the finite element {Vi > h) d

x

1S

( (5.3.53)

=0

+ bi(viih,PiihVh)

Ch.

5 ^

V Vj,h G

V qiih £ Qith

on T,

Stokes extension

of r)h to fl,-. Similarly, ( U * h , P i h ) 6

the finite element solution of the discrete problem Si(Ulh,Vi,h)

+ bi(vi
= Fi,h{yith)

V v,- A G

(V?h)d

{

Note that the definition of both S it h and X l h is independent of the choice of the extension operators TLi,hProceeding as in Lemma 5.3.7, the vector function A/, : = W A j r , where W / j is the discrete velocity, is the solution of the Steklov-Poincare equation (5.3.54)

G (Ah)d

: (ShXh,/ih)

having set Sh := Si
XH

V^

= (Xh,Hh)

G (Ah)d,

'•= XI,H + XI,/>•

Conversely, f r o m the solution \ h of (5.3.54) we obtain the solution of (5.3.48) by setting (5.3.55)

W

k | n

,:=U„A»

+

U-

A

Ph\Qi • = Pi:hXh + PIhBoth of the discrete operators Si,h and S 2 ,h is linear, hence continuous in (Ah) d (their continuity constants could, in principle, depend on h). Moreover, if we choose = Ui,hl*h i n (5-3.50), owing to (5.3.52) we obtain the identity {Si,hT)h,tih) = «i(Ui,fci?fc)U
Vti h ,Hh G (Ah)d-

Consequently, both operators Si^ and S2,h are also symmetric. Finally, due to the Korn inequality (5.2.5) in fl, and the trace inequality (1.2.5), the operator S2,h can be shown to be coercive in (Ah) d , uniformly with respect to h (recall that we have chosen a decomposition of Q, as in Fig. 5.3.4). In particular, S2th is invertible from (A'h)d to ( A h ) d . When <9fti n F n ^ 0, coerciveness also holds for the operator S i ^ , whereas, when d Q i fi r ^ = 0, we can only infer that Si t h is non-negative. In any case, the operator Sh is coercive in ( A h ) d , uniformly with respect to h. If the family of triangulations 7/j of ft is regular and the family of triangulations M.h of r is quasi-uniform, then both of the discrete operators SijL is uniformly continuous in (Ah) d • This can be proved as follows. Due to the assump-

r

§5.3

THE STOKES PROBLEM

177

tion (5.3.42), we can take in (5.3.52) the test function v ^ = Ui,h*)h ~ H-i,hVh' where r)h £ ( A h ) d and ~H.ithVh i s defined as in (5.2.21). It holds that Si(V

i i h

rj

h

,V

i ! h

ri

h

) =

Si(Uuht]h,Hiihrth)

+bi(Hi,hrih,Pi,hr)h) <[s i (U i , / l i7 A ,U l - A i7 A )] 1 / 2 [s I -(H i l A > 7 A ,H f i A J7 A )] 1 / 2

for a constant C > 0 independent of h. Using the inf-sup condition (5.3.44) it follows that II n II / 1 \\Pi,hVh\\o,n, < 7T sup P* Vi,he(v?h)d\o -

—

sup

bi(Vi,h,Pi,hVh) ' si(Uj,hWh>Vi,h)

li v i^l|l,fii

& vilhe(V?h)d\0

therefore

MU^ifo.Ui.fcifc)]1/2

<

C\\Hithr,h\\ltni.

Finally


Consider now the Dirichlet-Neumann iterative scheme for solving the multidomain problem (5.3.49): given X°h £ (Ah) d , for k > 0 find ( W ^ p f * Si(yvk+\vlih)

1

) G (Vhh)d

+b1(vl^p1+1)

x Q

u

=

(5.3.56) bi(W*+1,9lifc)=0 onF, then

Vgi.fcGQi.fc

: V vlih €

178

Ch.

OTHER BOUNDARY VALUE PROBLEMS

( find ( W ^ p ^

1

5 ^

) e {V2,h)d x Q2,h :

S2(W*+1,V2iA) + h i y ^ p l l

1

)

= ^2,„(v2,h)

V v2,h G ( V ^ ) *

(5.3.57)

= T2th{1l2,hnh)

+

-s1(Wk1+1,Tl1,hnh)

Fi,h(Ki,h»h) - b1(Hlihtih,pk1jl1)

V / i k e (A/i) c

and (5.3.58)

i k+1

on r.

Problem (5.3.57) has a unique solution provided that the inf-sup condition (5.3.44) is satisfied for the discrete spaces {V2th)d and Q2th- However, by proceeding as in the proof of Proposition 5.3.2, it can be shown that this is a consequence of (5.3.43), because we are dealing with discontinuous finite element pressures. Proceeding as in Section 5.3.1, it can be proved that the interface sequence {A^} obtained in (5.3.58) is actually equivalent to that produced by the preconditioned Richardson method (5.3.59)

A°,G(A h ) d \kh+x = \kh + es^h{Xh-sh\kh),

k> 0,

applied directly to problem (5.3.54). The convergence of the sequence X k , with respect to the norm of (A) d , follows from Theorem 4.2.2. In fact, we have already noted that both of the discrete operators 5i,h and S2,h is symmetric and continuous in (Ah) d , uniformly with respect to h. Moreover, Sh and S2,h are uniformly coercive in (Ah) d , hence all the assumptions of Theorem 4.2.2 are satisfied (see Remark 4.2.4). The rate of convergence is independent of h. A similar result holds for the Neumann-Neumann iterative scheme, basing the proof of convergence on Theorem 4.2.5. Remark 5.3.12 The matrix associated with the Steklov-Poincare equation is symmetric and positive definite. Therefore, this system can be efficiently solved by several methods, besides the preconditioned Richardson scheme that was outlined above. An a priori more efficient solver is the preconditioned conjugate gradient method. Several preconditioners enjoying optimal spectral properties can be proposed in order for the associated conjugate gradient iterations to converge at a rate independent of the number of grid points that are used in each subdomain: two examples are furnished by the Dirichlet-Neumann and Neumann-Neumann preconditioners.

r

§5.3

THE STOKES PROBLEM

179

In each iteration, computing the action of Sh on a given rfc will require solving two Stokes problems to find Ui^rj^, Pi^hVh- F° r more details on this approach see Quarteroni (1989). • In the case where Tjv ^ 0, the task of constructing the finite element vector function solution to (5.3.47) can be avoided, at the expense of solving, instead of (5.3.48), the following discrete problem: f find w h G ( V h ) d C (H£jn))d,

(5.3.60)

s(vrh,

+ b(vh,ph)

vh)

{b(wh,qh)

ph£Qh

=

:

V v/j G

= -b{$h,qh)

[Vh)c

VqhGQh,

where K(vh)

•= (f,vh)n

+ (
-

s(&h,vh).

Also, this problem can be solved by a Dirichlet-Neumann iterative scheme, which reads as follows: given G (Ah) d , solve for k > 0 find ( w f + S p f ^ 1 ) G

x Qlih

:

S i ( w { , r , v w ) + 6i(v1,h)pf+1) = ^r,ft(vliA)

V vlift G

[Vlh)d

(5.3.61) (

w

i V >

_ fk w k+1 1 ,h ~ <>h

=

V qi
, Qi,h)

G

Qlth

on r ,

then ffind(wk2%1,P^h1)e(V2,h)dxQ2,h : s 2 ( w * + 1 , v 2 i h ) + & 2 (v 2 , f l ,p*+ 1 ) = ^ , h ( v 2 l h ) &2(w^1,g2,h)

= -b2{$h\n2,q2,h)

V v 2 ,h G (V 2 ° h ) d

V q2,h G Q2,h

(5.3.62) S2(wk+1

,n2,hnh)

+

b2(n2,htih,Pk2%1)

-s1(wk+\n1^h)

- b1(K1,hfih,pk%1)

and (5.3.63)

th+1 =

+ (i

-

on r.

V fih

G (Aft)d

Ch.

OTHER BOUNDARY VALUE PROBLEMS

180

It can be shown that the sequence

:=

5 ^

— Wfcjp satisfies the recursive

equation 7kh+l=ikh-es^hshll where the Steklov-Poincare operators Sith are defined as in (5.3.50), and Sh : = converges to 0 (equivalently,

Si,h + S2,h- As a consequence of Theorem 4.2.2,

£h converges to w^|p = u ^ p - $/j|r) with respect to the norm of ( A ) d , at a rate that is independent of h. A similar result holds for the Neumann-Neumann iterative scheme, which reads: for i = 1,2, find ( w f t S p f t 1 ) 6 ( V t , h ) d x Qhh vilh)

+ biiv^p'X1)

:

= ^fcKfc)

V vhh

e

(V?h)d

(5.3.64)

k+1 _ (k w i,h — Sh

V qi>h G Qi,h

-bi(*h\Sii,Qi,h)

=

on r ,

then find

G ( V 1 : H ) D x QLTH

siWk1%\vi,h) + b 1 ( v 1 , h , 4 + 1 ) = 0 bi(4>i%\qi,h) = o (5.3.65)

:

V vlih £

(V°h)d

vqi,heQi,h

si(4>i%\Ki,h»h)+bi(Ki,h»h,4?h1) = -Flh(Ki,hHh)

-

Flh(1l2th»h) + 61

fc+l-l

+S1{WK1+1

,NLTHNH)

+S2(wk+1

,K2,hHh) + b2{n2, h^P^h)

v

A*h G (Aft)

THE STOKES PROBLEM

§5.3

ffind ( ^ . ^ J e M x f t , ,

(5.3.66)

s^X1

,n2,hfih)

+

:

= 0

+ b2(v2,h:nk2+1)

181

V v 2 ,/i G (V 2 ° h ) d

b2(K2,htih,nk2+1)

= Tlh{nlihtih)

+

Flh{K2,hHh)

-s1(wk+1,n1,htih)

-

b1(Klthfih,pk+h1)

-s2(wkX,n2,hnh)

- b2(H2^h,Pk2X)

V

G (Ah)d,

setting finally (5.3.67)

th+1

~ ^ j r )

on I\

This scheme is equivalent to the preconditioned Richardson iteration S-l

+

a2S-l)Shjk,

where y k ~ ^ — w/j|r and w;, is the solution to (5.3.60). Therefore, the proof of convergence, at a rate independent of h, follows from Theorem 4.2.5. Let us consider now the pure Dirichlet case, namely, let Tjy = 0. The finite element variational approximation of the Stokes problem (5.3.24) reads: find W h G CV h ) d C {Hl{Q))d, (5.3.68)

s(Wh,Vh)+b(vh,ph) b(Wh,gh)

PheQhD

=Fh(vh)

= 0

L20(Q) :

Vvfc G ( V h ) d

VqheQhnL2(ty,

where ^h(vh) ••= (f,v h )n -s(*h

+

9h,vh).

Note that the solution W / , to (5.3.68) indeed satisfies b(Wh,qh) = 0 for all functions qh G Qh- In fact, let qh be any function of Qh', its average is {qh)n ='• ~ f qhmeas U JN One has from (5.3.68)

182

O T H E R BOUNDARY VALUE PROBLEMS

b(Wft)

qh) = b(Wft,

Ch.

5 ^

- ( 9 h ) n ) + (ffft)nKWft, 1)

=

/ W h | 0 n • n* = 0. Jan

The last equality holds since ~Wh\dn = 0 when I V = 0. The corresponding two-domain formulation (which is the finite dimensional counterpart of problem (5.3.25)) is given by: find (With,Pi,h) £ (Vi,h)d x Qi,h, D (W2,H,P2,H) G (V2,H) x Q2,H such that f si(Wi,h)vi,h)+6i(vlih)p1,h) = ^ K f t ) 6 I ( W i i h , GI, H ) = 0

W1,h=W2,/l

V

G Q1>h N LG(FII)

onT

S 2 ( W 2 , f t , V2,FC) + b2(v2,h,P2,h)

(5.3.69) <

&2(W 2I FT, G2,H) = 0

s2{W2th,n2ihfih)

VVl,h G

= F2,H(V2,FT)

V V2,ft G ( V ^ J "

V q2th G
+

= F2,k{K2,hHh)

b2{K2,hHh,P2,h) +

h/ift)

- S l ( W i , h , H x , h H h ) -bi(Ki,hlih,Pi,h)

V/ifc £ (Aft,)d

/ Pi,h = - / ./fil •/ fi; where

and Tt^h denotes any extension operator from (Aft) d to i = 1,2. By following the guidelines stated in the proof of Lemma 5.3.9, we have the following result. L e m m a 5 . 3 . 1 3 The discrete Stokes problem (5.3.48) is equivalent to the twodomain formulation (5.3.49), in the usual sense that W/j|n ; = W ^ / , , Ph\n{ — Pi,h fori = 1 , 2 . Let us denote by (Ui, h r) h , P,hrih) € (V r i , h ) d x Q i , H N L § ( N ) and ( U ? h , P ? h ) G n t h e solutions to the problems (5.3.52), (5.3.53), respec(vi°h)d x tively, where the space of test functions Qi,h in the continuity equation has been substituted by Qith n Lo(fij), i = 1,2. The discrete local Steklov-Poincare operators S ^ and the linear functionals Xih-, i = 1>2, are still defined as in (5.3.50) and (5.3.51), respectively, with (U^jjh,Pi,hrih) and (U* ,P*) instead of (Uii?, Pit]) and (U*,P*).

THE STUKEB

183

FKUBUSM

In order to derive the Steklov-Poincare equation on T, it is useful to analyse the properties of the trace on T of the discrete solution W u of the Stokes problem (5.3.68). Denoting by \ U i the characteristic function of the solution W h satisfies b( Wh,Xni)

= 0,

as XQi <= Qh- Consequently, (5.3.70)

J W A , r - n = - b i ( W f c | 0 l , 1) = -b(Wh,xQl)

= 0,

because the solution W / 4 vanishes on 80, \ \ T (since PAT = 0). Therefore, we introduce the space (5.3.71)

l

h

: = j i f c e (A f t ) d \ f r V

h

n = o},

which is the trace space on P of the discrete functions v^ G (Vh) d satisfying the continuity equation b(vh,qh.) = 0 for each qh £ Qh,. In particular, as we have seen, e AhBefore stating the finite dimensional counterpart of Lemma 5.3.10, we note that the ( L 2 ( P ) ) d - o r t h o g o n a l complement of Ah in (Ah)d

is a one-dimensional

space. We denote by v*h the basis element of ( A h ) ± satisfying

meas We have L e m m a 5 . 3 . 1 4 If we setXh •'= then \h satisfies the following (5.3.72) Conversely, by setting

Xh e Ah

Steklov-Poincare

: (ShXh^h)

equation on T:

= (Xh^h)

V/ifcGAh.

from the solution Xh of (5.3.72) we recover the solution

(5.3.73)

where pith and p2}h

(5.3.74)

where CWh,Ph) w the solution to (5.3.68),

Ph\Qi •= Pi,hK

+ Pl*h

+Pi,h,

obtained by solving the linear \ Pl,h~P2,h I

=

system:

—^—={ShXh-Xh,v*h) measT

[ pi^meas fii + ^/jineas

02 = 0.

of (5.3.68)

184

Ch. 5 ^

OTHER BOUNDARY VALUE PROBLEMS

P r o o f We follow the guidelines stated in the proof of Lemma 5.3.10. The only point that deserves special attention concerns the verification that the functions given in (5.3.73) satisfy the interface relations stated in (5.3.69). Using (5.3.50) and (5.3.51) this reduces to proving that n =0

(Sh^h - Xh. Uh) - {Pi,h - P2,h) J^hfor each

G (Ah)d- If we define n,

(Hh)r •= — — p [ Ph meas F J r then the L2(T)-orthogonal

projection of FIH on AH is given by FIH - (PH)R^H- Then

we have {ShXh-Xh^h)

= {ShXh-Xh>Ph -

{pi,h

~P2,h)

-

(ma)vK)

J^Mh

+ (wOr(ShXh

~

Xh,K)

• n,

having used (5.3.72) and (5.3.74)i.

•

Also in this case, the Steklov-Poincare operators Sh, Si k and S2,h are symmetric, and continuous in (A^) d , uniformly with respect to h. Moreover, this time both operators Sith and S2,h are coercive, uniformly with respect to h (as in the case where Tjv ^ 0, with the decomposition of Fig. 5.3.4, left). The Dirichlet-Neumann scheme now reads: given XH £ AH, for /.: > 0 find ( W j + 1 , p f + 1 ) G ( V h h ) d x Quh

:

s i ( W f + 1 , v 1 ) h ) + b 1 (v 1 , h ,pf+ 1 ) = ^ i , h ( v l i h ) (5.3.75)

b1(wk1%1,qlih)

W f+ 1 = t h

/

yqi,h

G

QhhnL20(n)

on r

=

Jni

= o

Vvlih G ( V ^

Jns

r>k

followed by (5.3.57) and (5.3.58). This iteration is equivalent to the following Richardson scheme

(5.3.76)

K K+1

e = i

+ 0S-1h(xh-ShXkh),

k>

0,

r

§5.3

THE STOKES PROBLEM

185

whose convergence, with respect to the norm of (A) d and at a rate independent of h, is proved in the same way as for the scheme (5.3.59). The same result can be proved for the Neumann-Neumann iterative scheme.

Remark 5.3.15 Other approaches for the iterative solution of the Stokes problem in the framework of non-overlapping domain partitioning are proposed in Pasciak (1989), Bramble and Pasciak (1989), Fischer and R0nquist (1994), R0nquist (1996), Casarin (1996), Le Tallec and Patra (1997), Pavarino and Widlund (1997), and Pavarino (1998). For the construction of block-diagonal or block-triangular preconditioners based on overlapping subdomains we refer to Klawonn (1996,1998a, 6), Klawonn and Pavarino (1998a, b), and Klawonn and Starke (1999). • 5.3.4

Finite dimensional approximation: the case of continuous pressure

Let us consider now the case of continuous discrete pressures; namely, the case in which the finite element spaces satisfy Vh C ffpD(Q), Qh = X® C C ° ( f i ) for q > 1. Their subdomain and interface counterparts, denoted V, jt, V-)h, Ah, Qi,h, are defined as in (5.3.41). We still assume that (5.3.42) is satisfied, and set ^

3

Qi,h

~

{n*

e Qi,h

I ?i,fc|r =

0}

••= {qh\r]Qh £ Qh}Note that the space S/j is also the trace space on T of both <2;,/i and for each i = 1,2. We assume that the inf-sup condition (5.3.43) is satisfied for the finite element spaces ( V h ) d C ( # 0 1 ( f i ) ) d and Qh n Lg(f2) (when = 0), or that the inf-sup condition (5.3.45) is satisfied for the spaces (Vh)d C (Hp D (Ct)) d and Qh (when Tiv 7*- 0), with a constant (3 independent of h. Similarly, in the case where Tjv = 0 we assume that (5.3.44) is satisfied for the couple of spaces (V°h)d C (Hq (Oi))d and Q ° h fl L§(fij), as well as for the couple of spaces (Vi,/i) d and Qi,h, with constants Pi independent of h. In the case where TJV ^ 0 we assume that (5.3.46) is satisfied for ( V ? h ) d C ( ^ D u r ( f i i ) ) d and Q? f c and for (Vt,h)d and Qi>h, with constants Pi independent of h. As noted before, the inf-sup conditions (5.3.44) for ( V ° h ) d and are very often a consequence of (5.3.43) for (Vh) d and Qh H inf-sup conditions (5.3.46) for (V°h)d

Q°hnLl(Q,t) similarly, the

and Q° h, or (Vi t h) d and Qith are very often

a consequence of (5.3.45) for ( V h ) d and QhThe finite element approximation of the Stokes problem is still defined as in (5.3.48) (for TN ^ 0) or (5.3.68) (for VN = 0), but now the spaces Vh and Qh are chosen as indicated above. Denoting by pith any extension operator from 5/ t Qi,h, in the case where Fjv ^ 0 the two-domain equivalent formulation reads: find ( W i l f c ) P l , h ) G ( V l t h ) d x Qi,h, (W 2 ,h,P2,h) G (V2,h)d x Q2,h such that

186

OTHER BOUNDARY VALUE PROBLEMS Pi,h) = FiAvuh) 6i(Wi,hlgi,h) = 0

V vi,„ G

(V°h)d

WqhheQlh

W i , h = W 2 ,ft on

T

THE STOKES PROBLEM

§5.3

where denotes any extension operator from Sft to Q i ) h n L o ( f i i ) . N ° t e that the usual interpolation operator in fix, modified at one node to ensure that its mean value is zero, can be used as pi h-

r

S2(W 2 ,/ l ,V 2 ,/ l ) +& 2 (V2,/ l ,P2,h) = ^2,h(V2,h)

Proof Let us confine ourselves to the case where TN = 0, because the proof in the case where Tjv 0 is easier. Let ( W h , P h ) be a solution to (5.3.68): the proof that its restrictions With : = Wh\Qi and piyh := ph\n. satisfy (5.3.79) is easy. In fact, the local equations in fii; i = 1,2, are satisfied by extending by zero the local test functions v , ;, and qt j,. The two equations on T are satisfied because the momentum equation in (5.3.68) holds for every vh G {Vh)d\ in particular, for vh = 7Lnh, where fj,h G (Ah)d and

V v 2 , f t G (V 2 ° h ) d

M W 2 , h , <&,/.) = 0 s 2 (W 2 , / l ,72. 2 ^M/ 1 ) +

b2(K2,h»h,P2,n)

= F2,h(K2,hPh)

+ _

-SI(WI,H,WILH/IH) . MW

2

,

h )

P2,hiph)

B X {HI,HHH,JH.,H)

V / I F T £ (AFT)D

f « i ( W i , f c , v i , h ) + 6i(vi, f t ) pi,h) = Fi,h{vi,h) 6 i ( W i , h ) 91,h) = o Wi,h=W2,h

V 31,h G Q°lih n

in

in fi2 '

Ph^h • =

P\,h^h P2,h^h

in fti in fi2

Conversely, if ( W i } h , P i , h ) , i = 1,2, is the solution to (5.3.79), let us set ^ / Wi,i W2,h

onT

in fii in fi2 '

_ j p1
Ph

in fix in fi2 •

Clearly, W h G ( V h ) d and ph G Qh n L o ( f i ) . Proceeding as in the proof of Lemma 5.3.9 one sees that the only property that is not immediate is

on r

b2(W2,h,g2,h)=0 S2 ( W 2 , h ,

(^lV

L20(fh)

S 2 (W 2 ,h,V 2 ,ft) + & 2 ( v 2,/i,P2,h) =^2,hiV2,h) (5.3.79)

e

fti, hHh I ^ 2 ,h H h

J

and the continuity equation in (5.3.68) holds for every qh G Qh', in particular, for qh = Phi>h, where iph G Eh and

V Vh e Hft.

= -bi{Wi,h,Pi,hiph)

Similarly, in the case where TJV = 0 the two-domain equivalent formulation reads: find ( W l i h l f t i A ) G (Vi,/ i ) d x Q i , h ) ( W 2 , h ) p 2 , h ) e (Vb.h)" x Q2,ft such that

Pi,/i = P2,/i

187

L e m m a 5 . 3 . 1 6 When TN ^ 0, the discrete Stokes problem (5.3.48) is equivalent to the two-domain formulation (5.3.78). When TN = 0, the discrete Stokes problem (5.3.68) is equivalent to the two-domain formulation (5.3.79).

on r

Pl,h=P2,h

(5.3.78) <

Ch. 5

V v 2 i / l G (V2°ft)d

b(Wh,qh)=0

Vq2iflG

Vqh

To verify this, first take qh G Qh n Ll(fl) ^h '•= qh\r G 2/j, it follows that

h/lfc) + 6 2 ( ^ 2 ,

G

QhnL20(O).

such that qh\Ql G

fii).

= J~2,h (n2,hHh) +

-SI(WI,I,KI,A) b/2(W2,h,p2,h'>Ph) Pi,/i = - /

/fij

»=1

FiA'&i.hPh)

= bi (Wi,ft, qh\n1 ~pi,h^h) (A/L)D

bi{H1,hiih,pi,h)

= - M W i , ! , ? ! , ^ )

V ^ e H i

+b2(W2th,

= hiWi^PiM

+6i(Wi,ft,p1,/,Vft)

qh\n2 ~ P2,hfph) +

+ b2(W2,h,p2,hiph)

If now qh is an arbitrary function in Qh H Lg(fi), we set

b2(W2th,p2>hiph)

= 0.

Setting

Ch. 5

OTHER BOUNDARY VALUE PROBLEMS

188

=

- (gOnJ

-{qhhi

Jan

+ (QhhMWh,

1)

Wfcien • n* = 0.

The last equality holds since W ^ a n = 0 when T ^ = 0.

•

For the two-domain formulation (5.3.78) or (5.3.79), we can formulate the Dirichlet-Neumann procedure as follows. Let us consider only the case where Fjv = 0, the other being similar. Given A^ £ (Ah) d , find ( W f ^ . P ^ 1 ) e (Vx,h)d xQlih

e S j , for each k > 0

M^iji1)!!,/") (5.3.80)

= 0

WfV=A£

f Jch

=

nLKQt)

Another approach for the Stokes problem that is amenable to the use of domain decomposition algorithms is as follows. Taking for simplicity
(5.3.85)

-cAw = g

in Q,

w = 0

on dfl,

Up = G,

where U is sometimes called the Uzawa (or pressure) operator and is defined by U : = - d i v £ V , while G := -divQf.

nk ¥2,hi

Proposition 5.3.17 The Uzawa operator U is symmetric

then

space

•

(W\l\pk+h1)

E {V2,h)d

x Q2,h :

« 2 ( W £ + \ v2ifc) + h i ^ p ' X

{ s2(Wk2X,H2^h)

1

Proof

) = T2,h{v2,h)

+

The operator — div is the transpose of the operator V , and we have {Up, q) = ( - div GVp, q) = (V<7, QVp)

[ b2(Wk+\p2^h)

hCfl^^p'X1)

- b1(Tl1^h,pk1%1)

VfihG V

=

and finally set (5.3.82)

Vp,qe

(Ah)d

(DiW?,D, Wi

Moreover

G

(Uq,q) = v J 2 ||A ij =l Prom the estimate

A£ + 1

= 0 W * + ) r + (1 -

L2{0),

where (•,•) denotes the duality pairing between i J - 1 ( f l ) and Hq(Q). w p : = £ V p and w 9 : = QVq, we have that

{Up,q) = ( - M w ' . w " ) = !/ ^ -ai(Wk1%1,Kl,hMh)

and coercive in the

L20{Q).

V v 2 , h G (V 2 ° H ) D

V q2,h G

b2(Wk2%1,q2,h)=0 (5.3.81)

M e t h o d s based on the Uzawa pressure operator

the continuity equation d i v u = 0 in (5.3.2) provides the following equation for the pressure:

on r

- [ o

on P.

(5.3.84)

V vlifc 6

onF

fc+l _ „k Pljh PW

V qlih G Qih

5.3.5

:

^ ( W f + S v i . h ) + 6 i ( v 1 > h , p f + 1 ) = ^i, h (v 1 > f c )

189

The existence and uniqueness of the solutions to (5.3.80) and (5.3.81) is ensured by the validity of the inf-sup condition (5.3.44) for the discrete spaces (V°h)d a n d ( v 2,h) d a n d Q2,h, respectively. and Q°
and obtain = b(Wh,qh

fc+i

(5.3.83)

:= ^hL qhini b(Wh,qh)

T

THE STOKES PROBLEM

§5.3

9)\kh

on P

||Aw'||_i,n
Setting

O T H E R BOUNDARY VALUE PROBLEMS

190

Ch.

5 ^

we finally find {Uq,q)

>a2||WW?||21]n = a2||Vg||2_l!n >a 0 ||g||g in .

In the last inequality we have made use of the estimate ll?||o,n
VqeL20(n)

(see, for example, Girault and Raviart 1986, p. 20).

•

The finite dimensional counterpart of the Uzawa operator is therefore a symmetric and positive definite matrix. It is a full matrix due to the presence of the 'inverse' operator Q. A convenient approach for solving the associated linear system can therefore be based on conjugate gradient iterations. At each step the computation of the residual amounts to the solution of an elliptic problem like (5.3.84). In turn, the latter can be reformulated in a multi-domain fashion as in Chapter 2, and solved iteratively as illustrated in Chapter 3. We refer to Elman and Golub (1994), Bramble et al. (1997), and Maday et al. (1993) for other domain decomposition results for the Stokes problem based on the Uzawa operator.

5.4

The Stokes problem for compressible flows

We consider in this section a different type of Stokes problem that is related to the time-discretisation of the (linearised) Navier-Stokes equations for compressible flows. The differential problem is given by the following set of equations:

(5.4.1)

cm — divT(u, cr) = f

in ft

aa -+- div u = g

in ft

u =
on r . c

d

( 2v \ f ^ ( A w j + DjUi)n* + ( 7 - — j d i v u j j=1

^

-Penf

'

on I V , I = 1, • • •, d,

= (
where a > 0, f,
Iij(w,7r) : = v(DiWj + DjWt) +

where v > 0, 7 > 0 and /? > 0 are constants.

2v 7 - —

div w Si j -

PwSij,

§5.5

T H E S T O K E S P R O B L E M F O RINVISCIDCOMPRESSIBLEF L O W S

191

The model (5.4.1) has been introduced by Bristeau et al. (1987) as an intermediate stage of a fractional step algorithm to solve the Navier-Stokes equations for compressible flows. In this context, u denotes the fluid velocity, while a is the logarithm of the fluid density. A more familiar version of the momentum equation in (5.4.1) is obtained by noting that - d i v T ( w , 7 r ) = - i / A w - 7 * V d i v w + PVTT, where 7 * : = 7 + (d - 2)i//d. Setting, as usual, u; = U|Q. and Uj = a|Q., i = 1,2, problem (5.4.1) can be split in a multi-domain fashion, and the interface conditions on T become ui —

(5.4.3)

on F

d ( 2v\ 7/.1i ^An.-Iv ^ i D i U x j + DjU r i j + (i 7'V- —— ) div ui ni d j=1 ^

faint

( 2v \ = v ^ ( D i u 2 j + DjU 2 ,i)nj + f 7 - — J d i v u 2 n ; i=1 ^ ' -/3a2ni on T, I = 1 , . . . ,d, and impose continuity of the fluid velocity as well as that of normal stresses. Remark 5.4.1 Note that system (5.4.1) can also be regarded as a mixed-type formulation for the elasticity problem (5.2.1). In fact, setting a : = — d i v u , equation (5.2.1)! becomes —ftAu — AV div u + /tVo" = f

in ft

cr + div u = 0

in ft,

which has a structure very similar to that of the Stokes system for compressible flows (only the term au is missing in the momentum equation). Therefore, all the results that we shall present in this section can be extended to the elasticity problem as well, provided that T o ^ 0. • 5.4.1

W e a k multi-domain formulation and the Steklov-Poincare interface equation

We introduce now the bilinear form , v ) := J

a w • v + ^7 —

divw divv

d + ^ 2

{Diu>m + Dmwi)(Divm l,m—1

+ DMV{)

Assuming that f e ( L 2 ( f t ) ) d , f l e L2{Q.),
and
L2{TN))d,

192

OTHER BOUNDARY VALUE PROBLEMS

Ch. 5

ffind (w.cr) e [H^D{n))d x L2(Q) : s ( W , v) + f3b(v,a)

= (f,v)

(5.4.4) < + (<^V,V|RJRJV

V V G (I^D(ft))d

I A(A, TT) - b(W, TT) = (g, 7R) + 6 ( £ D , TT)

V TT G L 2 ( f t ) ,

where D. The existence of the solution (W,er) to (5.4.4) is a straightforward consequence of the Lax-Milgram lemma, because the bilinear form AW [ ( W ,

p), (V, TT)] : = S ( W ,

v) + 0b(v, p)

—/3&(W,7t) + (3a(p, 7r) is continuous and coercive in (H^

r(w, w) =

/ Jn

(ft)) d x L 2 ( f t ) . In fact, it can be proved that

a|w|2 + ( 7 - ^

) (divw)2

(5.4.5) + \ Z

(Diwi

+ Diwi

f

>K0||w||2 where K0 > 0 only depends on a, u, 7 , d and ft (see, for example, Secchi and Valli 1983). Introducing the local bilinear forms

»(wi,Vj) := (5.4.6)

/ awi - V i + ( 7 JUi + -

div Wi d i v v .

Y {Diwi>m + DmWi,i){Di Vi,m + l,M=1

DmVii)

and assuming for simplicity that I V = 0 and
^

§5.5

T H E S T O K E S P R O B L E M F O RINVISCIDCOMPRESSIBLEF L O W S

f Si (U!, Vi ) + / ? & ! ( • ! , <7!) = ( f , V l ) n i

V VI G [V?)d

a(cri 1 iri)n 1 - ^ ( u i , ^ ) = (5,-jri) ni ui = u 2 (5.4.7)

193

V TTI G L 2 ( f t i )

on T

s 2 ( u 2 , v 2 ) + (3b2(v2,cr2)

= (f,v2)na

V v 2 G (V2°)d

a{a2,ir2)n2

= (5,7r2)n2

V 7r2 G £ 2 ( 0 2 )

- b2(u2,n2)

s 2 (u 2 ,7i2M) + pb2(H2fi,(T2)

= {f,K2n)n2 + ( f . ^ M j o i

- S I ( U L , FCI/I) - /?6I (FTI/I, <TI )

V M G (A)D,

where, as before, 7ii denotes any extension operator from (A) d t o (Vi)d,i = 1,2, and the rest of the notation is as in Section 1.2. The last equation is the weak form of the interface condition (5.4.3) 2 (the continuity of the normal stress), which can be formally recovered by integration by parts. In the present case, the Steklov-Poincare operators can be defined as follows. First, denote by ( U P ^ r j ) G (Vi)d x L 2 (Oi) the extension of the interface datum rj given by the solution to r SiCufVve) (5.4.8)

= 0

a(P^r), n ) * - M u f V 7 ^ = 0 k

and by (

+(3bi(vi,P^rj)

U

U ^ t } = tj G

V Vj G (V°)d V TT, G £ 2 (n, : )

on r , x L 2 (Oi) the solution to

S i f u g , v,-) +

fibi^pM)

= (f, vOn,-

V vi G (V°)d

(5.4.9) T(") - ^(U^.TTi) = (ff.TTOn,

V 7T; G L2(Q,i).

For every i = 1,2 we introduce the bilinear form

(5.4.10)

• ^ [ ( w i i P O . t v i . T i ) ] : = Si(wi,Vi)

+0bi(vi,pi)

-Pbi(wi,TTi) + /3a(pi, TTi)ai

which is continuous and coercive on the product space (Vj°) d x L 2 (ftj), and the linear functional (5.4.11)

194

OTHER BOUNDARY VALUE PROBLEMS

Ch.

5^

The local Steklov-Poincare operators are then defined as (5.4.12)

(STV,

„> : =

KV^V,

P^v),

(U<"V

V)]-

It is worthwhile noting that, although the bilinear forms A ^ are not symmetric, the local Steklov-Poincare operators are indeed symmetric. In fact, from (5.4.8)2 we obtain therefore ( S ^ / x ) = ^ ( U ^ U ^ V ) +0a(P^ti,P^r})ni

=

(S^n).

A similar computation shows that

+/?(
V W

Moreover, by well known a priori estimates for the solution of (5.4.8), both of the local Steklov-Poincare S» operators is easily shown to be continuous in (A) d . Finally, owing to (5.4.5) and the trace inequality (1.2.5) they are coercive in (A) d . The solution (u, ex) to (5.4.4) (with ipD = 0 and TJV = 0) satisfies

u |fii =uWu |r + < ) , Setting A : = U|r, and inserting the latter expressions in the last equation in (5.4.7) (the stress balance condition), we easily obtain the following SteklovPoincare interface equation on F: (5.4.13)

SA = * ,

where we have set, as usual, S : = Si + 5.4.2

and x '•= Xi + X2-

Substructuring iterative methods

The Dirichlet-Neumann iterative scheme for solving (5.4.7) reads: given A0 G (A) d , for each k > 0 solve ' f i n d (uf + 1 ) < 7 1 f e + 1 ) G (Fi) d x L2(fii)

:

S i K + V i ) +pb1(vuaL;+1) = (f,Vl)ni

V vi S ( V f y

(5.4.14) ' a ( ^ + 1 , 7 r i ) n i - 6 i ( u f + 1 , 7 n ) = (5,7ri) n i

Vm G L

2

^ )

§5.5

T H E S T O K E S P R O B L E M F O RINVISCIDCOMPRESSIBLEF L O W S

195

then find (Uj L2

,A^)E(V2)DXL2(N2)

:

? 2 ( U 2 ^ 1 , V 2 ) + / 3 6 2 ( V 2 , ^ + 1 ) = (f,v2)na

V V2

a{a%+,ir2)n2

V

- b2{uk2+L, TT2) = (^,7R 2 )O 2

6

n2 G

L2(02)

(5.4.15) s2(v2+\K2H)+0b2(n2ti,<7L2+1) = (f,K2n)n2

+

(f,H1n){ll

-8l(vt+l,nlii)-l3bl(1llii,a*+l)

V/1G (A)d,

and finally set

XK+1 := flu^1 + (1 - E)\K

(5.4.16)

on r.

This iterative scheme is equivalent to the preconditioned Richardson method (5.4.17)

\K+1

=\K

+ ES^IX

-

SXK)

for the Steklov-Poincare equation (5.4.13), as can be shown following the guidelines as in Section 1.3. The convergence of this scheme is again a consequence of Theorem 4.2.2, because we have already noted that both Steklov-Poincare operators are symmetric, continuous, and coercive in (A) d . A similar result can be obtained for the Neumann-Neumann iterative scheme, which can be easily defined following what was carried out in Section 1.3 for the Laplace operator. It can be equivalently formulated as an iterative procedure on the interface P: Afc+1 = x

t

+

e(aiS-i

+

ff2s~

1)(X _

sx").

All the assumptions of Theorem 4.2.5 are satisfied, therefore the convergence in ( A ) d of X k follows at once. 5.4.3

T h e finite dimensional approximation

The finite element approximation of (5.4.4) can be based on the usual finite element spaces (Vh) d C ( - f f o ( f i ) ) d introduced in (2.1.3) for the velocity field, and the corresponding local and trace spaces (Vi t h) d , (^ 0 /i) d and ( A h ) d (see (2.1.7) and (2.1.8)). This choice corresponds to the use of continuous piecewisepolynomial finite elements of degree r > 1 for the velocity components. For the approximation of the logarithmic density a we consider the finite element space consisting of discontinuous elements (5.4.18)

r —1

:=

{TTH

E L2(N) I wh\K E P r _ i (K)

V

K

r>

1,

Ch. 5

O T H E R B O U N D A R Y VALUE PROBLEMS

196

and their restrictions YJ^ 1 to Qj, for % = 1,2. The discrete problem corresponding to (5.4.4) (with tpp = 0 and 17jv = 0) is given by f find (uh,oh) (5.4.19)

e (V^)d x Y^" 1

s( U / l , v A ) + (3b{vh,ah) = (f, v A )

V v t £ (V/,)11

^ a((7/i, 7T/J) - 6(U/1,7T^) = (ff, TTft)

V 7Th G V^" 1 .

The finite element two-domain formulation corresponding to (5.4.7) can be easily defined, as well as the finite dimensional counterpart of the SteklovPoincare interface equation (5.4.13). For the sake of exposition, let us give a precise definition of the discrete Steklov-Poincare operators. First, for each r)h 6 (Ah)d (Vi,h)d

x

Y [ -

x

and i = 1,2, we denote by

the solution to v i i h ) + /3bi{vilh, P^f)h) <x(P!l]vh,fi,

(5.4.20)

P^rih) £

[V^Vh=r)h

= 0

- M U & V =

0

V v i i h G (VftJ r-1

V 7Tiih e YT-

onT.

The discrete Steklov-Poincare operators are defined as

(5.4.21)

We have already observed that properties 1 and 2 in Remark 4.1.7 hold, therefore these operators are symmetric and coercive in [Ah)d, uniformly with respect to h. On the other hand, the uniform continuity of the Steklov-Poincare operators can be proved as follows. Taking as test functions in (5.4.20) vi,h

= U$t?h

- Hi,A«7h)

7Ti,ft = /3Pjj^r)h,

where Hi^rjh has been introduced in (5.2.21), and adding the two resulting equations we find that SiCU%r, hl U $ i , h )

P^nhi

Therefore (5.4.22)

\\V^r,h\\lttli + WP^VhWo^ < CWTL^r,^

V r,h £ ( A , ) d ,

^

§5.5

T H E S T O K E S P R O B L E M F O R INVISCID C O M P R E S S I B L E F L O W S

197

where the constant C > 0 depends on the coefficients a and j3, the continuity constants of the forms si and bi, and the coerciveness constant of the form si, but is independent of the grid size h. Consequently, from (5.4.21) and Theorem 4.1.3 (Silhr)h,Hh)

< C||Hiifci|A||I,Ni||HiiA/iA||Iini < C||;7a||A|K|]A,

provided that the family of triangulations Th of ft is regular and the family of triangulations M h of F is quasi-uniform. The Dirichlet-Neumann iterative scheme (5.4.14)-(5.4.16) on the finite element subspaces turns out to be equivalent to the preconditioned Richardson method for the discrete Steklov-Poincare equation, with S2,h a s a preconditioner. Concerning its convergence, we can therefore resort to Theorem 4.2.2. W e have already noted that all of its assumptions are satisfied, uniformly with respect to h\ thus, the Dirichlet-Neumann iterative scheme on the discrete problem (5.4.19) converges, and its rate of convergence is independent of the grid size h. A similar results holds for the Neumann-Neumann iterative scheme, using Theorem 4.2.5 instead of Theorem 4.2.2.

5.5

The Stokes problem for inviscid compressible flows

This problem is the inviscid counterpart of the Stokes problem for compressible flows (5.4.1) considered in the previous section (that is, the one in which v = y — 0). The new problem reads:

=f

in ft

acr + div u = g

in ft

cm + /3V.7

(5.5.1) u • n* = a -ip

N

on T u on V N ,

where a > 0 and (3 > 0 are constants, f is a given vector field, and g, ipD and (pM are given scalar functions. Note that the system above is no longer elliptic. Following the general principle presented in Remark 1.1.1, the interface conditions are determined by requiring that the solution u belongs to a suitable space of functions defined in ft, and that it is a distributional solution of the given problem in the whole ft. From (5.5.1), it is apparent that the velocity field does not need to belong to the usual energy space ( H l ( f t ) ) d , because only the operator div acts on u. W e thus introduce the Hilbert space (5.5.2)

H{div; ft) : = { v G ( L 2 ( f t ) ) d | div v G £ 2 ( f t ) } ,

and require that u G # ( d i v ; f t ) . Let us also recall that, if v G # ( d i v ; f t ) , then its normal component v • n is continuous across every interface.

O T H E R B O U N D A R Y VALUE PROBLEMS

198

Ch.

5 ^

We are now in a position to state the interface conditions for u and a. Setting : = Ui and cr|n; : = cr,, i — 1,2, they read ui • n = uo • n

(5.5.3)

<j\ = o2

on P on r .

The continuity of a across T corresponds to the continuity of the normal stress. In fact, for the inviscid problem, the stress tensor T(w, TT) reduces to Tij(w,n)

:=

-(3n5ij,

and the condition T ( u i , o i ) • n = T(U2,CT2) • n reads —(Tin = -o"2H

on P,

which is (5.5.3)2. Remark 5.5.1 System (5.5.1) can be regarded as a mixed-type formulation for the elliptic problem —eAcr + cr = g

in 0,.

In fact, setting u : = — V
{

a + e div u = g

in fi

u + V<x = 0

in fi,

which has the same structure as the Stokes system for inviscid compressible flows. Therefore, all the results that we shall present in this section can be extended to the elliptic problem (5.5.4). • 5.5.1

Weak multi-domain formulation and the Steklov-Poincare interface equation

Taking for simplicity ipo = 0 and I\v = 0 (and referring to Quarteroni and Valli 1990 for the general case), and assuming that f £ (L2(fl))d and g £ L2(Q), the weak formulation of (5.5.1) is: find (W,cr) e # 0 (div; ft) x L2{0) (5.5.5)

{ a ( W , v ) +/Bb(v,a)

= (f, v)

a(a,Tr) — &(W,7r) = (g,w)

:

V v G Ho (div; fl) V tt G L2{Q),

where H0(div; fi) : = {v G H(div; n ) | (v • n*) ! a n = 0}.

§5.5

T H E S T O K E S P R O B L E M F O R INVISCID C O M P R E S S I B L E F L O W S

199

The existence of a unique solution ( W , a) of (5.5.5) follows from the L a x Milgram lemma, by considering an equivalent formulation for the coercive bilinear form associated with the operator al — / ? a _ 1 V d i v (see Quarteroni and Valli 1990). The multi-domain weak formulation reads as follows. First, let us introduce the spaces Wi (5.5.6)

= {vi £ H(div; Qi) | (v; • n*)| a f i n a f i i = 0 }

W? = # o ( d i v ; $ =

: r -)• R I xp = (v • n)|r, v £ ff 0 (div; ft)}.

The space $ is included in the interface T has no boundary (see If we set u, u ^ and a j : = following one: find (UI,CTI) £ Wi

dual space of A, and coincides with it if the Girault and Raviart 1986, p. 27-9). cr|n,, then problem (5.5.5) is equivalent to the x L 2 ( f t i ) , (u2,a2) £ W2 x L2(fl2) such that

a ( u i , v i ) n i +/3b1(v1,a1)

= (f.v^n,

V v i £ W?

"(0-1,^1)0! - &l(Ul,7Tl) = ui • n = U2 • n (5.5.7)

on r

a ( u 2 , v 2 ) n 2 + (3b2{v2,a2) a(u2,7r2)o2 a(u2,K2iP)n2

V 7T1 £ X 2 ( f t 1)

- b2(u2,n2)

V v 2 £ W2°

= (f,v 2 )ti a = [g,ir2)n2

+ (3b2(K2^,a2)

V ir2 £ i 2 ( f t 2 )

= (f,^2^)n2

+

where we have denoted by f i i any extension operator from $ to Wi satisfying • n)j r = ip for each ip £ "F. T h e last equation of (5.5.7) is the weak form of the interface condition (5.5.3) 2 (the continuity of the normal stress that in the present case coincides with the continuity of the logarithmic density), which can be formally recovered by integration by parts. {Hiip

For each ip £ ^ we can introduce the solution (

U

€

to f a ^ V . w O n , - +/?6I(WI,PFV) = 0

V w< £ W ?

(5.5.8) a(P^)rt>,Wi)ni-bi(vf^,iri) = 0 ( U f ^ - n ) | r =xj>

on r.

V tt, £ L 2 ( f t ; )

W{ x L 2 ( f t ; )

O T H E R BOUNDARY VALUE PROBLEMS

200

Moreover, we denote by ( U ^ , f a(ufi,

(5.5.9)

Ch.

5^

) £ W® x L 2 ( f i j ) the solution to

WL)N,

+

= (f,WI)N,

FLBI(WI,

V wT E

W?

I U ^ U i t a

- ^ ( u g . T T i ) = (g^ifot

V7r,ei2(fi,).

The bilinear forms associated with (5.5.8) and (5.5.9) are continuous, but they are not coercive in Wf x L2(Q,i). Nonetheless, we have already noted that the existence of a unique solution can be proved by resorting to an equivalent coercive problem associated with the operator al — / ? a _ 1 V div. Moreover, one has (5.5.10)

H U f V l k d i v i n o + II^(0V||o,Q; <

cmw.

Denoting by A t h e bilinear forms introduced in (5.4.10) (where we have taken p = j = 0), the Steklov-Poincare operators Si are defined in the trace space as follows:

and the linear functionals Xi £

are

given by

(4>,Xi) : = ( f . u f V k +

V)o,.

Arguing as in the previous section, the Steklov-Poincare equation reads (5.5.13)

S\ = x

onT,

where we have set as usual S := S\ + S2 and x : = Xi + XiNote that, similarly to all the preceding cases, we have transformed the multidomain problem into an equivalent problem on the interface T which expresses the continuity of the normal stress. In the present situation, this means the continuity of the logarithmic density. Clearly, one could proceed in the opposite way, coming to an interface equation where the trace of the logarithmic density is the unknown, and the continuity on T of the normal trace of the velocity has to be imposed. The Steklov-Poincare operators Si and S2 are both symmetric, and continuity in $ follows at once from (5.5.10). Finally, taking in (5.5.7) 7Tj = div(U^°'?/j) one has jldivCU^VOIkn, < C o | | / f V l | o , n i . Since the trace inequality states that

§5.5

T H E S T O K E S P R O B L E M F O R INVISCID C O M P R E S S I B L E F L O W S

201

IMI*
Hence both Steklov-Poincare operators are coercive in it. 5.5.2

Substructuring iterative m e t h o d s

The Dirichlet-Neumann iteration scheme for problem (5.5.7) reads: given A0 £ it, for each k > 0 solve find ( u i + 1 , < T i + 1 ) £ Wy x L 2 ( f i i ) : a

( u i + 1 j vi)fh +

(vi, ctj + 1 ) = ( f , V l ) 0 l

V

Vl

£ W?

(5.5.14) aKfc+1,7ri)ni

-6i(u{+1,7Ti)

u i • n = \k

= (g,ni)0l

V^e

L2(Qi)

on T,

then find (uk2+1,
£ W2 x L 2 ( f i 2 ) :

a ( u * + \ v 2 ) f i 2 + (3b2(v2,ak+1)

(5.5.15) <

a(ak+1,w2h2 a(u

= (g, 7r 2 ) fi2

~ b2(uk+1,7r2)

V v 2 £ W2°

= (f,v2)fi2 V

£ L2(ft2)

k+1,K2^)+(3b2(K2ij,ak+1)

= (f,n2^h2

+

{f,K1ip)nl

and finally set (5.5.16)

: = 0(vi k 2 +1 • n ) ] T + {I - 6)ip k

on T.

By proceeding as in the previous sections, the Dirichlet-Neumann iterative scheme turns out to be equivalent to a Richardson procedure for the interface equation SA = x , with S2 as a preconditioner. The convergence of these iterations can be proved by using Theorem 4.2.2, because we have already noted that all its assumptions are satisfied. Similarly, as a consequence of Theorem 4.2.5, the Neumann-Neumann iterative method applied to the solution of (5.5.7) is convergent.

OTHER BOUNDARY VALUE PROBLEMS

202

5.5.3

Ch. 5 ^

The finite dimensional approximation

For the approximation of problem (5.5.1) we can rely on its analogy with the mixed-type formulation (5.5.4) of the Poisson problem. For the latter, classical finite elements like the Raviart-Thomas finite elements if d = 2 (see Raviart and Thomas 1977), or the div- conforming Nedelec finite elements if d = 3 (see Nedelec 1980, 1986) can successfully be applied. The latter elements were introduced in Section 4.1.2 and the associated finite element space was denoted by N f . Let us denote the finite dimensional space used for approximating the velocity u by Wh •= n //o(div; ft). The approximation of o is again based on the space Y^" 1 of discontinuous finite elements introduced in (5.4.18). Setting W h h = {Vfcifii I vfc e Wh}, i = 1,2 = WithnH0{div; = ( K

ft,),

-n)|r | v h G Wh},

for each iph € ^ h we introduce the solution ( ' a(ugVh, (5.5.17)

i = 1,2

U

x Y?,11 to

+Pbi(vi,h,P^h) = 0

< a ( P ^ i P h , n h h h , ~ bi(XJ^h,iri,h) = 0 .

£

- n ) , r = tph

V viifc G W ° h V

G Y^1

onT.

The discrete Steklov-Poincare operators S ^ are defined as

(5.5.18) These operators are symmetric; proceeding as in the infinite dimensional case, it can be shown that they are coercive, uniformly with respect to h. In particular, taking 7ri)ft = div(U^tph) in (5.5.17), it follows that (5-5.19)

||div(U^)i|o,Q, < C 0 | | / f M l l o , n ; -

Proving the uniform continuity is a more delicate task. For each tph £ &h i = 1,2 introduce the discrete function Q i ^ h

G

as in (4.1.27). Taking

Vi,h = U f l ^ h ~ Qi,hiph and m, h = fiP-^iph in (5.5.17), it follows that T(0)

= a ( U f h tp h , Q i ^ h h , + 0bi{Qi, h iP h ,P$xp h ) Therefore,

and

§5.6

FIRST-ORDER EQUATIONS

203

The uniform continuity of the operators S it h thus follows from Theorem 4.1.9, which, under the assumptions that the family of triangulations Th of fl is regular and the family of triangulations Mh of F is quasi-uniform, shows the uniform continuity of the operators Q^/j. The finite dimensional Steklov-Poincare equation corresponding to (5.5.13) can be solved by means of a preconditioned Richardson procedure, with S 2 ,/i as a preconditioner. Theorem 4.2.2 ensures the convergence of the iteration, at a rate independent of h. The same is also true for the Dirichlet-Neumann iterative scheme corresponding to (5.5.14)—(5.5.16), because it is equivalent to the Richardson scheme with preconditioner S2,h for the Steklov-Poincare equation. As a consequence of Theorem 4.2.5, an analogous result holds also for the Neumann-Neumann iterative scheme, which is equivalent to a Richardson method for solving the Steklov-Poincare equation with (a\S^ + o ^ S ^ ) - 1 as a preconditioner.

5.6

First-order equations

Several mathematical problems in fluid dynamics or wave propagation are described by non-linear advection equations of the following type ^

+ div[F( U )] = / .

Applying a time discretisation based on an implicit method, followed by a linearisation procedure, we are thus led to the first-order linear advection equation (5.6.1)

LQU : = d i v ( b u ) + CLQU = f

in fl,

which is the simplest model of a hyperbolic equation. In the discretisation process described above, the coefficient ao is proportional to (At)"1, the inverse of the time step. The case of hyperbolic systems will be considered in Section 7.2. Let us denote by n* the unit outward normal vector to and by (5.6.2)

<9f2in : = { x €E dfl | b ( x ) • n * ( x ) < 0 }

the inflow region of f2 with respect to the advective field b . The Dirichlet boundary value problem associated with LQ is given by

{

LQU

= /

u = ipjj

in fl on

dflln.

The interface conditions require that the normal flux b • n u is continuous across T; namely, denoting by Ui the restriction of u in fli, i = 1, 2:

r

204

Ch.

OTHER BOUNDARY VALUE PROBLEMS

(5.6.4)

b| ni • ntt: = b|n2 • n u 2

5^

on T.

In an equivalent way, provided that b • n is continuous across T, the continuity of u on the inflow region of fii as well as on the inflow region of has to be imposed: (5.6.5)

u!=u2

onrinurout.

Here, as usual, we have denoted by n the unit normal vector on P directed from Qi to and we have set Tin r

out

:= { x G T | b(x) • n(x) < 0} : =

{

x

e

r

| b ( x ) • n(x) >

0}.

We also define dnoat

:= ( x £ an I b(x) • n*(x) > 0},

and, for i = 1, 2, we put := dflin n dtli, 5.6.1

dQ°iUt •= 900ut

n ddi.

Weak multi-domain formulation and the Steklov-Poincare interface equation

With the aim of deriving the weak formulation of (5.6.3), we start with some notation. First, assuming that be(l°°(Q))d,

Djh €(L°°(Q))d,

j =

l,...,d

(in particular, the vector field b is continuous in fi), let us define the space L2(dOin (5.6.6)

U <9f20Ut) := (77 : dQin U <9fiout R| L s f / |b-n*|772 < o o L

endowed with the norm

IMkcHiinuan-' : = ( [ lb"11*!7]2) \Jd Q^uan0"' /

,

and denote by Xf, the Hilbert space (5.6.7)

Xb := {v G L2(fl) | div(bv) G L2(Q),

endowed with the norm

v G L2b(dQin U <9fi out )},

§5.6

FIRST-ORDER EQUATIONS

205 1/2

\v\\xh ••= (iMIo.n + II div(biO||g,n + |MI&,an'»u0no«.) Then, assuming that a 0 £ L°°(Q), form C(w,v)

~

(5.6.8)

for each w,v £ Xb we define the bilinear

\ f [div(bui) v — div(bu) w] 4- f + -

2

( d i v b + ao^l w v

i f

2Jsi

JDWUT

b - n *wv

—~

2 JQQin

b-n*wt>.

The variational formulation of (5.6.3) reads (5.6.9)

fmdueXb

: C{u,v)

= (f,v)

+ (
V v £

Xb,

where we have denoted by (•, Oft.an'- the scalar product in £j;(<9fi m ), and we have assumed that / £ L2(Q) and ipD £ L2b(dQ,in). It can be proved that problem (5.6.9) has a unique solution u (see, for example, F. Gastaldi et al. 1990), provided that the condition (5.6.10)

^ d i v b ( x ) + o 0 ( x ) > ^o > 0

is satisfied for almost all x £ f2. Moreover, the solution satisfies u £ Introducing for i = 1, 2 the local spaces

Ll(dflout).

XIIB := {VI £ i 2 ( 0 , ; ) | div(bvi) £ L 2 ( f i ; ) , V{ £ Lg(3fij n U d f t ° u t ) , (5.6.11)

Vi £ i 2 ( r i n u r o u t ) } b

n

X1of,:={Wl£X1,„|^1|ri„=0} X 2 % := {v2 £

1 f2|rout = 0 } ,

and the local bilinear forms defined in Xi, b by Ci(wu v ^ : = \ j 1

+ / (5.6.12)

[div(bwj)

- d i v ( b v j ) Wi]

J n;

I - div b + a0 I Wi Vi j i

+ - / b-n*WiVi-~ h-n*WiVi 1 Jdn?ut ^ J dQf + - / b-nWiVi-- / b-nWiVi, L JYout £ Jrin the multi-domain weak formulation of (5.6.3) reads: find such that

£ Xiib

and u2 £

X2ib

Ch.

OTHER BOUNDARY VALUE PROBLEMS

206

Ci(ui,ui)

= (/>i)ni

+ {
C2(u2, v2) = (/, v2)n2 +

^isfi-Oft.ani,"

V«i6 V

5^

X°b e

(5.6.13) (U^pin - U2|ri„,/i)&irin +(u2|r-t-ui|r-t,/i)ft,r-« = 0

V » £ L2{rin

U rout).

Note that L 2 b (T in ) is the space of traces on Tiri of X l i b , as well as I ^ ( r o u t ) being the space of traces on T o u t of X2
KlV£Xltb

: C1(K1r],v1)

= (i],vllr^)btrin

V^Gly.

Similarly, K2r] is defined in Q2 and satisfies (5.6.15)

K2r, £ X2
: C2(K2rj,v2)

= (rj,v2lTo,t)bX^

V v2 £

X2ib.

The solution Kir] can be shown to satisfy the a priori estimate (5.6.16)

W>||^i7?!|02,ni + l l l K i v W l r o ^ u a n r

-

and similarly (5.6.17)

/io||*2»7||§,na + l \ \ K 2 v \ \ l r ^ u 9 n r <

(see, for example, F. Gastaldi et al. 1990). Introduce, moreover, the solution u* of (5.6.18) u* £ Xitb

: Ci(u*,Vi) = {f,Vi)ni

Denoting by A the trace on p i n u p o u t seen that the restriction U|n£ satisfies (5.6.19)

u\Qi=Ki\

+ {(fD,vildnin)bgnin 0f

V Vi £

Xiib.

the solution u to (5.6.3), it is readily

+ u*.

Therefore, imposing that KiX + u* and K2X + u 2 are the solutions to (5.6.13), we find the Steklov-Poincare equation:

§5.6

FIRST-ORDER EQUATIONS

(5 6 20)

(A

"^

2 A )

l

r i n

"

u2|rin>

207

»hr>*

+ ( A - ( A T i A ) i r o u . - u * l r o » t , M ) b i r o u t =0

V /j e L2b(Tin u r o u t ) ,

which in a strong form reads j A-(iT 2 A)|rin = ^ , r i „

on T in

1 A - (ifiAJirout = u* |rout

on r o u t .

Defining in £ 2 ( r i n UT o u t ) the Steklov-Poincare operator S = Si+S2,

{S2r],p) := (ri,n)b,=

and the function X

where

( ( i ^ i r ' " , r -

Xi + X2> where (Xi.M) : = K i r ° » t >/*)&,r°"' . 11 \X2,W == K| r in,M)b,rf,

5-6.22

the Steklov-Poincare equation can be written as (5.6.23)

\EL2(RINUROUT)

: (S\,M) = (X,M)

vAie £g(rinurout).

From (5.6.16) and (5.6.17), we see that the operator S is continuous in L 2 ( F i n u p o u t ) M o r e o v e r , u s i n g ( 5 . 6 . 1 4 ) a n d ( 5 . 6 . 1 5 ) , it c a n a l s o b e w r i t t e n as 2 »=i The operator Si satisfies

(SIV,V) = ITOLLER'"~

+ \\\KiV\\l,r^ -

{{KiT])lroui,ri)b!r^,

having used (5.6.16). We also have ^\\Kir]\\lrout

therefore

-

((Kir])lrout,r])btrout

=

- K^Wl^

-

^ | h i »

208

(5.6.24)

Ch.

OTHER BOUNDARY VALUE PROBLEMS

(Siv,v)

>

+ f^WKivWlfl,

+

5 ^

\\\Kir]\\laar

Similarly, we find that

(5.6.25)

{S2V,V) >

1

+ l*o\\K2T)\\ln2 + \\\K2r)Wl9U?t 2

6,rin

_

hence (577,7/) >yuo||^i»?llo,n1 + ^\\K2v\\ln2 (5.6.26)

1

Therefore, the operator S is positive, i.e. {Sri, V) > 0 5.6.2

for

77

^ 0.

Substructuring iterative methods

The following iterative method has been proposed by F. Gastaldi and L. Gastaldi (1994, 1995): given A?n £ L^(r i n ), for each k > 0 solve ' finditj+1 £ Xi,b : (5.6.27)

t C i ( u f + 1 , u i ) = (f,v!)Ql

+

(vz3,«i|an»)fc,an'-

then

(5.6.29) In (5.6.27) we are imposing the Dirichlet condition = Afn on T 1 ", while in (5.6.28) we are imposing the Dirichlet condition u 2 + 1 = u k + 1 on r o u t . For each rj £ L 2 (F l n U r o u t ) denote by r]m and r^out its restrictions to F in and r o u t , respectively, and introduce the operators J\ : L 2 (F i n ) - 5 > L 2 ( r o u t ) and J2 : L 2 ( F o u t ) L g ( r i n ) as follows:

§5.6

FIRST-ORDER EQUATIONS

(5.6.30)

Jirjin •= (Kir] in )\ r au t ,

J

2

W

:=

209

(^2%ut)|r'»-

The iteration operator associated with (5.6.27)-(5.6.29) is given by J 2 J i , and in F. Gastaldi and L. Gastaldi (1994, 1995) it has been proved that it is a contraction in L ^ ( r l n ) . Therefore, the iterative method (5.6.27)-(5.6.29) is convergent in L 2 ( F i n ) , and the limit function Ain is equal to U|r;n, where u is the solution to (5.6.9). Using a similar proof the map J\ J2 is also shown to be a contraction in L 2 ( r o u t ) . The dual space of L2b(Tin U r o u t ) is given by L 2 1 / b (T i n U r o u t ) : = jr? : T i n U T o u t

R |

/ |b • n| _ 1 rf < 0 0 yrinUrout J

>.

It is easily seen that the Steklov-Poincare operator in u r o u t ) s: £^(r in u r o u t ) -»• L\ -1/6/b{r

can be rewritten as ^

on r m on r o u t

f - b • nr7in + b • n 2 t ^ oJu \ - b • n Ji^ in + b • Vno u t

Introducing the operator Q : L{/b(Y'm

(5-6.31)

'

U T o u t ) -s- i ^ ( r i n U r o u t ) defined as

=

b • n

\b•n

J

we find that

=

:"rl •

Hence, the operator I - QS is a contraction in I<j(r i n U r o u t ) . As a consequence, the operator QS is coercive in the same space; in fact, from the inequality Ik - <35r/||^ rinurout < AT2||?7||2rinurout, where 0 < K < 1, it follows that 2(QSr],ri) b i r in u r o«t > (1 - K2)\\r]\\2br in ur out

rmurout

-

The iteration procedure (5.6.27)-(5.6.29) can be seen as a preconditioned Richardson method without relaxation (i.e. for 0 = 1) applied to the operator S , with Q-1 as a preconditioner. Since the operator QS is coercive, other iterative procedures could be devised as well. Note that, unlike for elliptic equations, relaxation is not mandatory at the interface.

210

OTHER BOUNDARY VALUE PROBLEMS

Ch.

5 ^

The finite element approximation of (5.6.9) and the finite dimensional iteration procedure corresponding to (5.6.27)-(5.6.29) have also been studied by p. Gastaldi and L. Gastaldi (1994, 1995). Their approximation is based on discontinuous piecewise-polynomial finite elements (see Lesaint and Raviart 1974; and Quarteroni and Valli 1994, p. 487). The rate of convergence of the iteration-bysubdomain procedure is proved to be independent of the finite element mesh size.

5.7

The time-harmonic Maxwell equations

The Maxwell equations describe the wave propagation phenomena in electromagnetism. They read: ? ot

= rot U - J

dB

where £ and % are the electric and magnetic field, and V and B are the electric and magnetic induction, respectively, and J is the density of the electric current. The following constitutive relations are assumed to hold:

V = e£, B = iM (where e and // are the dielectric permittivity and magnetic permeability, respectively), as well as Ohm's law J = a£ (where a is the electric conductivity). The quantities e, [i and a are, in general, symmetric matrices, depending on the space variable x, and are assumed to be uniformly positive definite in a conductor. In a perfect insulator a is zero, while e and /j, are positive constants. For a complete presentation and for several theoretical mathematical results and approximation methods for the Maxwell equations see Duvaut and Lions (1976) and Bossavit (1993). The so-called time-harmonic Maxwell equations are derived from the complete Maxwell system when one seeks solutions of the form (5.7.1)

£(t,x)

— Re[E(x) exp(iai)],

H(t,x)

= Re[H(x) exp(iat)],

where i is the imaginary unit and a / 0 is a parameter (the angular frequency). The unknowns in the resulting problem are the complex vector fields E and H , and these have to satisfy

(5.7.2)

rot H - ieaE - crE = 0

in fi

rot E + ijiaH = 0

in fi

, n x E =

on dfl,

r

§5.7

THE TIME-HARMONIC M A X W E L L EQUATIONS

211

where ft is a bounded domain of R 3 , and the tangential vector field is a given datum defined on 3ft. Rewriting the system above in terms of the unknown E, we find that C rot(/^ - 1 rot E) - a2sE + iaaE = 0 in ft I inxE = $ on 3ft.

(5.7.3)

Setting u := E - E^, where E ^ is defined in ft and satisfies n x E ^ .= VP on dQ, problem (5.7.3) can be finally rewritten as { r o t ( | i - 1 rot u) — o?eu + iacru. = E n x u = 0

in ft on 3ft,

where F ;= - rot(yU_1 rot E>j.) + a 2 e E ^ - iaaE^,. The two-domain formulation of (5.7.4) is as follows. Denoting by u« the restriction of u to the subdomain ft;, i = 1,2, in each subdomain Uj satisfies (5.7.4). In addition, the interface conditions are given by (see, for example, Alonso and Valli 1997) n x ui = n x u.2

(5.7.5) 5.7.1

n x (/i

1

rot ui) = n x

on T (/j, 1 rotu2)

on T.

Weak multi-domain formulation

Introducing the Hilbert spaces '

H(rot;ft) := { v G (L 2 (ft)) 3 | r o t v £ ( L 2 ( f t ) ) 3 } #O(rot; ft) := { v G H{rot; ft) | (n* x v)| Sn = 0 } ,

and the bilinear form (5.7.7)

m(w, v ) : = I (u 1 rot w • rot v — a2ew • v m(w,v):=^y +iacrw -v) V w, v 6 / / ( r o t ; ft),

the weak formulation of (5.7.4) reads (5.7.8)

find u G / f 0 ( r o t ; f t ) : m(u,v) = (F,v)

V v G # 0 ( r o t ; ft).

Let us now restrict our analysis to the case of a conductor, so that e, n and <7 are assumed to be symmetric matrices, uniformly positive definite in ft. The case of an heterogeneous medium will be considered in Section 8.5. Under these assumptions, the bilinear form m(-,-) is continuous and coercive in i f (rot; ft); therefore, the existence and uniqueness of a solution follow from the Lax-Milgram lemma. In fact, we have that

212

O T H E R BOUNDARY VALUE PROBLEMS

Ch.

5 ^

/

rot v • rot v — a 2 I e v • v Jn

Mv,v)|2 =

+ ( a I <7V • v J nQ

2 / rot v | + [ a 2

.Jn

« Jn

\ 2 ev • v

- i t s 1 rot v • rot v | | aJn / 2

.In

ev • v

crv • v

Noting that 0,

2AB < 5A we find, for 5 < 1,

|m(v, v)|2 > (1 - <5) ( / fx rot V ' r o t v 'n \l-S)e2 21 +a 1 J crv • v Jn Therefore, coerciveness follows on choosing a2e2 a2ej

+ (Tq

Note that in the so-called low-frequency term

<5<1. case that corresponds to omitting the

— I a2sw•v Jn in (5.7.7), the coerciveness result still holds with an easier proof. To derive the equivalent two-domain formulation, for j = 1, 2 let us denote by mj(-, •) the restriction of m(', -) to f l j and introduce the following spaces:

(5.7.9)

Zj

= ( v j e i?(rot; Qj) | (n* x Vj)|afinan3- = 0 }

Q

= Ho(rot\ Clj)

Xv = {ip : r ->• R 3 I ij> = (n x v)|r, v G ff0(rot; ft)}. The variational formulation of the two-domain problem reads: find ( u j , u 2 ) £ Z± x such that

r

THE TIME-HARMONIC M A X W E L L EQUATIONS

§5.7

fmi(ui,vi) = (F,vi)ni n x ui = n x U2

VviGZ?

on T

m2(u2,v2) = (F,V2)o2

(5.7.10)

m2( u 2 , ^ V 0 =

213

V v2 £

(F,^)n

Z\

2

-mi(

U l

,^)

V^GAr,

where we have denoted by 72.* any operator from X? to Z:t satisfying (n x H*ip) |r = V* f°r 5.7.2

each

V* S <*r-

The finite dimensional Steklov-Poincare interface equation

We start by considering the finite element approximation of problem (5.7.8) and of its two-domain formulation (5.7.10). It is based on the so-called rot -conforming Nedelec finite elements, introduced in Section 4.1.3 and there denoted by M Let us denote the finite element space used for the approximation of the unknown u by Zh := Mrh n H0(rot; fl), and set = {VFTIN,- | V A £ Zh},

Z

lh

:=

Zj,h H Ho(rot\ flj),

j =

1,2

j — 1,2

Xr,h •= { ( n x vi i A )| r | vi i f t £ Z1>h] = { ( n x v2>/l)|r | v 2 ) f t £

Z2,h}-

The finite element approximation of (5.7.8) is given by (5.7.11)

find

u

h

eZ

h

: m(uh,vh) = (F,vh)

V v f t £ Zh.

On the other hand, the induced approximation of the two-domain formulation (5.7.10) reads: find ( u ^ , u 2>/l ) £ Zith x Z2jh such that ' mi (ui i f t , vi i f t ) = (F, v l i A ) n i n x ui i A = n x u2)ft (5.7.12)

m2(u2,h,v2,h)

V vi, ft £ Z l

h

on T

= {F,v2ih)a2

V v2,h £ Z°2 h

m 2 (u 2 , h ,Hl h tl> h ) = (F,Kl h il> h ) Q 2 + ( F , K l h $ h ) n i - m i ( u h h , T V l hiph)

XTth,

where we have denoted by h any operator from Xr,h to Z j j , satisfying (n x ^ I h ^ h ) ^ = iph for each iph eXr,h. Since the bilinear forms m,j(-,-) are continuous and coercive in H(rot; flj), j = 1,2, we can introduce for each rjh £ Xr,h the finite element solution £j,hVh £ Zj,h to

OTHER BOUNDARY VALUE PROBLEMS

214

i{£j,hri h , Vj,h) = 0

Ch. 5 ^

V vj ) f c G Z°h

(5.7.13) ( n x £j,hVh)\r = Vh and the solution £* (5.7.14)

h

£ Z°h

onT,

to = (F, v M ) n .

mj(£;ih,Vjih)

V vjih

£

Z°h.

The finite dimensional Steklov-Poincare equation now reads (5.7.15)

(S h X h ,fM h ) = (Xh,fih)

where Sh :=

5.7.3

Xr>h,

+ S2,h, Xh := Xi,h + X2,h and

(5.7.16)

(5.7.17)

V fi h £

(Sj.hVki Ph) == m j i S j ^ h i S j ^ H h )

V r} h ,l*h £ Xr.h,

(Xj,h,l*h) •= ( F . ^ f c ^ J n , - - m ^ * ^ , ^ h )

V/ihe

XT>h.

Substructuring iterative m e t h o d s

The Dirichlet-Neumann iterative scheme for solving the two-domain problem (5.7.12) reads: given X°h £ <%r,/i> for each k > 0 solve ' find u ^ 1 £ Z:l,h (5.7.18)

m i ( u f + 1 , v i i f c ) = (F,vlth)Ql [ ( n x u t + 1 ) | r = A^

V v 1 > h £ Z[ y h

onf,

then find MIX1 £ Z2th m2{uk+h1,v2,h)

= (F , v 2 ; h ) n 2

V

£ Zlh

(5.7.19) m 2 {vL k + h \Hlh^h) = ( F , n , h 4 > h h * + mi(u\X\KIM

V^G^r.ft,

and finally set (5.7.20)

k+ls

on P.

This iterative scheme is equivalent to the following preconditioned Richardson iterations for the Steklov-Poincare problem (5.7.15)

r §5.7

THE TIME-HARMONIC M A X W E L L EQUATIONS

(5.7.21)

Xkh+1 = Xkh + 0S~l(X

- ShXkh)

215

on T.

Its convergence can be proved by resorting to Theorem 4.2.10 and Corollary 4.2.11, because we are dealing with complex non-symmetric Steklov-Poincare operators. The complex matrices associated with the discrete Steklov-Poincare operators are defined as ((£,-,hi?,/i)) :=

(Sj,hrih,nh),

where rj, (J, £ Cff are the vectors corresponding to the tangential traces r)^, /j,h £ -^r./i, respectively, and ((•, •)) denotes the scalar product in CMh. We want to prove that (4.2.14) is satisfied, and that the constants Co and Co in (4.2.15) and (4.2.16) can be bounded independently of h. Let us now restrict ourselves to the low-frequency case; that is, suppose that the frequency a is so small in comparison with the other coefficients and ao that the coefficient a2s in (5.7.4) can be omitted. In this case, the operators Sj,h, j = 1,2, satisfy l(SMVh,Vh)l

(5 7 22)

+ 11rot£j,hr}h\\l^j)

> ci{\\£j,hr)h\\l>u. >c 2 ||%|| 2 * r

VIJ„ £ X R , H ,

where we used the coerciveness of the forms mj(-,-) and the continuity of the tangential trace operator from H(rot; Q.j) into X-p that was proved in Alonso and Valli (1996). Moreover, Re(Sj,hrih,r)h)

=

(5.7.23)

/

n'1 n

Im(Sj,hr)h,r)h)

=a

r o t ^ t ^ • rot£jthf^ > 0

y

JQj

a£j,hr)h ' £j,hVh > 0-

Hence Re ( ( S / i » 7 , 7 7 ) ) R e ( ( E 2 , ^ , r ? ) ) +Im((S/i7j,T})) Im((E 2) / l »7,T))) = Re((Si, A i7 ) i7))Re((E 2 , h f7,i7)) +Im((E 1 ,^,R ? ))Im((S 2 ,FTJ ? , ) 7 )) + |((S 2 , /i J 7 , )? ))| 2 > |((E 2 i ^, J ? ))| 2 , and (4.2.14) and (4.2.15) are satisfied, with C0 = 1. The proof that (4.2.16) holds with a constant Co independent of h is more involved. First of all, from (5.7.22) and the continuity of the form m i ( - , - ) one has |((Sl7,l?))| miV,f)))\ I \£l,hWh 11 Jf (rot; fii) |((S 2 i 7) I ? ))| The estimate

+

|((E2f?,»?))| -

+

3

II^H^

216

O T H E R BOUNDARY VALUE PROBLEMS

(5.7.24)

Ch.

5 ^

| \£l,h,Wh I (//(rot ;Hi) <
is obtained as follows. For each j = 1,2, let us first introduce the discrete functions J"ithVh a s i*1 (4.1.39). Taking in (5.7.13) the test function v^/j = £j,hVh ~ •F].hT)h, j = 1,2, the coerciveness and continuity of the form m , ( - , •) yields I \£j,hWh I (//(rotiOj) < C^W^Fj,h^h\\H(vot-,Qj)Finally, the inequality I

11 /f(rot;Q_,) < C6 |\Vh I \xT >

is proved in Theorem 4.1.11, under the assumptions that the family of triangulations Th of fl is regular, the family of triangulations M h induced by Th on F is quasi-uniform, and T is a convex portion of <90 j . For the proof of the convergence of the Dirichlet-Neumann iterative scheme (5.7.18)-(5.7.20) we need that (5.7.24) is satisfied only for £\ i h t] h , and not for £2,hVh- Hence, we obtain the result under the assumption that F is a convex portion of d f i i . A similar procedure can be used for showing the convergence of the NeumannNeumann iterative scheme applied to the solution of (5.7.12). As usual, it can be seen that it is equivalent to the Richardson iterative scheme for solving the Steklov-Poincare equation (5.7.15) with ( a i S ^ + ^ S ^ h ) - 1 a s a preconditioner. We have already verified that assumptions (4.2.20) and (4.2.21) of Theorem 4.2.13 are satisfied (see (5.7.22) and (5.7.23)). Moreover, a bound on the constants Ci and in (4.2.22), uniform with respect to h, is a consequence of (5.7.24) for £i,hVh and of the analogous estimate for £2,/i»?/,,. Hence the convergence result follows under the assumptions that the family of triangulations Th of fi is regular, the family of triangulations M.h induced by Th on F is quasiuniform, and T is a convex portion of both <901 and d$l2; namely, a plane surface. For the approximation of Maxwell equations by means of the additive Schwarz method see Toselli (1999). In Table 5.7.1, on p. 217, we summarise the interface conditions associated with the Laplace operator and with the other partial differential equations considered in this chapter.

r §5.7

THE TIME-HARMONIC M A X W E L L EQUATIONS

Table 5.7.1

217

The interface conditions for different partial differential

Laplace

Advectiondiffusion

Elasticity

-

operators

P.D.E.

Dirichlet continuity

Neumann continuity

-A u = f

u

du/dn

u

du/dnL

u

CT(U) • n

Dt(aijDju) + div(bit) + aou = /

-diva(u) = f [aij := fi{DiUj + DjUi) +Adivu<5/j]

Oseen (Stokes: u* = 0 )

f —divT(u,p) I +(u* • V ) u = f [ divu = 0

u

T(u,p)-n

u

T(u, a) • n

[Tu := v(DiUj + DjUi) -Vhj\

Compressible Stokes

f au - divT(u, a) = f \ acr + div u = g [Tij := v{DiUj + DjUi) -/3aSij + { j - 2v/d) d i v u % ]

Compressible inviscid Stokes

f au + /3V(T = f | au + divu = g

Advection

div(bw) + aou = f

Maxwell time-harmonic

rot(jit -1 rot E ) - a 2 e E + iota E = f

u •n

a

b • nil

n x E

n x {p"1

rotE)

6

ADVECTION-DIFFUSION EQUATIONS This chapter deals with a class of elliptic equations that are not symmetric, due to the presence of first-order terms in the differential operator. These operators describe advection-diffusion processes, and typically arise in fluid dynamics. Our emphasis is on the case in which the transport phenomena, driven by the advective terms, are dominant with respect to the diffusive ones, which are related to the principal second-order part of the operator. The case of a non-symmetric advection-diffusion equation with dominant diffusion was considered in Section 5.1. When these equations are used as computational kernels in the simulation of Navier-Stokes problems, the ratio between convective and diffusive terms is expressed by the flow Reynolds number, which may be very large in many applications. We investigate the solution of advection-diffusion equations in the framework of non-overlapping multi-domain partitions. Domain decomposition methods based on the Schwarz overlapping technique are described, for example, in Cai (1990, 1995), Widlund (1992), Garbey (1996), and Garbey and Kaper (1997) and the references therein. Domain decomposition methods based on substructuring iterations are very effective when the diffusive part is dominant; whereas, if the convective part becomes more relevant, then the natural interface conditions may produce instabilities. A first strategy for avoiding these instabilities is based on imposing interface conditions that are consistent with the hyperbolic limit of the advection-diffusion equation, taking into consideration the direction of the characteristic curves on the interface. The interface is split into subsets according to the direction of the flow related to the convective part, and the interface conditions are accordingly devised. Some iterative schemes have been proposed in this spirit (see Carlenzoli and Quarteroni 1995; Nataf and Rogier 1995; and Auge et al. 1997). This approach avoids the creation of internal layers and then makes the iterative procedure more stable. We present several methods stemming from this idea, which are based on different treatments of interfaces between adjacent subdomains. A second strategy makes use of different interface conditions that are more germane to those of the symmetric elliptic equations. Splitting the operator into its symmetric and skew-symmetric part, the associated bilinear forms turn out to be coercive in each subdomain ftj, without requiring any special adjustment of the interface conditions according to the direction of the convective field on T. The interface conditions are invariably of the Dirichlet and Robin type, and can

ADVECTION-DIFFUSION EQUATIONS

220

Ch.

6^

be used to introduce a Steklov-Poincare equation on T, equivalent to the original problem. For this approach we prove convergence of the subdomain iterations; in some cases, at a rate independent of the mesh size. Our discussion begins with the analysis of several multi-domain formulations of advection-diffusion equations. Although equivalent at the level of the differential problem, the different sets of interface conditions behind these formulations give rise to various iterative substructuring methods.

6.1

The advection-diffusion problem and its multi-domain formulations

We consider the boundary value problem

{

Leu := —eAu + div(bu) + a^u — f u = 0

in ft on <9ft,

where ft is a bounded domain in R d , d = 2,3, s > 0 is a constant diffusion coefficient, b = b(x) denotes the given flow velocity and ao = ao(x) an absorption term. Finally, / = / (x) represents a given body force. The characteristic quantity UJ : = (2e) - 1 |b| (essentially the analogue of the Reynolds number for Navier-Stokes equations) will play an important role in what follows; in particular, we will be primarily concerned with the advectiondominated case, in which LO 1. For the existence and uniqueness analysis of (6.1.1) it is useful to assume that b, ao and / satisfy b G (L°°(ft)) d , a0 £ L°°(Q),

d i v b £ L°°(Q.) f £

L2(n)

and (6.1.2)

^ divb(x) + ao(x) > 0

for almost every x G ft.

Under this restriction, it is known (see, for example, Quarteroni and Valli 1994, Chapter 6) that there exists a unique solution to the following weak form of (6.1.1) (6.1.3)

u £ Hq(Q)

: a°(u,v)

= (f,v)

Vwg^ffi),

where we have denoted by (6.1.4)

a°(w, v) :=

/ [eVw • Wv + div(bw) v + a0 w u] w,v £ Jn

H1^),

the bilinear form associated with the operator Le. This bilinear form is continuous and coercive in i?o(ft) owing to (6.1.2).

§6.1

M U L T I - D O M AIM F O R M U L A T I O N S O F A D P R O B L E M S

221

As in Chapter 1, let us consider a partition of Cl into two non-overlapping open subdomains Cli, i = 1, 2, and denote by Y : = ftiflf^ the common boundary between Cli and Cl2, and by n 1 the unit outward normal vector on T, setting n := n 1 . Under these assumptions, problem (6.1.1) can be reformulated as follows: for j = l , 2 find Ui = U|n; such that (LeUi = f

(6.1.5)

in Cli,

i = l,2

Uj = 0

on<9f2nd£}i,

Ui = u2

on r

dui du2 £——=£-— an on

„ on r .

This is called the Dirichlet-Neumann We now set, for 2 = 1,2,

i = l,2

(DN) formulation.

Vi := {vi G iJ 1 (Oj) | UjiaQngQ; = 0 } and for each Wi,Vi G Vi (6.1.6)

a°i(wi,Vi) : = / [eVw, • Vv, + div(bw,) Vi + a0 Wi Vi]. J Qi

Proceeding as in Lemma 1.2.1 it is readily seen that (6.1.4) is equivalent to the following multi-domain problem: find ui 6 Vi, u2 G V2 such that f a ? ( u i l t ; 1 ) = (/,r; 1 )n a

(6- 1 - 7 )

Ui = u 2

on r

a°2(u2,v2)

= (f,v2)Q2

2

V^

6/^(Oi)

V « 2 e H^(Cl2) 2

J 2 a°i(ui, K m ) = i=i

KiPh,

V n G A,

4=1

where A is the space of traces on T of the functions of Hq (O) (see (1.2.4)), and TZ, jj.. denotes any possible extension of /i to Cli (for example, TZtn — H-iji, its harmonic extension in Cli, see (1.2.14)). We assume that T is a Lipschitz interface and we distinguish three subsets

(6.1.8)

r°

: = { x G r | b ( x ) • n(x) = 0 }

T in

:= { x G T|b(x) • n(x) < 0}

rout

._

G

r | b(x) • n(x) >

0},

222

ADVECTION-DIFFUSION EQUATIONS

FIG.

6.1.1. Partition of the d o m a i n

Ch.

6

fi.

which are identified through the local direction of the flow field b ( x ) at the subdomain interface (see Fig. 6.1.1). If we assume that F° has zero surface measure, then we may replace (6.1.5) by the following equivalent Robin-Neumann formulation (RN): in fij, i = 1,2

f LeUi = f Ui = 0 (6.1.9)

dui s on dui

on dttndQi, du2 b • nui = e — on du2

dn

dn

b • nU2

i=

1,2

on T on r .

The new interface conditions derive from a different weak formulation of (6.1.1), which is alternative to (6.1.3), (6.1.4). The difference stems from the fact that this time we also integrate the convective term by parts, and obtain: (6.1.10)

find u 6 Hq(Q)

: aR{u, v) = (/, v)

VvG

where

i(w,v)

:= I [sVw • Vv — wh • Vv + Jn

aowv].

The variational formulation of (6.1.9) takes a form similar to (6.1.7) and reads: find « i 6 Vi, u 2 £ V2 such that

^

MULTI-DOMAIN FORMULATIONS OF A D PROBLEMS

' A°(UI,WI) = ( F , V ! ) Q L

2 (6.1.11)

223

VVIGFOHOI)

2 KiM) = £ ( / ,

J2 affa, i= 1

Vn e A

i=l

o§(«2,v2) = (/,u2)na 2

V » 2 £ H£(Sl 2 )

2

J2a°i{ui,nifi) = Y/{f,TZip)Qi i=l

V / i G A,

i=l

where (6.1.12)

af-{wi,Vi)

: = / [eVifj • Vt/j JUi

b • Vvi + a0 Wi v^,

i = 1,2.

Note that af(wi,Vi)

= af-{wi,Vi) - J b n l W i V i ,

Wi,Vi£Vi,

hence the two bilinear forms a°(-,-) and a f (•, •) are coincident on the space Hq (ft;). For this reason a,({ could be replaced by a f in the first equation of (6.1.11), and a2 by a f in the third equation of (6.1.11). Finally, for any type of T 0 (that is, even if it has positive surface measure), a third set of interface conditions can be used. The associated multi-domain problem, alternative to either (6.1.5) and (6.1.9), is called the (3-Robin-Neumann (RpN) formulation, and reads in ft«, i = l , 2

' LeUi = f Ui= 0 (6.1.13)

dui

s-^ on

on <9ft fl dftj, du2 pux = £— an

dm du2 £-p— = e—— an on

pu 2

on

i = l,2

r

on T,

where (3 E L°°(T) is a given function, almost everywhere different from 0. In particular, note that (6.1.9) is obtained for (3 = b • n. Its variational formulation is given by: find u\ E V\, u2 E V2 such that

Ch.

ADVECTION-DIFFUSION EQUATIONS

224

( Qi(ui,vi) — (f,vi)a1 2

(6.1.14)

'

a°2(u2,v2) 2

6^

V ^ e ^ f O , )

2

= (f,v2)n2

V^eFo1^) 2

1=1

where (6.1.15)

: = a f ( w i , i ; j ) + / ( b • n - /?)n • n l r

v{.

In conclusion, we can consider either one of the multi-domain forms that are given by equations (6.1.5), (6.1.9) and (6.1.13). Any such choice can be rigorously justified and the equivalence between them can be proved (see F. Gastaldi et al. 1996). However, although all sets of conditions DN, RN and R^N are equivalent to one another, they will generate different iterative substructuring procedures between fii and The reason for choosing a specific set of equations (out of the three possibilities) depends primarily on the data for the problem under consideration. T w o further iterative procedures will be introduced in Section 6.4, based on multi-domain formulations that are different from (6.1.5), (6.1.9) and (6.1.13).

6.2

Iterative substructuring methods for one-dimensional problems

The previous multi-domain formulations suggest the application of suitable iterative methods that reduce problem (6.1.1) to a sequence of mixed boundary value problems on each subdomain. Before introducing these methods in the general case, let us consider the simple (albeit meaningful) one-dimensional problem Lsu := —£Uxx + bux + a^u = f

in 17 = (0,1)

(6.2.1)

where e > 0, b 0 and ao > 0 are constant coefficients. In F. Gastaldi et al. (1996) the following iteration-by-subdomain method to solve (6.2.1) has been analysed: given A 0 , solve for k > 0

§6.2

ONE-DIMENSIONAL PROBLEMS

f Leuk+1

225

in f2i = (0, c)

= f

0) = 0

(6.2.2)

then Leuk+1 (6.2.3)

=f

in Cl2 = (c, 1)

uk2+1(l)=0 eutHc)-

+

uk+1(c)=euk+1(c)-(\b

+ B^I

uk+'(c),

and finally set (6.2.4)

A fc+1 = e u ^ i c ) - Q f c + a ) u k + 1 ( c ) ,

where 0 < c < 1, and A and B are real parameters, with A ^ B. In fact, the cases A = ± o o and B G R , or A £ R and B = ± o o can also be considered. The first choice corresponds to a Dirichlet boundary condition at x = c in (6.2.2), the second one to a Dirichlet boundary condition at x = c in (6.2.3). A straightforward computation shows that this method converges provided that (6.2.5)

\pe(A,B)\
where PE(A,B) r-

T coth(rc) - B/E r c o t h [ r ( l - c)] + A/E t coth(rc) - Aje

and

r c o t h [ r ( l - c)] + B/e'

\Jb2 + 4eao

T :=

2s

(see F. Gastaldi et al. 1996, in which a = \ b + A and /3 = \ b + B). Introducing a relaxation parameter 6 ^ 0, we can consider a more general iterative scheme, still based on (6.2.2), (6.2.3), where now (6.2.4) is replaced by (6.2.6)

Xk+1 = ,

EUkt\c)-[^b

+

A\uk+l<

In this case we have convergence if (6.2.7)

\l-0[l-pe{A,B)]\
+

{i~e)\k.

226

Ch. 6 ^

ADVECTION-DIFFUSION EQUATIONS

Noticing that pe(A,B)

1 for A ^ B, this yields the following limitation on 9:

However, since the focus here is on advection-dominated problems, we are looking for methods that converge when 9 is chosen independently of e as e -> 0+. A direct calculation shows that (6.2.9, hence, the choices A = |6|/2 and B = —16|/2 (which would be forbidden in the limiting case e = 0) are expected to lead to an inefficient scheme even for £ / 0 but small. These choices correspond to imposing the value of the normal derivative ux at the inflow region, or the value of the conormal derivative eux — bu at the outflow region. From this analysis we can easily conclude that the sign of b needs to be taken into account when one chooses the type of interface conditions to be used. The methods we shall present in Section 6.3 will be tailored in a way consistent with the hyperbolic limit of the convection-diffusion equation; namely, the Dirichlet (or Robin) boundary condition is imposed on the inflow boundary, while the Neumann condition is enforced on the outflow boundary. Let us also point out that, when the asymptotic reduction factor po{A, B) belongs to the interval (—1,1), the relaxation parameter can be chosen in the whole interval (0,1], leading to efficient iterative schemes. From (6.2.9) it is seen at once that the choice A < 0 and B > 0 (with A ^ B) implies that - 1 < p0(A,B) < 1. The condition A < 0 and B > 0 is closely related to the coerciveness of the bilinear forms associated with the boundary value problems (6.2.2) and (6.2.3). In fact, these forms are given respectively by a i ( w i , u i ) := / (ewi,® vi,x + i o » i vi) •'o 1 f° + b{v! wi>x - wi vi<x) 1 Jo

Awi(c)vi(c)

2^2) ••= / (sw2,x v2,x + fflo W2 v2) 1 +

b(V2W2'x

~w2v2,x)

+ B w2(c)

v2(c);

hence, they are coercive in H1(0, c) and Hl(c, 1), respectively, for ao > 0 and for every choice of the parameter e if and only if A < 0 and B > 0. In other words, coerciveness implies that —1 < p0(A,B) < 1, and consequently the iterative

§6.3

ADN, ARN AND AR^N METHODS

227

method converges for every relaxation parameter in the interval (0,1], This analysis for the one-dimensional case leads us to propose, also for the general case (6.1.1), iterative methods based on bilinear forms that are coercive in iJ 1 (S7i) and H 1 (Cl2), respectively: this will be done in Section 6.4.

6.3

Adaptive iterative substructuring methods: ADN, A R N and ARgN

Now we return to the multi-dimensional problem (6.1.1). As mentioned in the previous section, the difficulties arising when the parameter e is small can be given a heuristic explanation in the limiting situation e —> 0 + . W h e n providing each subproblem in Oj and n 2 with a boundary condition at the interface, one should account for the local direction of the characteristic curves in order to be consistent with the hyperbolic limit. Hence, the Neumann interface condition has to be enforced at the outflow boundary of a subdomain, whereas the Dirichlet condition has t o be enforced at the inflow boundary. Let the boundary <90 be partitioned as dtt = dflm U d f l o u t U dflo, where (6.3.1)

dQia := { x e dQ | b ( x ) • n * ( x ) < 0 }

(6.3.2)

dfl° := {xedn\

b(x) • n*(x) = 0}

and (6.3.3)

dfiout := { x 6 d n | b(x) • n*(x) > 0},

where n* is the unit outward normal vector to dfl. Then we set, for i = 1,2,

(6.3.4)

dn™ : = dnin

n

dfy.

If we want to consider the hyperbolic limit case of (6.1.1) (taking formally e = 0), following what we have done in Section 5.6.2, we should operate as follows: for given values A0 and pi0, we look for a sequence {uk+1 ,uk+1} with k > 0 such that: '

L0uk+1 =

on d n ?

= A*

on r i n

L0U2+1 (6.3.6)

l

,k+1 _

2

in

=0

uk+1

,

/

= f

in 0 2 on dnii on pout

228

ADVECTION-DIFFUSION EQUATIONS

Ch. 6 ^

with (6.3.7)

(6.3.8)

\k+1

:= duk2+} + {1 - d)\k

= 0uk+l + (1 - 9)pkk

pk+l

onri:

onrout.

To be consistent with the hyperbolic limit (6.3.5)-(6.3.8), we will therefore split the interface conditions for the advection-diffusion problem (6.1.1) according to the local flow direction. The idea is to define a sequence { u k , u k } where uk satisfies Leuk = / in Ui, along with suitable boundary conditions at the subdomain boundary F that depend on the local direction of the flow field b(x). These conditions at the interface are selected from the set DN, RN or R^N, so that one of them is associated with uk, the other with uk. The corresponding algorithms will be called adaptive, because the role played by the boundary condition on T varies according to the flow conditions there. Consequently, we will have Adaptive Dirichlet-Neumann (ADN), Adaptive Robin-Neumann (ARN), or Adaptive /?Robin-Neumann (AR/jN) algorithms depending upon the choice of interface conditions. These methods have been proposed by Carlenzoli and Quarteroni (1995), motivated by the observation that, for the one-dimensional problem (6.2.1) with do = 0, they guarantee a convergence rate of order exp(—b/e). They have been further developed by Ciccoli (1995), Trotta (1996) and F. Gastaldi et al. (1996). In order to keep our discussion as simple as possible, our approach will be based on the differential (instead of variational) formulations. The variational formulation of ADN and A R N methods for the finite element approximation is presented in Section 6.5. Let us start by presenting the Adaptive Dirichlet-Neumann given uf in Ui, i = 1,2, solve for each k > 0 Leuk+1 uk+1 (6.3.9)

<

= 0

on dU n sni on r i n u r °

u k + 1 = Ak £

and

in Qi

=f

duk+1 dn

==

£

duk dn

-

on r o u t

algorithm (ADN):

ADN, ARN AND AR^N METHODS

§6.3

C

(6.3.10)

T „,*+! f LeU2 = J

in 0,2

4+1

on on n dn2

= o

U^1 =

-

duk+1 2

dn

on r o u t

^

du\+1

-= -£ -

229

1

dn

_

on r i n u r°,

with (6.3.11)

X

:=0

+ (i - e')uk,r i„ u r o

on r 1 u r

and (6.3.12)

: = '"tijfiut + (1 - 0")«2|r-t

on r o u t ,

9' and 9" being two positive parameters that are used to allow possible underrelaxation (if needed) to ensure convergence. Typically, a single parameter 9 suffices (two parameters allow more flexibility in achieving an optimal convergence), and sometimes 0 = 1 (no relaxation at all) is a possible choice. Should the sequence {uk,u2} converge in a suitable sense, its limit { « i , 112} would be the desired solution, because it would satisfy (at least formally) the differential problem (6.1.5). The proof of the convergence of the A D N iteration has been given in F. Gastaldi et al. (1996), for a problem set in the square (0, l ) 2 , with a constant advective field b = (b, 0) and a constant absorption term a0. On the set one can either impose a Dirichlet or Neumann condition. Clearly, once the choice is made, on the complementary domain, say n 2 , one has to impose the condition that has not been enforced in fii (see Fig. 6.3.1 for a geometric representation of the role of the interface conditions). The formulation (6.3.9)-(6.3.12) is sequential, because it yields the solution of problem (6.3.10) in fi2 only after having solved problem (6.3.9) in Although this sequence guarantees the quickest convergence, it might not be the best option in view of parallelism, especially when many subdomains are used. W i t h this aim, an obvious modification of the previous algorithm consists of solving again problem (6.3.9) in fii, and, simultaneously, the following modified problem in n 2 :

230

ADVECTION-DIFFUSION EQUATIONS

Ch. 6

ADN, ARN AND AR^N METHODS

§6.3

231

and f Leuk+1

= f

in Q2

,k+1 _ (6.3.16)

iP(uk+1)

on <9fi n d02 =

on r o u t

flk+1

duk+1 duk+1 £ — ~ — = e- 1 dn dn

on r i n ,

with dv ib(v) :— £-T dn

(6.3.17)

FIG. 6.3.1. The role of Dirichlet and Neumann conditions at the subdomain interface in the A D N method. r Leu

= /

(6.3.13)

in 0,2

and

on d f i n 80,2

(6.3.19)

on r o u t k du k 2 +1 = e du —± dn on

A* : = 0 > ( U 2 f c ) | r i „ + ( l - 0 ' ) ^ i ) | r i t >

pk+1

: = 0 ' V « + 1 ) | r o U t + (1 -

on r u

e")^{uk)|r„ut

on r°

Again, we are enforcing a Neumann condition on the outflow part of the interface, whereas a Robin (rather than Dirichlet) condition is imposed at the inflow. The convergence of the A R N scheme will be proved in Section 6.4 as a particular case of a family of iteration-by-subdomain methods (see Theorem

on r i n u r°,

6.4.2 and R e m a r k 6.4.3).

with (6.3.14)

(6.3.18)

b • ni)

vk

=

6"uk]rout

+ (1 - 0")u£|rw

out

on F

The adaptive (3-Robin-Neumann algorithm (AR^N) can be obtained f r o m (6.3.15) and (6.3.16) by simply replacing the flux in (6.3.17) by the modified flux:

This approach generalises straightforwardly to the case in which fi is partitioned into many subdomains.

(6.3.20)

Iterative algorithms based on other interface conditions can be defined similarly. Let us state here that based on Robin-Neumann matching conditions (6.1.9). The Adaptive Robin-Neumann algorithm ( A R N ) reads: assuming that r ° has a zero surface measure, given in fi,, i = 1, 2, solve for each k > 0

In this case, the assumption on F° can be omitted, and, for instance, on T 0 we may impose the same condition as on P " . T o ensure the solvability of (6.3.15) and (6.3.16) in the AR^N scheme we have to assume that

in fii

fieuf+1 = / u

=

0

on <9fi n d f i i

(6.3.15)

on r i n du\+l dn

duk2

= £•

dn

(3 < l b • n on T i n U T°,

on

- (3v.

(3 > ^ b • n on F o u t .

As noted for the A D N algorithm, a 'parallel' version can be easily generated by s i m p l y replacing ( 6 . 3 . 1 6 ) 3 , (6.3.16) 4 and (6.3.19) respectively b y (6.3.21)

rout

dv •4>{v) : - £—

iP(uk+1)=pk

onrout

ADVECTION-DIFFUSION EQUATIONS

232

3u\+1

(6.3.22)

(6.3.23) 6.3.1

e — - —

dn

pk

9u{

e — -

=

dn

Ch. 6 ^

on r u

:=9"Muk1)\rm«+(l-e")il>(u*)\T.

on r

out

T h e d a m p e d form of the iterative algorithms: d - A D N , d - A R N and d - A R ^ N

W h e n to 1, namely, advection is strongly dominating the diffusion, the domain decomposition algorithms previously introduced can be modified as follows on the outflow part of the interface boundary. The Neumann condition expressing the continuity of normal derivatives is replaced by a homogeneous Neumann condition. The corresponding algorithms will be referred to as the damped forms of their predecessors, and indicated by the acronyms: d - A D N , d - A R N and d AR^N. The d - A D N algorithm consists therefore in solving the following problems: in Oi

Lsuk+1 =f ,fc+1 = 0 (6.3.24)

on dn n dn1

,fc+i _ \k M

+

1

dn

on r i n u r °

0

on r o u t

and r Leuk2+1 ,k+I

(6.3.25)

= f _

on dn n dn2

uk+1 = pk+1 duk+1 dn

in n 2

= o

on r o u t on r i n u r°.

A similar modification can be carried out on both of the A R N and A R ^ N methods, obtaining the following d - A R N algorithm (if ip is as in (6.3.17)) and d - A R ^ N algorithm (if ip is as in (6.3.20)):

ADN, A R N AND AR^N METHODS

§6.3

in fii

f LeU\+1 = f ,k+1 _ (6.3.26)

on dn n afii

(uf+1) = Xk

C

233

duk+l

on pout

=- 0

±

dn

on r i n

and LeU2

fc+1

,k+1

(6.3.27)

in n 2 on dn fl dn2

= 0

tp{uk+1)

for1 dn

= nk+1

on T o u t

= 0

on r m .

The damped algorithms weaken the coupling between ux and u2 at the interface F, thus allowing a faster convergence of the corresponding sequence. In the particular case when b(x) • n(x) > 0 for all x G F, we have that T = r o u t , and the d-ADN algorithm (without relaxation) becomes

(6.3.28)

LeU\ = f

in fii

mi = o

on on fl <9fii

du\ „ „ e ~ = 0 on T on and f Leu2 = f (6.3.29)

in n2

u2=0

on dn

U2 = U1

on r .

n

dn2

Clearly, in this case ui is independent of u 2 , and u 2 c a n be obtained in a single step after solving (6.3.28). For the same case, the d-ARN is obtained from (6.3.28) and (6.3.29) provided the third equation of (6.3.29) is replaced by the Robin condition: (6.3.30)

du2 dn

b • nu2 = e^-' 1 — b • n « i (= —b •nui) dn

on r .

Note that, in general, the limit function of the damped iterations differs from

234

A D V E C T I O N - D I F F U S I O N EQUATIONS

Ch. 6

the solution of the original problem, because we have modified the continuity equation for the normal derivative. However, the difference between the two functions can be estimated in suitable norms and turns out to be small for small values of £ (see Ciccoli 1995 and F. Gastaldi et al. 1996). Moreover, unlike the original undamped solution, the limiting damped solution for the ARN approach fails to be continuous at subdomain interfaces. However, the jump discontinuity is small in the advection-dominated case OJ 1 (see Ciccoli 1995 and Trotta 1996).

6.4

Coercive iterative substructuring methods: 7 - D R and 7-RR

Other iterative methods can be proposed that do not necessarily pay attention to the local direction of the advective field b on T. Instead, what is required is that the bilinear form associated with each individual boundary value problem set in fii or ft2 for the subdomain iterations is coercive in 1) or Hl(f22), respectively, under the sole assumption that | d i v b + ao > po > 0 in Cl, without requiring the boundary value to vanish on any part of dfli, i = 1,2. In this respect, it is worthwhile to note that the bilinear forms •) and a ^ - , •) introduced in (6.1.6) are coercive in V\ D 1) and V2 H H^,oul (fi 2 ), respectively, and the bilinear forms a f (•, •) and •) introduced in (6.1.12) are coercive in Vi n 1) and V2 n # r i „ ( f t 2 ) , respectively, but not in i f 1 (ft 1) or H 1 { 0 , 2 ) With respect to the adaptive iterative substructuring methods introduced in the preceding section, the drawback is that we will now use somewhat more complicated bilinear forms (see (6.4.4) below) compared with (6.1.6) or (6.1.12), and a further parameter 7 has to be considered besides the relaxation parameter 9. On the other hand, no attention has to be paid to the flow direction, and the methods can be easily extended to systems of equations. 6.4.1

T h e 7 - D R iterative method

The following scheme has been proposed by Alonso et al. (1998), and was given the name of 7-Dirichlet-Robin (7-DR) method: let A0 be given, for each k > 0 solve Lev![+1 (6.4.1)

,k+1 _ ,k+l =

= f

in Oi

0

on dfti n 80,

\k

on r

^

r

§6.4

7-DR AND 7-RR METHODS

f Leuk+1

235

in 0,2

= /

,fc+i _ (6.4.2)

M

on o n 2 n d n

+ l

^ b - n + 7 ) uk+1

dn

and set on r ,

Xk+1 := 6uk+1 + (1 - 0)A*

(6.4.3)

where 0 ^ 0 is a relaxation parameter. In (6.4.2) 7 = 7 ( x ) is a given non-negative function belonging to L ° ° ( r ) , which, in general, influences the rate of convergence of the method. If applied to the one-dimensional problem (6.2.1), the 7 - D R method corresponds to the scheme (6.2.2)-(6.2.4) with the choice A = —00 and B = 7. Let us introduce the local bilinear forms a^(wi,Vi):

= J

eVwi • Vu; + ^

d i v b + a 0 ] w%Vi

(6.4.4) + ^ f {vih -Vwi-wih 1 Jn,

- Vvi).

Note that, owing to (6.1.2), there exist constants /?• and a\, i = 1,2, such that (6.4.5)

fliK.vO

< ) 3 j IKIIi.0.. Huilli,^

. V ^ D j e f f

1

^ )

and (6.4.6)

a^vuvi)

> c^ | N l i , n ;

V^etf

1

^).

Define Vi, V°, i = 1,2, and A as in (1.2.4). The variational formulation of the 7 - D R iterative scheme (6.4.1)-(6.4.3) reads: ' find uk+1 (6.4.7)

G Vx :

aS(uf + 1 ,t; 1 ) = (/ ( t; 1 )n 1 "lir 1 = A*

Vt^eVf

Ch. 6 ^

ADVECTION-DIFFUSION EQUATIONS

236

,k+1

find

4(uk+1,v2)

eV2

:

= (f,v2)n2

Vv2eV2°

(6.4.8) a\{uk+\%2^)

+ jf

7u*+V

-a\(uk+1,K1fi)

=

(f,K2fj,)n2

+

{ f , H

i^K

A,

+J^u^v

setting finally (6.4.9)

A * + 1 : = FLU'F1 + ( 1 -

6)\K

on r ,

where 7Zi denotes any extension operator from A to Vi. Problem (6.4.8) can be rewritten in the equivalent form find (6.4.10)

,fc+i G v2

ab2(uk+1,v2)

+ J 7«2jr 1 , ; 2|r = (f,v2)n2 -a\(uk+1,11^)

+ (/,ftn>2|r)n,

+ ^ 7«fjrlu2|r

V t;2 G Va,

which is coercive in V2, for any 7 > 0. Hence, the iterative scheme (6.4.7)-(6.4.9) is correctly defined. These iterations are different from the A D N scheme, because the Dirichlet boundary condition is imposed on the whole interface T, no matter whether it is an inflow or an outflow boundary. However, in the particular situation when the flow has the same direction on the whole set of T, say b • n < 0 on T, by choosing 7 = — \ b • n we recover the A D N scheme. We also propose a modified algorithm, which is slightly more complicated to implement, but enjoys better convergence properties. It is obtained by replacing the integral on T by the scalar product in the trace space A, say ((?7,JU))A (for a similar choice, see also Dryja 1982 and P.-L. Lions 1990). Thus, we solve, instead of (6.4.10), the following problem: f find u k + 1 G V2 : (6.4.11)

ab2(uk+1,

v2) + 7((«5jr , "2|r))A = if, v2)n2 -a\(uk+1,7^2|r)+7(K£\«2|r))A

for a constant 7 > 0.

+ (/,

Tl^rh, Vv2eV2,

r

§6.4

6.4.2

7-DR AND 7-RR METHODS

237

Convergence of the 7 - D R iterative m e t h o d

For proving the convergence of the scheme (6.4.7), (6.4.11), (6.4.9) we need some preliminary results. First of all, for i = 1,2 and for each 77 e A, let us introduce as in (5.1.10) the solution E\r) G V, of the Dirichlet boundary value problem:

(6.4.12)

( a\(Eitv,Vi) = 0 { [ (E\t])\y = V-

V

<E V°

By well known a priori estimates for elliptic problems, the extension operator E\ A Vi is continuous; that is, there exists ki > 0 such that (6.4.13)

I I ^ N l k n i
Moreover, the trace inequality (1.2.5) yields (6.4.14)

VtjgA.

For each r|,/j£ A and i = 1,2 let us define now the Steklov-Poincare operators Si : A -)• A' as in (5.1.9): :=aUElV,Etij). Moreover, let us set ,R , , (6.4.15)

<^7)*?> H) == a\ (Eirj, E\p) - 7 ( ( „ ,

M))A

( S ^ r j , fi) : = 4 ( ^ 7 7 , E & ) + 7((»?, /"))A, and define s = s1 + s2 = Both operators S\y\ i = (6.4.13) we have (6-4.16)

s['l)+s™.

1,2, are continuous, because from (6.4.5) and

{^(7)77,Ai)<(^fci+7)IM|A|H|A.

Moreover, S ^ is coercive for each 7 > 0, because from (6.4.6) and (6.4.14) we have (6.4.17)

<^

7

W>[c4(C

2

T

2

+ 7 ] IMIa-

Proceeding as in Section 1.3 for the Poisson problem, the iterations (6.4.7), (6.4.11), (6.4.9) are equivalent to a preconditioned Richardson method for the Steklov-Poincare operator S , with Si, 7 ' as a preconditioner. W e are now in a position to prove the following convergence result.

Ch. 6 ^

ADVECTION-DIFFUSION EQUATIONS

238

T h e o r e m 6 . 4 . 1 There exists 7* > 0 such that for each 7 > 7* and for each A0 6 A the iterative scheme (6.4.7), (6.4.11), (6.4.9) is convergent in A, provided that the relaxation parameter 0 is chosen in a suitable interval (O,0 7 ). P r o o f Owing to (6.4.16) and (6.4.17), assumptions (a) and (6) of Theorem 4.2.2 are satisfied provided we set a2 := ab2(C;)~2

+ 7, ft : = # * ? + 7 ,

»=

U .

W e shall prove that there exists 7* > 0 such that for each 7 > 7* assumption (c) of Theorem 4.2.2 is also satisfied; namely (6.4.18)

(S^r,,

+ (Sri,rt) >

(S^y'Sr,)

V r, e A.

W e have ( S ^ i S ^ S r , )

:(7)i + (Srj,rj) = 2(Sr,tr,) + ( S ^ ^ S ^ S v ) > 2(5*7?, rf) -

(S^Srj)

-

(Sr,,r,) -

(Sr,,r,)\.

Setting p = ( S ^ ) ~ ~ l S r i , we obtain (S^r'Sv)

- (Sw)I

/

= Ia\{E\r,,E\p)

- a!>2(E^, E^rj)|

b • (ElfiVE*r, - Eb2VVEb2p)

<2\\b\\L~in2)\\Eir]\\1,U2\\EU\i,n2

Prom the definition of S and (6.4.5), (6.4.6), (6.4.13) and (6.4.14) it follows that (Sri,ri

+ 01%)

M\lMl

(Sr,,7,)>[ai.(Cf)-2+ak2(C2*)-2)\\v\\l Therefore, setting f3 := p\k\ + (3\kl and a : = a[{C^)~2

+ 4 ( C 2 * ) " 2 , we have

( S ^ V , ( S ^ r ' S r ! ) + ( S w ) > 2 a M 2 A - 2||b||ioo(Q2)fc2-f | a2 = 2 ^a - ||b||Loo(n2)fc| Condition (6.4.18) is satisfied provided that k* : = 2 ^ a - | [ b | | L 0 0 ( f i 2 ) f e 2 2 £ j i.e.

>0,

NIA-

r §6.4

7-DR AND 7-RR METHODS

239

a 2 > ||b||L~(n2)fc|^. Recalling the definition of a 2 it is therefore sufficient to take 7 > 7 * , where 7* = 0

if a 2 (C 2 *)~ 2 > ||b||L~(n2)A;22^

7* > ||b||L~{na)k%£ - 4 ( c ; r

2

if

< l|b||L~(n2)^2 ~

Since all the assumptions of Theorem 4.2.2 are satisfied, the conclusion follows. •

It is worthwhile to note that the rate of convergence of the iterative scheme (6.4.7), (6.4.11), (6.4.9) depends only on the parameters (3\, a\ in (6.4.5) and (6.4.6), ku C* in (6.4.13) and (6.4.14), i = 1,2. 6.4.3

T h e 7 - D R iterative method for systems of advection—diffusion equations

The 7 - D R iterative method can be easily extended to the case of advectiondiffusion systems of the form

(6.4.19)

. - e A u + V Di ( B ( j ) u) + A 0 u = f { fy

in ft on 3ft,

u = 0

where j = 1 , . . . ,d, and AQ are q x q symmetric matrices. In particular, a splitting procedure a la Chorin-Temam applied to the linearised non-stationary Navier-Stokes equations for incompressible flows (see Section 7.3.1) can yield a system of this type. We assume that the coefficients of B ^ and A 0 belong to £ ° ° ( f t ) , and that the coefficients of • DjB^ belong to L°°(ft). Moreover, we require that the matrix d (6.4.20)

Af(x) :=

+ A0(x) 3=1

is positive semi-definite for almost all x € ft, which corresponds to the coerciveness assumption (6.1.2). We can introduce the associated bilinear form: a b (w, v (6.4.21)

) := J

Jn" +

x

eVw • V v + (M w) • vj f d

/ 2 ,

2 JJn (l

> [(i?"-1 v) • DjW — (B{j)Djv) „•_•, i=i

• w],

ADVECTION-DIFFUSION EQUATIONS

240

Ch. 6 ^

which can be used to rewrite the Dirichlet boundary value problem (6.4.19) in the variational form ue(H10(n))d

(6.4.22)

: a\u,v)=

f f v Jn

V v G (Hq(Q))°

Let 7 be a q x q positive semi-definite matrix with constant coefficients. Similarly to (6.4.7), (6.4.11), (6.4.9), the 7 - D R scheme for problem (6.4.22) reads: find u* + 1 6 [VxY • a^uf+SvO = (f.Vita

(6.4.23)

V V l e (H^n!))«

k+1 "i|r

find u* + 1 e (V2)« c4(u*+\v 2 ) = (f,v 2 )n a

V v 2 e (Hi(Cl 2 ))«

(6.4.24) ^ ( u k 2 + \ K 2 r i + ((711ft 1 , m))a = (f,*2M)n 2 + (f,ftiM)ni V / i 6 (A)«,

-0,1 u and finally (6.4.25)

XH+1 := 9uk+} + (1 - 9)\k

on T,

with obvious meaning of notation. The convergence of this iterative scheme can be shown as in Section 6.4.2. More precisely, the 7 - D R scheme is proved to converge provided that the matrix 7 satisfies

for a suitable 7* > 0, depending on the data of the problem. Substituting in (6.4.24) the scalar product in (A) 9 with the (L 2 (r)) 9 -scalar product, one can also consider another 7 - D R scheme for problem (6.4.22), corresponding to (6.4.7), (6.4.8), (6.4.9). 6.4.4

The 7 - R R iterative method

In this section we present and analyse another iteration-by-subdomain procedure, proposed by Alonso et al. (1998), and called j-Robin/Robin (7-RR). It extends to the non-symmetric case the Robin method described for symmetric elliptic operators in Section 1.3. The scheme reads as follows: given A0 in L2(T), for each k > 0 solve

r

7-DR AND 7-RR METHODS

§6.4

in Qi

Leul+l

on Sfii n do

= 0

(6.4.26)

241

b• n- 7

dn

fc+i k u

\k

on r

and f Leuk+1

in 0,2

= f

,fc+i _ , (6.4.27)

on d02 n dO

duk+1 dn

fc+i 2

+

duk+l

b • n + 7 I uk+1

dn

on F,

where (6.4.28)

vfc+l : = e du

/1 x fc+1 - b . n - 7 ) ^ 1

k+1

on r ,

7 = 7 ( x ) being a given function in L°°(r) satisfying 7(x) > j > 0 for almost all x € T . In the one-dimensional case, this corresponds to the scheme (6.2.2)(6.2.4) with the choice A = —7 and B = 7. Noting that

(6.4.29)

Afc+1

=

duk+1

- O

= Xk +21(uk+1

*+7)

u

"+1+2^fc+1

-uk+1),

we can rewrite the scheme above in the variational form ' find u G (6.4.30)

Fi : + f = / M
then

IVipVi|r + / Afc«i|r ./r

V d i G Fi,

Ch. 6

ADVECTION-DIFFUSION EQUATIONS

242

f find uk+1 a\{uk+1,v2)

G V2 : +

J

7-DR AND 7-RR METHODS

3.4

/ fR-2Vi\r -a2(u\n2, J n2

243

+ / 7"|r^i|r Jt

U^r)

j^uk+lv2\T

(6.4.31) = f fv2+ Ja2

[ /7li«2|r Jch

+ J-ru^v2\T

and finally

Setting now, for each k > 0, ek := uk -

and

du 1, u)k : = \k - I s^f- - ^-h-nu dn 2

V«2ei/2l

+ yu

ir

the error equations can be written as (6.4.32) - u ^ 1A)0 £ L2on Xk+1 = x k + 2l(uk+l where a\ have been introduced in (6.4.4). Since {T) r ,and 7 G L°°(T), Xk +1 e for each k > 0

then

Let us emphasise that the bilinear forms

(6.4.33)

a\(e*+1,v1)

+ f

-ye^vnr

V vx G V,

= j ^ v ^

and \{ek+\v2)

+

(6.4.34)

J-le2\rv2\T

= -a\{ek+l,n1v2{T)

V v2 G V2,

+ J jek+rlv2lr

which are used in (6.4.30) and (6.4.31), are coercive in Vi, for every 7 > 0. 6.4.5

where

Convergence of the 7 - R R iterative method

We obtain the following convergence theorem, which is proved in Alonso et al. (1998), and is inspired by the results of P.-L. Lions (1990) and Nataf and Rogier (1995). Theorem 6.4.2 Assume either that Q is a Lipsehitz polygonal domain or that 3fi is regular enough, say dfi G C2. Suppose, moreover, that b|r G ( L c o ( T ) ) d . For each G £ 2 ( F ) and for each i = 1,2, the sequences uk converge in H1(Q,i) to the restriction u^ of the solution u of (6.1.1). Proof

Taking

-e^1).

ujk+l =u>k + 27{ekp

(6.4.35) Vl

= e\+l

(6.4.36)

in (6.4.33) and v2 = ek+1 a\(ek+\ek^)

= f

y

in (6.4.34), we have that

-

7efjt1)cf+1

and

The exact solution u of (6.1.1) satisfies ai(u|fii>wi)

+ / 7"|r^i|r = / fvi+ Jr J ni

(6.4.37)

a\{e2+l

VviGVi,

Choosing

= T h e k ^ in (6.4.33), we also obtain

(6.4.38)

and + / Jr

7u|r^2|r

-aU^inn^i^ir)

= / fv2+ Jn2

Jn1

+ j

^

- ek^)ek^

f%2«i|r

Jn2

~a2iu\n2,TZ2V1\T) + J 7u|r^i|r

a\(u\nv,V2)

1e2+1) =

a\(ek+\TZiek+l)

=

-

7e*+1)

ak+1

"2|r '

fHiw2|r

+ J 7"|r^2|r

and using this equality in (6.4.37) we have Vv2eV2.

From well known regularity results for elliptic equations (see, for example, J.-L. Lions and Magenes 1972 and Dauge 1988), the solution u belongs to HZI2+S{Q) 2 for a suitable 5 > 0, and consequently G L (T). Therefore we can also write

(6.4.39) Adding (6.4.36) and (6.4.39) we find that

2|T "

Ch.

ADVECTION-DIFFUSION EQUATIONS

244

= / V ^ j F - e f t «/ r /

1

) -

6^

^(efp-ejf1)2]

Recalling (6.4.35), we finally obtain (6.4.40)

+

Summing up from k = 0tok

(6.4.41)

+ ^ i - ( ^ ) = M-l,

= jf

i-(^)2.

it follows that

M

r i

iS

^ 47

hence the series

2

$oo> > ( e f ) e f ) + 4 ( e * , e * ) ]

is convergent, and, as a consequence of the coerciveness of a-(-, •) in Vi, e\ converge to 0 in H1^), i = l,2. • Remark 6.4.3 It is worthwhile to note that the 7 - R R method generalises several other methods proposed recently. For instance, we have ' =

• n|

ARN method (without relaxation)

|i/|b • n|2 4- 4ao£

Nataf and Rogier (1995)

§\/|b • n|2 + 4KE, K > 0

Auge et al. (1996).

Therefore, Theorem 6.4.2 yields, in particular, the convergence of the unrelaxed ARN method introduced in Section 6.3. Conversely, this is not the case for the AR^N method. • Remark 6.4.4 The 7 - R R iterative method can also be easily extended to advection-diffusion systems (see Alonso et al. 1998). •

6.5

The finite element realisation of the iterative algorithms

Adopting the notation of Chapter 2, the Galerkin finite element approximation of the boundary value problem (6.1.1) reads

THE FINITE ELEMENT APPROXIMATION

§6.5

find uh G Vh : a°(uh,vh) = (f,vh)

(6.5.1)

245

V vh G Vh.

By analogy with what was carried out in Section 6.1 for the differential problem, (6.5.1) is clearly equivalent to the following multi-domain formulation: find Ui^h G Vi,h, u2,h G V2,h such that (

^(UI.FT,^,/,) = ( / ,

Ui,h

(6.5.2)

VUI,H G

V°h

V v2,h

G V°h

= u2,h On r

a°2(u2sh,v2,h) = (f,V2,hha

a2(u2ih, 'R-2,hP-h) = {f,fc2,h,Hh)n 2 + ( / , -A°(UI,H,T^I,HLJ-H)

V IIH G A/J.

As usual, denotes any extension operator from A a to V^/,, i = 1,2. In practice, one often chooses the interpolant namely, the element of the finite element space V^h that assumes the value of /i/,, on T and vanishes at all the other finite element nodes in fij. Problem (6.5.2) can be regarded as the finite element counterpart of (6.1.7). It provides the basis for the iterative algorithm ADN for finite elements: given

u°l h G VI,H and u°2h G V2,h,forall k > 0 ' find 4+1

(6.5.3)

G Vhh

•ft 1 =

:

on rin u r°

+a "

v^r eAr and r find u ^ 1 G V2jh a:

on pout

(6.5.4) 0(

/c+1 T>'n,0

in,0\

/ r -r>in,0 in,0\ 0/

~al(ui]A

. / r i->in,0 in,0\

/i->in,0 in,0\

)

w

in.O

VaV

a in.O

G Ah- ,

246

Ch. 6

ADVECTION-DIFFUSION EQUATIONS

where Ar

:=-Kir--'K

e

Ah'°

'•= {%!r-ur°

T

THE FINITE ELEMENT AFmUXlMATiUiN

5.5 find

m

2, h

GV2,h

Vh}

: V v2,h G V2°h

= (f,v2ih)aa

\ vh G Vh},

247

(6.5.6)

and fcfft denotes any extension operator from A°hut to Vith, i = 1,2, vanishing at the nodes internal to T i n U T 0 . An analogous definition holds for H ^ f • In a similar manner, assuming that the surface measure of T 0 is zero, the A R N algorithm is defined as (6.5.3), (6.5.4) provided the Dirichlet conditions on T i n for w j t 1 and on Po u t for u k + ^ are replaced by the Robin conditions:

~ a i [ u k + h \ nlthfih)

+ ( f o u r t h s ~ + (1 - 9 ' ) a * ( U l h , K i > ' , n )

^ K / ^ / X " ) ]

v nhe

Ah,

setting finally (6.5.7)

= *'[(/,

+ J 7u k X} r Ph

Xk+1

•.= euk+]T

+

(l-0)Xk

on r.

The second step (6.5.6) can be substituted by

V rf G A" 1 r find u ^ 1 £ V2,h

and ^2{uk2%l,v2,h)

V v2,h G V2°h

= (f,v2,h)a2

(6.5.8) < a2{u2^h

-afK+s^vr)] +(i -

v

ft^vr)

a S K + S ^ V D

= ( / . W ) o i

VM r

e A°r

and a.0 / k + 1 T-)in,0 in,0\

in,0

/ i

in,0\ in,U\

in.O _ w ,,m,u

.»it: in,0

+ l((ukX]T, -ab1(uk+1

e A-S

respectively. The damped forms of the algorithms A D N and A R N are obtained by omitting in (6.5.3) and (6.5.4) the contribution from the complementary domain in the equation at the outflow (i.e. on F o u t for the domain fii, and on F n U for the domain U 2 ). The last equations in (6.5.3) and (6.5.4) therefore become

'^^Ph)

Ph))A = (f,ft2,hVh)n2 ,TZhhPh)

+ 7((«fSr»A»ft))A

+ (/.ftl./^/Ofii V Ph G Ah,

for a constant 7 > 0. As we have already noted, the iterative scheme (6.5.5), (6.5.8), (6.5.7) converges for 7 > 7* (see Theorem 6.4.1), and the rate of convergence only depends on the parameters a\ in (6.4.5) and (6.4.6), ki} C* in (6.4.13) and (6.4.14), i = l , 2 . In the finite dimensional context, all these constants, possibly except k,t, are independent of the total number of degrees of freedom. Therefore, the rate of convergence is independent of h, provided that we can find a uniform bound for hi. In other words, it is necessary to prove the uniform extension result (6.5.9)

\\EihVh\\i,a,
V Vh G Ah,

where for each rj G A, E\ hrj is the finite dimensional counterpart of the extension respectively, which are the weak form of the homogeneous Neumann condition.

E\r) introduced in (6.4.12). This result has been proved in Section 5.1, assuming that the family of triangulations Th of Q, is regular, and the induced family of

The finite element formulation of the 7 - D R scheme reads: given A° G Ah, for each k > 0 solve ' find uk^ (6.5.5)

G Vlih

a\(uk+h\vlth) ,.k+1

Ul,h\T

_

~

Xk

Ah

:

= (f,vlih)ai

V vlih

G V°h

triangulations M h of T is quasi-uniform. The convergence of the iterative scheme (6.5.5), (6.5.6), (6.5.7) can be proved by a similar argument. In fact, for discrete functions all the norms are equivalent; hence, there exists a constant K h > 0 such that KhWmWl <

;

r

V rih G Ah-

By using this estimate, we only have to substitute the constant a2 := a2(C2) 7 in (6.4.17) with

2

+

Ch. 6

ADVECTION-DIFFUSION EQUATIONS

248

a2,h •= 0.2(02)

2

+ 1Kh,

and convergence is ensured provided that inffj 7 > 7^ : = 7 * / K h Therefore, in this case we are not in a position to prove that the rate of convergence is independent of h. However, the numerical results obtained in Alonso et al. (1998) suggest that this is in fact the case. R e m a r k 6 . 5 . 1 Though the convergence result in Theorem 6.4.1 holds only for 7 sufficiently large, numerical evidence shows that the 7 - D R iterative scheme converges for any 7 > 0; in particular, for 7 = 0 (see Alonso et al. 1998). Moreover, an efficient choice of the relaxation parameter 6 is achieved by taking it into the neighbourhood of 1/2. • The finite element 7 - R R scheme reads: given A0 £ L2(T), f find uk+hl £ Vlih +J

(6.5.10) =

for each k > 0 solve

: 7«fSr«i,h|r

/ fvi,h+ J n,

/

JT

XkvlMT

V vx,h £

Vi,h,

then ( find u^l1

€ V V22 ,h

ab2(uk2+1,v2 ,h) + j

luk-y-]TV2,h\T

(6.5.11) -

Jfi2

fv2,h+

J fii

fKi,KV2,h\r ~ a\{u\+h ,Tli,hV2,h\v)

+ J 'yui%1\rv2,h\r

V v2,h € V2,h,

and finally on r.

(6.5.12)

The convergence result in Theorem 6.4.2 holds true for this scheme. In fact, for each fj,h £ Ah we can write / fR<2,hVh ~ab2{uh\Q2,Tl2,htJ.h) + / luh\Tfj,h Jn2 Jr = J 9hVh for a suitable gh £ Ah, and the proof of convergence follows as in the infinite dimensional case.

§249.5

THE FINITE ELEMENT APPROXIMATION

245

Though we have no information about the rate of convergence, which, in principle, can depend on h, the numerical results in Alonso et al. (1998) show that this is not the case, and the number of subdomain iterations is independent of the mesh size, for suitable choices of the parameter 7 (possibly depending on h).

R e m a r k 6 . 5 . 2 A possible strategy for choosing the parameter 7 is reported in Alonso et al. (1998). • At this point, we would like to point out a potential advantage of using the domain decomposition iterative approach. In the advection-dominated case (u> 1), the standard finite element approximation (6.5.1) of the advectiondiffusion problem might be unstable, unless suitably stabilised by an upwinding mechanism, such as those yielding the S U P G or G A L S methods (see, for example, Johnson 1987; Franca et al. 1992; and Quarteroni and Valli 1994, Chapter 8). The stabilisation procedure, which should be used in the whole domain fl when using a standard single-domain finite element approach, can be limited solely to those subdomains that contain boundary or internal layers. In all other subdomains the less expensive, better conditioned Galerkin finite element method can be used. Since the stabilisation terms do not affect the subdomain boundaries, the interface treatment in the previous iterative procedures will not change. W h a t should change is the bilinear form a°(-, •) (or af (•, •), a-(-, •)), due to the inclusion of the stabilising terms. For a more detailed discussion on this issue see Trotta (1996). There is another, less conventional, way to combine finite element and iterative domain decomposition algorithms. This time a single-domain, stabilised finite element approximation of the whole advection-diffusion problem (6.1.3) is carried out first, yielding an algebraic system TIU — f whose matrix is sparse, positive definite, with a large condition number. Typical methods for solving this kind of system require G M R E S or B i C G S T A B iterations and a suitable preconditioner. The latter can be sought in the class of algebraic preconditioners (ILU or polynomials preconditioners), which, however, are unsuitable for parallel implementations. Otherwise, we could use few steps of any of the iterative domain decomposition algorithms presented in this chapter as a preconditioner of the residual at each step of the outer G M R E S or B i C G S T A B iteration. Obviously, by this second approach we will find in the limit the original finite element solution expressed by the vector u of its grid values. The reader may find a more detailed discussion of these issues concerning parallel preconditioners for nonsymmetric accelerators, for instance, in Quarteroni and Valli (1994) and in the references quoted therein.

7 TIME-DEPENDENT PROBLEMS In this chapter we illustrate how the domain decomposition techniques that have been presented so far can be applied in the process of solving time-dependent problems. As usual we assume that fi is a bounded domain in R d , d = 2,3, with a Lipschitz boundary. The initial-boundary value problem to be considered will have the abstract form

(7.1)

— + L{u) = f

in QT := fi x (0, T)

with suitable boundary conditions for each t > 0 on the whole boundary <9fi or on a subset of it, depending upon the nature of the differential operator L. In (7.1) / and Uq are given functions, while the operator L can be either linear or non-linear. In Section 7.1 we will address the parabolic case (when L is a second-order linear elliptic operator), while hyperbolic equations (of both first-order and second-order) are discussed in Section 7.2. After reformulating (7.1) in a variational way, we will introduce its semidiscrete continuous-in-time approximation using the Galerkin method for the spatial discretisation. For the time discretisation, different kinds of approaches (based on either finite differences, or finite elements, or fractional steps) can be adopted, yielding either explicit or implicit schemes. In the latter case, at every time level we are left with a fully coupled boundary value problem that can be faced by the domain decomposition iterative techniques previously introduced. However, the convergence behaviour of these iterations will benefit from the special structure of the associated spatial operator, which depends on the time-step parameter At: indeed, in the limit as At — 0 , the resulting operator approaches the identity. An intermediate situation between the fully explicit and fully implicit is one in which the interface variables are the first to be updated at the new time level. They provide, therefore, the new boundary values that make the individual subdomain problems independent of each other. In Section 7.3 we will consider the case of non-linear operators L(u), with the purpose of illustrating how domain decomposition concepts can be conveniently adapted to the time-advancing schemes that are most commonly used for equations of this kind. In particular, we address the Navier-Stokes equations for both incompressible and compressible flow problems, and the Euler equations of gas dynamics.

Ch. 7T§7.3NON-LINEARTIME-D

TIME-DEPENDENT PROBLEMS

252

7.1

Parabolic problems

Consider the second-order linear symmetric differential operator d (7.1.1)

Lw := — ^ Di(aijDjiv) l, 3 = 1

+ a0w,

where aij = aji for each l,j = 1 , . . . , d. Let us recall that L is elliptic if there exists a constant ao > 0 such that d (7-1.2)

£

>a 0 \£\ 2 ,

1,3 = 1

for each £ £ R d and for almost every x £ fi. When L is elliptic, the operator (7.1.3)

§-t+L

is parabolic. A classical example is provided by the heat equation — Aw = / , in which case L = — A . For simplicity, we only treat the case of a homogeneous Dirichlet boundary condition on dQ. We are therefore dealing with the following initial-boundary value problem: 011 — + Lu = f

in QT : = fi x (0, T)

u = 0

on<9fix(0,T)

(7-1.4)

_ W|t=o = u 0 in fi, where / = / ( x , t) and UQ = tto(x) are given data. As in the time-independent case considered until now, a weak formulation can be provided. Let V = HQ(Q) and denote by L2(0,T; V) the space L2(0, T; V) : = j v : (0, T) ->• V \ v is measurable in (0, T) and J

|M£)||i dt < o o j .

Similarly, the space C°([0, T]; i 2 ( f i ) ) is defined as C ° ( [ 0 , T ] ; L 2 ( f i ) ) : = {u : [0,T] -> L2(Q)\v

is continuous in [ 0 , T ] } .

The weak formulation of (7.1.4) reads as follows: given / £ L2{QT) L 2 ( f i ) , find u £ L 2 ( 0 , T ; V) fl C°([0,T]; L 2 ( f i ) ) such that

and UO £

HYPERBOLIC

§7.2

(7.1-5)

f £(u(t),v)+a*(u(t),v)

<

dt

PROBLEMS

= (f(t),v)

253

VveV

{ u{0) = u0, where (•,•) denotes the scalar product in L2(ft), the bilinear form a*(-,-) has been introduced in (1.4.3), and the above equation has to be intended in the sense of distributions in ( 0 , T ) . The existence and uniqueness of the solution to (7.1.5) is proved, for example, in J.-L. Lions and Magenes (1972), under the assumption that the bilinear form a*(-, •) is continuous and weakly coercive in V; that is, there exist two constants a > 0 and A > 0 such that (7.1.6)

a*(v,v)

+ A|M|2 > alMI 2

V v G V.

Very often, this inequality is satisfied with A = 0; namely, the bilinear form a* (-, •) is coercive. For example, this is the case for the heat equation. In particular, continuity and weak coerciveness are satisfied if the coefficients aij and ao belong to L°°(fl)] in this case, (7.1.6) holds for A > ||ao||z,°°(n). Let us note, moreover, that, if we introduce the change of variable u\(x, t) : = e~~xtu(x,t), where u is the solution of (7.1.4), the new unknown u\ satisfies ^+Lux

+ Xux = e-xtf

in QT.

If (7.1.6) holds for a*(w,v), the bilinear form a*x(w,v) := a*(w,v) -I- A(w,v) associated with this last problem is coercive; that is, it satisfies (7.1.6) with A = 0. Therefore, if we replace / with e~xtf and L with L + XI, I being the identity operator, without loss of generality we can assume that the bilinear form associated with the initial-boundary value problem (7.1.4) satisfies (7.1.6) with A = 0. This will always be assumed in the rest of this chapter. 7.1.1

Multi-domain formulation and space discretisation

We adopt the same notation as in Section 1.2. Upon setting u, U\q. for i = 1,2 and all t > 0, problem (7.1.5) admits the following equivalent twodomain formulation: find m G L 2 ( 0 , T ; K ) n C°([0, T]; L2^)), i = 1,2, such that

254

TIME-DEPENDENT PROBLEMS

Ch. 7T§7.3NON-LINEARTIME-DE

(7.1.7)

- — {uiiUin)^ - al(ui,Kin) dt

V n £ A,

where a*(-, •) are defined in (1.4.7), 71i denotes any possible extension operator from A to Vi, and the equations have to be satisfied in the distributional sense on (0, T ) . Clearly, the initial condition Uj(x, 0) = u 0 |n ; ( x ) h a s also to be imposed. The proof that (7.1.7) is equivalent to the single-domain problem (7.1.5) follows from that for the steady case (see Lemma 1.2.1), and is left to the reader. We note that the interface condition (7.1.7)4 can be equivalently reformulated in the more 'physical' way (see (1.4.5)): (7.1.8)

dui

du2

driL

driL

on F,

for each t > 0

this equality being satisfied in a weak sense. Now a semi-discretisation (continuous-in-time) can be defined via a Galerkin approximation of the space V by finite element subspaces i7),.; for instance, those introduced in (2.1.3). Assuming that ft is a polygonal domain with a Lipschitz boundary, the semi-discrete approximate problem reads as follows: given that / £ L2(QT) and uo,/i £ VH being a suitable approximation of the initial datum uq, find Uh{t) £ Vh such that, for each t £ (0, T ) ,

dt (7.1.9)

(f(t),vh)

Vvh£Vh

. uh{0) = u0,hProblem (7.1.9) is a system of ordinary differential equations. Writing

3' 3

where {(fij}, j = 1 , . . . , Nh, is a basis of Vh, and u0,/1 = J2j ZojPj, problem (7.1.9) can be written as

HYPERBOLIC

§7.2

M - m

(7.1-10)

+ Am

PROBLEMS

255

m

=

£(0)=£o, where Mij := ((fi,
Aij :=

a*((fij,ipi),

Since the mass matrix M is positive definite, there exists a unique solution £(t) of (7.1.10). Proceeding as for the continuous problem, this semi-discrete problem can be formulated in the following equivalent way, upon setting Ui,h := uh\Q., i = 1,2 for each t > 0: ,h, Vi^hi = u2,h

(7.1.11) <

dt

+ a-t(ui,h, Vi,h) = { f , Vi,hh!

V vlih e V°h

on T

[u2,h,V2,h)n2 + « 2 (u2,h,V2, h) = (f,v2th)n2

V v2,h e V°ih

n2 + a2{u2,h,TZ2,hlJ-h)

dt d where denotes any possible extension operator from Aft to V^/,. The initial condition ^ ^ ( x , 0) = ito^in^x) has also to be imposed. The continuous-in-time problem (7.1.11) is at the heart of the waveformx (0,T), relaxation domain decomposition algorithms. Denoting with '•= i = 1,2, the cylinder identified by the ith subdomain flj, one could devise the following iterative algorithm: given A^ £ Aft, find for k > 0 the solution uk+1 (/,) £ Vi ,h of

dt (7.1.12)

(uk;%\vlth)Ql = Xkh

onT

liMt=o~u°'h

then

+al(uf^1,«i,h)

in

= (/,wi,fc)ni

V«i,h

€ V?ih

257

Ch. 7

TIME-DEPENDENT PROBLEMS

+a5(«2,V>v2,h)

dt -(uk2+\n2
= {f,TZ2,hPh)u2

+ ^(u^1, +

V v2,h £ V2°h

= (f,v2,h)n2

T

(7.1-16) {unh+1,vh)+Ata*(ul+1,Vh)

(f^l.hPh)^

onT.

+ (l-8)\k

Ll%un+1

:= (I +

AtLh)u n+l

u " + At f n + 1

at all nodes internal to 0 , where Lh is the finite difference discretisation of L and u n denotes the vector of grid values at the time level tn. The eigenvalues rjJE of L\\t satisfy

finally setting :=duk2+]r

V vh e Vh.

= (unh,Vh)+At(r+1,Vh)

Obviously the same problem would have been obtained if we had started from (7.1.9), and applied the backward Euler method. At each step (7.1.16) yields a system whose matrix is : = M + At A, M and A being the mass and stiffness matrices, respectively. Using finite differences rather than finite elements for the space discretisation of (7.1.15) would give the following fully discrete problem:

n2,hfih)

(7.1.17)

(7.1.14)

273

NON-LINEAR TIME-DEPENDENT PROBLEMS

§7.3

(7.1.18)

Cxha( 1 + At) < r]JE < C2hd( 1 +

Ath~2)

At each step one has to solve one parabolic problem in the cylinder QI,T with a Dirichlet boundary condition on F, and one problem in the cylinder Q 2 ,T with a Neumann boundary condition on T. Note that k is an iteration level, and not a time level. For every k, continuous-in-time problems in QXiT and Q2,T need to be fully discretised in time by either implicit or explicit methods.

(see, for example, Quarteroni and Valli 1994, Section 6.3.2). Similarly, the eigenvalues ?7j D of satisfy

A dual approach to that previously introduced in (7.1.9) is a continuous-inspace time discretisation of the original parabolic problem. Using for the sake of simplicity the first-order backward Euler scheme, denoting by At > 0 the time-step, tn := nAt the nth time level, and using the superscript n to refer to the nth time level, we obtain the following problem for each n > 0:

Therefore, the condition number fch,At of both A t h ~ 2 ) . If At is small, e.g. At = 0(h) or At smaller than the condition number of Lh-

LAtun+1

:= {I + At L) un+1

=un

+At

fn+1

in 0

(7.1.15) un+1 = 0

on df1,

uq. For each n we have a time-independent elliptic boundary value where u problem, whose weak formulation (for either a single- or multi-domain framework) can be easily devised. The new operator L&t is better conditioned than L owing to the presence of the identity operator I and the factor At. This feature enhances the rate of convergence of any substructuring iterative method that is used for solving (7.1.15) in a multi-domain framework. This desirable property is obviously reflected by the space discretisation of (7.1.15), as discussed in the following section. 7.1.2

Implicit t i m e discretisation and subdomain iterations

The actual solution of the parabolic initial-boundary value problem is accomplished by a full space-time discretisation. Still for the ease of exposition, let us consider the backward Euler method for time discretisation. We can start from (7.1.15) and approximate the space operator L& t by means of the finite element subspace Vh- This yields the fully discrete problem: given u°h : = u 0 ,h, for each n > 0 find ul+l £ Vh such that

(7.1.19)

C r ( l + A t ) < 7?f D < C 2 ( l +

Ath~2). and = 0(h2),

is of order 0 ( 1 + then «ft,At is much

This more favourable condition number has interesting consequences for domain decomposition algorithms that are used to solve (7.1.16). Precisely, as pointed out by several authors (see, for example, Chan and Mathew 1994a and the references therein), in these cases a coarse grid problem may not be required for global communication of information. Equivalently, the singular perturbation character of the operator Lh,At is used to suppress the global communication within each time-step. For the case of overlapping domain decomposition, observing that the entries in the /th row of the discrete Green function Gh,At '•= ( I + A t L h ) - 1 decay rapidly as the distance between the nodes { x s } increases; precisely |(Grft,At)is| < e when |xz - x s j > CyrAilog(e"1) (see Kuznetsov 1988, 1991; and Meurant 1991), it is possible to estimate the number of Schwarz iterations that are needed to guarantee that the convergence error is less than e, a tolerance that equals, for instance, the local truncation error (see Kuznetsov 1991). In particular, one finds that if the minimum size of overlap among subdomains is 0 ( \ / A £ I o g ( e - 1 ) ) , then just one subdomain iteration of the Schwarz method is enough to guarantee that the error is 0 ( e ) . If e = 0(h2) and At = 0(h), then the extended subdomains must have a minimum overlap of the size 0 ( v / h | log h\) (see Kuznetsov 1988). If the Schwarz method is used as preconditioner P a s (see (3.5.8)) for problem (7.1.17), then Cai (1991) has proved that ^(P^1 Lh.At) is bounded by a constant

Ch. 7T§7.3NON-LINEARTIME-DEP

TIME-DEPENDENT PROBLEMS

258

independent of h, H and At, provided that At < CH2. For larger At, the same estimate is obtained for K(P~J:LH,At) (where PCAS has been defined in (3.6.11)). Coarse grid correction is therefore needed to maintain a constant rate of convergence when At is large. Similar results hold for the multiplicative Schwarz preconditioner. Consider now the case of the decomposition of fi into M subdomains fij without overlapping. We assume that there is a black-white ordering, with black subregions holding a Neumann interface condition and white subregions holding a Dirichlet condition. Let fix and fi2 denote the union of white and black subregions, respectively, and T denote the union of all subdomain interfaces. Following Section 2.3, the algebraic system associated with (7.1.16) can be represented as follows:

(7.1.20)

iun+1 =

where A := M + AtA

/ i n 0

0 A22

iir\ A2r

/

Wn

ir2

ArrJ

\"r+1

= L^EAt, Ari = Afr,i

u2"+1

= /

/ g™+1 \ g£+1 , \gr+1

J

= 1,2, and g n + 1 := M u " + A£f™ + 1 .

The matrix A n represents the coupling between pairs of degrees of freedom associated with the set fli, ylir the coupling between pairs associated with fii and the interface F, and the other entries of A are defined similarly. Let S/j denote the Schur complement of A with respect to ^4rr- Then (see (2.3.9), (2.3.10))

with Si,h

-AnA^Air.

The matrices App are such that yipp = j4pp + 4 p p ; for the stiffness matrix A, they have been introduced in (2.3.3). In the two-dimensional finite element case (7.1.16), Dryja (1991) has proved that (7.1.21)

K f S - i S ^ C ^ l + log^)

,

where C > 0 is a constant independent of h, H and At. The right-hand side of (7.1.21) governs, therefore, the rate of convergence of the Dirichlet-Neumann subdomain iterations (see Sections 2.3 and 3.2). As pointed out in Section 2.3.1, knowledge of the preconditioner S 2j /i for E/, allows the setting up of a preconditioner QF, for the whole matrix A. Following Section 2.3.1, the construction of QH is carried out as follows. Since A = LU, where

rAKABOLlC rHUBLKMb

§7.1

(

259

h

0

0\

L:= \ 0 \inixl1

I2 Ar2A^

0 IT J

(A U :=

0

u

0 V 0

iir \ i2r Si,h + S 2 , h /

i22 0

we can define Qh := LU,

(7.1.22) with (7.1.23)

U:=

/ i n 0 V o

0

iir

A-22 0

i2r

The preconditioned stiffness matrix Q h 1 A has the same eigenvalues as the matrix S ^ S h , plus the eigenvalue 1 (see Section 2.3.1), therefore K{Q~h1A)
+ \ogI^j

.

Using the preconditioner Qh in a conjugate gradient method ensures that a solution with accuracy e can be obtained after an order of (log j ) ( l + l o g ^ ) iterations. For solution algorithms to be used at each iteration see Dryja (1991). Remark 7.1.1 Substructuring methods for space-time discontinuous finite elements are reported in Lube et al. (1998). A Neumann-Neumann iterative method is described in Dryja (1991). For other implicit methods see Israeli et al. (1993). •

Alternative approaches to fully implicit methods stem from the idea of computing first the interface values u ^ 1 on the interface T, using an explicit scheme or even an implicit scheme in a small neighbourhood of P. Then these boundary values are used at the new time level to provide Dirichlet data for independent subproblems to be solved in each subdomain. We term these methods predictorcorrector domain decomposition methods. Perhaps the most natural predictor-corrector domain decomposition method is easily illustrated on the two-domain formulation (7.1.11). For i = 1,2 denote by , s = 1 , . . . , N{, the basis functions supported in Cli, and by ipf ,1 = 1 , . . . , Nr, the basis functions associated with the nodes on T. Given defined in Cl, for i = 1,2 seek the solutions Nr

(7.1.24)

Ni

+ 1= 1

S= 1

TIME-DEPENDENT PROBLEMS

260

Ch. 7 T §7.3 NON-LINEAR TIME-DE

of 2

/,-,"+! _ vn —

\ 71'i./iMh

At

^i.hVh) / Q; »=i

together with the matching condition (7-1.26)

^ S r - « 2 . t | r = 0.

where iTithtJ-h denotes the interpolant of ^

in fij, i = 1,2.

These equations provide a reduced system whose solution yields the set of new variables u ^ 1 ( = U j t j r

=

"ltjr)

advance (7.1.11)i and (7.1.11)3 and use

on

r-

u^J 1

After this predictor step, we can as the Dirichlet datum for the two

subproblems. The splitting introduced in (7.1.24) permits us to decouple the computation of the solution within the subdomains. However, the solution generated by this scheme does not coincide with that of the original finite element problem (7.1.16). For the implementation of a similar predictor-corrector domain decomposition method in the context of finite differences, see Kuznetsov (1988). On the basis of this idea, instead of (7.1.25) Dawson and Du (1991) and Dawson et al. (1991) use a similar equation in which the test functions are defined on a larger stencil than that of fj,h- This is an explicit scheme on a coarser mesh. This enlarged support guarantees a better coupling and, consequently, a stability condition on the time-step less strict than that of the fully explicit scheme. Another predictor-corrector domain decomposition procedure, proposed by Blum et al. (1992), is concerned with overlapping partitions. Setting Tj : = dfli \ dQ, the predictor step consists now of generating preliminarily the values u ^ 1 , i = 1,2, at the new time level tn+1, by either a linear extrapolation /..n+l . _ 9 n

_

n-1

or by a quadratic extrapolation uh\Ti

•-

6uh\ri

6uh\Vi

+

uh\Vi

from previous time levels. Then a second-order Crank-Nicolson scheme is used to discretise equations (7.1.11) in each subdomain, using the previously computed values as Dirichlet data. This step is completely parallel. Finally, at a third step, a global single-valued function € V/, is constructed from the patchwise 1 through a suitable averaging process between the two solutions solutions computed in the overlapping region. This method is stable under a step-size condition of the form

HYPERBOLIC PROBLEMS

§7.2

At <

a

261

-Kgh2,

where Kp ~ K0/32 |log/3|"2 is independent of the number of subdomais, Ko ~ 2, f3h is the size of the overlapping zones and a is the coerciveness constant of the bilinear form a*(-, •).

7.2

Hyperbolic problems

Hyperbolic equations provide the mathematical model for the description of wave propagation processes in many different areas of application; notably, gas dynamics, geophysics, electromagnetism, and the dynamics of elastic structures. A mathematical feature common with hyperbolic problems is that information propagates along characteristic curves. Partitioning the spatial domain into subdomains is especially well suited for dealing with heterogeneous media, because the latter yield sudden changes of the wave speeds when crossing the interfaces of different materials. We illustrate the basic mathematical concepts behind domain decomposition, first for an advection equation, then also for problems arising from propagation of acoustic and elastic waves. For each problem we describe the associated initialboundary value problem, as well as its multi-domain formulation. Then we show how domain decomposition algorithms can be derived after time discretisation. The first initial-boundary value problem we consider is a simple scalar conservation law (see also Section 5.6) ' du — + div(bu) = / (7-2.1)

in QT : = ft x (0, T) on <9ftin x (0 , T )

\u=
in ft,

where T > 0 is an upper time level, / ,
m =

it follows that jtMt(t),t)}

= f m , t ) -

(divb)(*(*),*)u(f(*),*)•

262

Ch. 7T§7.3NON-LINEARTIME-DEP

TIME-DEPENDENT PROBLEMS

In the case in which / = 0 and b is divergence free, the solution u is constant along the characteristics and it is simply given by the wave travelling with local speed b. The unknown u can represent different physical variables, such as, for example, the concentration of a certain quantity. We now consider the case of a second-order hyperbolic equation that models acoustic waves. In a bounded inhomogeneous medium fi the acoustic dilatation is represented by the solution u(x, t) (the pressure field) to the governing equation

supplemented by the initial conditions (7.2.3)

u\ t=0 = M0,

1J7 = v0 at }t=o

in fi,

and by suitable boundary conditions on the whole boundary of fi, which, for simplicity of exposition, we will assume to be of a Dirichlet type, i.e. u = (pD

(7.2.4)

on <9fi x (0,T).

In (7.2.2) p(x) is the density, c(x) the wave velocity, and / ( x , t) the source forcing term. Finally, uo(x), vo(x) and
e(w) = (ey(w)) = (^(DlWj

+ D ^ ,

l,j =

l,...,d,

the strain field and by
§

- div
in QT

HYPERBOLIC PROBLEMS

§7.2

._ ( d i v
263

d<7tj{w)

Y ^ 3= 1

The constitutive law provides the stress components as

(7.2.7)

aij(-w) = 2fj,£ij(w) + XdivwSij, l,j =

l,...,d,

where p, > 0 and A > 0 are the Lame's constants (see Section 5.2). The differential equations (7.2.6) need to be supplemented by initial conditions, say (7.2.8)

, U|i=o = U0 and 1

9u —

dt

|i=o

in fi,

= v0

and by boundary conditions that for simplicity we will assume to be of a Dirichlet type u

(7.2.9)

=

ipD

on <9fi x (0, T ) .

One of the most common solution strategies leads to the so-called displacement formulation of the stress analysis problem (7.2.6), taking into account (7.2.5) and the constitutive equations (7.2.7). The resulting system reads

d2u dt2

(7.2.10)

-t-Lu = f

in

QT

with Lw : = —/iAw — (/t + A)V divw.

(7.2.11) 7.2.1

M u l t i - d o m a i n formulation

For the sake of exposition we still assume that fi is partitioned into two subdomains. Having defined, as in Section 5.6, that <9fi-n : = <9fi i n fl5fij, a multi-domain formulation of problem (7.2.1) reads as follows: the restriction m of u to fij satisfies '

duj

dt (7.2.12)

+ div(b Ui) = f

Ui=
infijX(0,T) on dflf

x (0, T)

in fii

for i = l , 2 , as well as the following matching condition at the interface T: (7.2.13)

(b - n)ui = ( b • n)u 2

on T x (0, T ) ,

as for the steady equation (see (5.6.4)). This condition implies that u\ = u2 at all points of T except those where the flow field b is tangential to the interface.

TIME-DEPENDENT PROBLEMS

264

Ch. 7T§7.3NON-LINEARTIME-D

Referring for notation to Section 5.6, the weak multi-domain problem corresponding to (7.2.12), (7.2.13) reads: find u; e C°([0, T]; Xi]b) such that ^{ui(t),Vi)

n; +Ci{ui{t),Vi)

=

(f(t),Vi)u;

+(vr>W.«i|ani»)6,sn';» (7.2.14)

v vi e

xlb

( U l | r i n (t) - u 2 \T^{t),p)br ia + ("2|r°» t (0

v p e Lg(rinurout), _ U t ( 0 )

=

u

0

\

Q i

=°

v t £ [O,t]

,

where equations (7.2.14) have to be understood in the sense of distributions in (0, T). The interface equation in (7.2.14) 2 is equivalent to the flux-continuity condition (7.2.13), this latter being satisfied in a weak sense. The multi-domain version of the acoustics problem (7.2.2)-(7.2.4) is formulated as follows:

I G?T5 ! )- d w Ol=' i""-x(0'T) (7.2.15)

Wj|t=0 = "0, 1 Ui -

(fi

d Ui -KT, = at |t=o

in Qj on ( a f i i n a f t ) x (0, T)

D

for i = 1,2, supplemented by the following transmission conditions at subdomain interfaces, which are inherited from the (elliptic) structure of the spatial operator -div(IV): U\ =

(7.2.16)

1

on T x (0, T)

U2

du\

Pin,, dn

1

du2

p\ci2 dn

on T x (0, T).

This time, besides the continuity of the solution (see (7.2.16)i), we have required continuity of its normal flux as well (see (7.2.16) 2 )- Note the analogy with the interface conditions (1.1.1)3, (1.1.1)4 for elliptic operators. Assuming for simplicity that
HYPERBOLIC PROBLEMS

§7.2

265

Qi = (f(t),vl)n, U1{t)

on r ,

= u2(t)

VuiGV;0

\/te[0,T]

(7.2.17) H)

+ I —Vui(t), VTli/i QI

\P

J QI

2

= ui(0) = u0|n;,

dui - ^ - ( 0 ) = V0|Q. ,

V/ieA

i=1

where Vi, and A have been defined in (1.2.4), and Hi indicates any possible extension operator from A to Vi. Equations (7.2.17)i, (7.2.17)3 have to be understood in the sense of distributions in ( 0 , T ) . fn particular, (7.2.17)3 is equivalent to the normal flux condition (7.2.16)2Finally, let us turn to the linear elasticity problem. Denoting by Uj the restriction of the horizontal displacements u to fti, i = 1,2, it can be written in the following equivalent multi-domain form: d2Uj dt2 (7.2.18)

ui|i=0

Uj =

- Lu,; = f — UJ 0, 5

du i = v0 rv , at |t=o

in ^

x (0,

T)

in ft, on (dfti n d f t ) x (0, T)

tpD

for i = 1,2, with the interface transmission conditions (see (5.2.2)) on T x (0, T)

ui=u2 (7.2.19)

d

d

(ui )rij = ^ c j i i ( u 2 ) n i j=l

onFx(0,T),

l=

l,...,d,

j=i

which enforce the continuity of displacements as well as that of normal stresses at the interface of the subdomains. In order to restate (7.2.18) in a weak form, we introduce the following definition (for i = 1,2):

(7.2.20)

e»(wi, Vi) := 2£t y 2 / , Jfti 1,3 ==1i

+ A /

divWidiv

266

TmE-JJEFEjNJJEjNT

JmUtSbEMS

Uh.

7

Note that e;(-, •) is the bilinear form associated with the operator L (see (7.2.11)) restricted to f V If we assume, for simplicity, that t f D = 0 on dQ in (7.2.9) (i.e. we consider the homogeneous Dirichlet boundary condition), then problem (7.2.18), (7.2.19) can be reformulated as follows: find U i <E (C°([0, T]; Vi) n C 1 ([0, T]; L 2 ( ^ ) ) ) d such that d? — (u i{t),Vi)Qi ui

2

(7.2.21)

£i=i

+ei{ui(t),Vi) onT,

(t) = u 2 ( t ) 'd2

d t 2 KUi(t),1liH)a.

= (f

{t),Vi)Qi

V Vi G ( V0\d i)

V i e [0, T]

+ei{ui{t),Tlifi)^

2

1=1

U I ( 0 ) = U0|!

duj dt

(0) = v 0 1 n 4 ,

where H i indicates any possible extension operator from (A) d to (Vi) d , the spaces VI, V° and A are defined in (1.2.4), and equations (7.2.21)i, (7.2.21) 3 have to be understood in the sense of distributions in ( 0 , T ) . For each i = 1,2, equation (7.2.21)i is equivalent to (7.2.18)i, while (7.2.21) 3 is the counterpart of the stress continuity equations (7.2.19)2-

Similarly to what we have done in Section 7.1 for parabolic problems, a spatial discretisation of the problems above can be realised, for example, by Galerkin or finite difference methods. The continuous-in-time, semi-discrete version of the multi-domain formulations are straightforward to deduce. 7.2.2

Implicit t i m e discretisation and subdomain iterations

W e are interested in the case of implicit time discretisation of the semi-discrete problems that we introduced at the end of the previous section, and their solution in the framework of subdomain iterations. As before, it is customary to use finite difference schemes for the temporal derivative; however, owing to the hyperbolic nature of the original problem, discontinuous finite elements with least-square stabilisation offer an interesting alternative, see, for example, Shakib and Hughes (1991), and also Lube et al (1998). As usual, let us introduce a time-step A i > 0 and the time levels tn = nAt with n — 0 , . . . , Af and t^ — T, and consider first the advection problem (7.2.12). W i t h the aim of simplifying our notation, we do not use the superscript n to identify the current time level, and simply denote by Ui the function . Advancing from tn to tn+1 (n > 0), (7.2.12) becomes

HYPERBOLIC PROBLEMS

§7.2

(7.2.22)

aoUi + div(buj) = F

in

Ui = p a

on<9fimndfii,

, (b • n ) u i = ( b • n)u 2

267

i — 1,2 1 = 1,2

on F,

where ao > 0 is proportional to the inverse of At, and F depends on / and u; at previous time levels. The coupled problem (7.2.22) can be solved iteratively as illustrated in Section 5.6.2 by alternating a boundary value problem in fli with one in fi2. Perhaps the simplest way is to construct two sequences of functions { u f } and {u2}, ^ > 0, that satisfy: ,k+1 aQu'- ' * + d i v ( b j u f + ) = F

in Qi on dflin n dfli

(7.2.23) Uk+1

on r: n

=

for « = 1,2, where r *i H : =

(2 ( l

if i = 1 if i = 2 '

and r1/1 is the portion of T on which b is pointing into fi,. For each k we have, therefore, two independent inflow-outflow boundary value problems to be solved, one in fii, the other in fl2As for the parabolic case, the smaller At, the larger ao, and therefore the convergence rate is higher. Had fl been decomposed into M > 2 subdomains, the conclusion would be the same; namely, M (rather than two) inflow-outflow independent subproblems have to be solved at each iteration step. Remark 7.2.1 An additive Schwarz method for the Galerkin approximation of the advection equation (7.2.1) is analysed in Wu et al. (1998), where convergence is proved under the time-step limitation At = 0(h). • Let us turn now to the acoustic wave problem (7.2.15), (7.2.16). The secondorder time derivative can be discretised by implicit finite differences, e.g. the two-step, second-order backward differences (Gear 1971) or the family of onestep Newmark schemes (e.g. Raviart and Thomas 1983) that include either firstor second-order methods. In all cases, after advancing from tn to tn+1, if we keep denoting by w, the updated function Ui(tn+1), then we are left with the new problem:

268

TIME-DEPENDENT PROBLEMS

aoUi — div

1 - Vitj VP /

= F

in fij,

Ch. 7T§7.3NON-LINEARTIME-D

i = 1,2

Wj = VD

on dQiHdQ,,

Ul =

on r

(7.2.24) 1

dui

Pjni ^n

1

i = 1,2

on r

P|q2 dn

where ao > 0 depends on the finite difference scheme that is used for the discretisation of the time derivative, and is proportional to ( A t ) ~ 2 . At each time level this two-domain formulation is clearly equivalent to a second-order elliptic boundary value problem in Q, which can be therefore solved by any one of the domain decomposition algorithms introduced in Sections 1.3 and 1.4. If conforming finite elements are used for the space discretisation, then the matrix associated with (7.2.24) is L ^ = M + a^A, where M and A are the mass and stiffness matrix, respectively. Its eigenvalues are bounded as in (7.1.18), with At replaced by a^ 1 . Thus, the spectral condition number of At is bounded by C ( 1 + ( A t ) 2 h ~ 2 ) , and no coarse grid is needed, as already noted for parabolic problems. An alternative point of view consists of reformulating the original initialboundary value problem (7.2.2)-(7.2.4) as a first-order hyperbolic system with constant coefficients. Before analysing the specific system obtained by (7.2.24), let us consider the following one-dimensional problem i n f i x (0,T),

(7.2.25)

where ft is an interval, U = (U\,..., Up) and A is a p x p matrix with constant coefficients. This system is said to be hyperbolic if A is diagonalisable with real eigenvalues, so that one can consider the decomposition (7.2.26)

A =

L~lKL,

where A : = d i a g ( A i , . . . , A p ) is the diagonal matrix of eigenvalues, and L is the matrix whose rows are given by the left eigenvectors of A, i.e. \rA = A r r , Introducing the characteristic

r = 1,...

,p.

variables U : = LU in (7.2.25) gives

(7.2.27) This decouples into p independent scalar advection equations

HYPERBOLIC PROBLEMS

§7.2

(7.2.28)

~~df +

Xr~d7

=

( i F ) r

'

r =

269

1'-'-'P-

The curves X(t) — x0 + A r t of the (IE, i)-plane satisfying X'(t) = A r are the rcharacteristics. Any characteristic variable Ur is constant along each corresponding r-characteristic, hence the value of Ur at any point ( x , t) can be recovered from the data of the problem tracing back the characteristic line passing through M Let us return to problem (7.2.24). Introducing the new set of unknowns U : = (with Dt

(DtU,D!U,...,Ddu)

I = 1,...

,d, Dt

:=

setting F

:=

( / , 0 , . . - , 0) and assuming for the sake of simplicity p = c = 1, we obtain d

flTJ

(7.2.29)

-

+

= F

in QT,

i=I

where entries: a ^

= 1 , . . . , d, are (d+1) x (cJ+1) symmetric matrices with the following 1

=

ai\+i

= ~~ L a rs = 0 otherwise. Initial and boundary conditions

are derived accordingly. The multi-domain version of (7.2.29) can be easily obtained after generalising (7.2.12), (7.2.13), which were derived for the scalar advection equation. Let us present this procedure for a general hyperbolic system of the form (7.2.29), where A(l\ I = 1 , . . . , d, are p x p matrices (possibly with variable coefficients) such that for each direction £ £ R d the matrix is diagonalisable with real eigenvalues (this is surely the case if each matrix A ^ is symmetric). For any point x G T define the characteristic matrix C = C ( n ) = J^i , where, as usual, n is the unit normal vector on T, directed from fii to fl->- By the hyperbolicity assumption, C can be diagonalised as A = LCL-1, where A = diag (A r ) with Ar G R , r = 1 , . . . ,p, and L is the matrix whose rows are the left eigenvectors of C. W i t h the usual notational convention, the restrictions U j of U to fi,, i = 1, 2, satisfy rjTT y +

(7.2.30)

(7.2.31)

d in fi^ x (0, T ) ,

C(n) Ui = C(n) U2

i = 1,2,

onrx(0,T).

Since we can rewrite (7.2.29) in the conservative form

^

+

in i=I

QT,

i=i

the interface condition (7.2.31) is indeed a matching condition for the normal flux.

270

TIME-DEPENDENT PROBLEMS

Ch. 7T§7.3NON-LINEARTIME-DE

When we iterate between the subdomains, this condition ought to be split into incoming and outgoing characteristics. For this reason, for i = 1, 2 we introduce the variables Ui : = L U ; (which coincide with the characteristic variables if C is a constant matrix), and distinguish between positive and negative eigenvalues. Assume, for example, that Ar > 0 for r < q and Ar < 0 if r > q for a suitable 0 < q < p (in particular, C is non-singular). Then (7.2.31) can be written equivalently as Ui

(7.2.32)

_ '

r

U2,r

= =

. ' Ui,r

T

for r >

q

for

q,

r <

where Ui 0 is the subdomain iteration counter, while we have omitted the superscript n + 1 that indicates the time level):

D,U*+1 = Gi

(7.2.33)

infli

1=1 on T ,

(7.2.34)

a 0 U 2 fc+1 + 5 3 A® D,U*+1 z=i U2T

= U£t

= G2

r >

q

in o n r>

r < 9,

where a0 := 1 / A i and G j : = F i ( t n ) + a 0 J J i = 1,2, U " being the approximation of U i ( t n ) obtained at the previous time-step, as the limit of the iteration by subdomain described above. Note that for both problems (7.2.33) and (7.2.34) we are providing the values of the incoming 'characteristic' variables on T. These conditions, together with the boundary conditions prescribed on dfl, provide the correct number of boundary conditions to solve both subdomain problems (7.2.33) and (7.2.34). The convergence of the sequence { U ^ } to Ui as k —> 00, for i = 1,2, has been proved in the one-dimensional case for a constant matrix A, by analysing the behaviour of the corresponding characteristic variables U^ = LUf (see Quarteroni 1990; see also Bj0rhus 1995). For a different way of enforcing interface conditions, based on the penalty residual in the framework of finite difference approximations, see Carpenter et al. (1997).

HYPERBOLIC PROBLEMS

§7.2

271

Turning now to the elastic wave problem addressed above, it is clear that all methods discussed for the problem of acoustic waves apply to this as well. The change of notation is obvious and the conclusions are quite similar (see, for example, Faccioli et al. 1996, 1997). For this latter case, let us also show how an explicit time-advancing scheme can be introduced and implemented in the multi-domain framework. We assume that problem (7.2.21) is discretised in space by a Galerkin finite element method. Formally speaking, this can be easily accomplished if VI, V-' and A are replaced by suitable finite element subspaces, say Vith, Vl°h, and Ah- The easiest way to generate these spaces is to start from a master space Vh, which is a classical finite element subspace of V, and then proceed as in (2.1.7) and (2.1.8). Then Uj(£), Vj and fi are respectively replaced by ui,h(t), Vi,h and fih, while H i / i should be replaced by Hi}hHh G V,;^,., a finite element extension of fih- With the aim of simplifying the notation we will drop the subscript h everywhere and therefore we will refer to (7.2.21) as the finite element problem. Concerning the discretisation of the temporal derivative, a classical instance of an explicit scheme is provided by the second-order, leapfrog method, which transforms at each time level tn the problem (7.2.21) into the multi-domain boundary value problem (7.2.35)

( 5 « ) , V i ) Q i + e i « , V i ) = (f(r),Vi)Q;

(7.2.36)

u

n+1

_

„n+l

Un

V v< e (V°)d,

on r

(7.2.37) Y ^ M ^ K i r i a , + e i ( u f , n i t i ) } = £ ( f ( t n ) , ^ / i ) n j i=1

i = 1,2

V fi G (A) d ,

i=1

having set .= < + 1 - 2 u r + u r 1 ' [At)2 An alternative approach consists of coupling (7.2.35), (7.2.36) with the new interface equation V

1'

2

(7.2.38)

i=1

2

= ^(f(r+1),^/i)ni i=i

V/iG(

A)d,

where 2 u " + 1 - 5u™ + 4 < " 1 x , n+1. 5 lSb(u™+1) : = — 1 2

<~2 5

(At)

l i c r r o + iri^ + iAn of of now denotes the second-order backward implicit discretisation

' dt2

at t =

tn+1.

Ch. 7

TIME-DEPENDENT PROBLEMS

272

Since the values u " + 1 are available at all internal finite element nodes after applying (7.2.35), the new 'implicit' equations (7.2.38) yield for each finite element node x; on T a simple algebraic equation that provides the common value <+1(xi)=ur1(xOTherefore, (7.2.35), (7.2.36) and (7.2.38) provide a 'semi-explicit' method that has roughly the same computational complexity as the fully explicit method (7.2.35)-(7.2.37). A dual approach is inspired by the predictor-corrector method illustrated in Section 7.1.2. It consists of first advancing the interface equations (7.2.37) in which u " + 1 is replaced by u™+1 (the latter vector is defined in analogy with (7.1.24)). Together with (7.2.36), this yields the updated values of u n + 1 on T, which are then used as boundary data for the implicit equations n+l

i) =

(f(tn+1),Vi)

v

Vi

e

(vnd,

T

NON-LINEAR TIME-DEPENDENT PROBLEMS

§7.3

Other approaches can exploit the specific form of the non-linear problem (7.1) in order to linearise it by semi-implicit time discretisation or fractionalstep methods. Hence, one or more linear boundary value problems are derived at every time level, and suitable domain decomposition algorithms can be applied directly to these subproblems. In what follows we briefly address three different types of non-linear equations that are commonly used to model fluid flows: the Navier-Stokes equations for viscous incompressible flows; the Navier-Stokes equations for viscous compressible flows; and the Euler equations for inviscid compressible flows. 7.3.1

Navier-Stokes equations for incompressible flows

The Navier-Stokes equations for a viscous incompressible fluid confined in a region tt of R d , d = 2,3, read (see Landau and Lifshitz 1959; Ladyzhenskaya 1969; and Temam 1984): r du

for i = 1,2. Remark 7.2.2 The domain decomposition approach for problems of wave propagation is also well-suited to dealing with material inhomogeneity. Many illustrative examples for the cases of both acoustic and elastic waves are provided in Seriani and Priolo (1994) and Faccioli et al. (1997). •

7.3

(7.3.1)

dt (7.3.2)

Non-linear time-dependent problems

LAt(un+1)

:= un+1

+ AtL(un+1)

=

gn+1

ft,

with u° := u0, and the boundary condition inherited from the original initialboundary value problem. Domain decomposition methods can be called into play for problem (7.3.1) according to two different paradygms. 1. Problem (7.3.1) can first be linearised by a Newton method (/ + AtL'(un+1'k))

(un+l'k+1

-un+1'k)

= gn+1

— i/Au + V7r = f — (u • V ) u in

u = 0

on dQ x (0, T)

QT

in fl,

where v > 0 is the viscosity coefficient, £(x, t) and u 0 (a;) are given data, while u denotes the velocity field and n the pressure. These equations can be advanced in time by a fully implicit method (for example, the first-order backward Euler scheme), then formulated in a multidomain context with interface conditions that coincide with those for the Stokes problem (see Section 5.3). Then we proceed as indicated in 1. However, most often the complexity is reduced by suitable time-advancing schemes with the aim of decoupling the convective term from the other terms, or the computation of the pressure from that of the velocity field. An example of the first kind is provided by the Crank-Nicolson/AdamsBashforth scheme, which approximates (7.3.2) by f

-LAt(un+l'k)

for k > 0, with := u n , then at each step k the associated linear boundary value problem can be solved by a Krylov iteration process using a domain decomposition preconditioner of either the Schwarz or substructuring type. 2. Problem (7.3.1) is reformulated as a multi-domain problem with the correct interface conditions, then suitable iterative substructuring methods are applied to generate a sequence of problems in the subdomains that are still non-linear and need, therefore, to be linearised at the subdomain level.

in QT := Q x ( 0 , T )

div u = 0

„ u|i=0 = U0

The time discretisation in (7.1) can be based on implicit, explicit or semi-implicit schemes. Any implicit method yields at each time level a non-linear boundary value problem resembling (7.1.15) of the following form:

273

un+l

un

At

1 1 + - V A ( u " + 1 + u " ) + ^V(7T n+1 + 7Tn) z z

= -(f"+1 + n

(7.3.3)

- ~ ( u " • V)u n + - ( u " - 1 . V ) u71 —1 in n

div u n + 1 = 0

in 0

u n+l

on d n .

=

o

In this way we find at each time level a Stokes problem that can be solved using the algorithms analysed in Section 5.3.

TIME-DEPENDENT PROBLEMS

274

Ch. 7

A different way consists of using a projection method a la Chorin-Temam (see Quarteroni and Valli 1994, Section 13.5; Prohl 1997). In this case, the momentum equations are advanced first to generate a preliminary velocity field u " - 1 / 2 that is not divergence free:

At

u n+l/2

(7.3.4)

u") -

n+l/2

=

o

in ft on

aft.

This scheme for the momentum equation corresponds to a skew-symmetric treatment of the convective term (u • V ) u . At this stage, the pressure term is either disregarded (ir" = 0) or treated explicitly (it" — 7r"), while the convective term is treated explicitly ( u " = u") or semi-implicitly (u™ = u™ + 1 / 2 ). In the former case, we obtain a second-order symmetric elliptic equation for each velocity component, which can be treated as shown in Sections 1.3 and 1.4, while in the latter we obtain a linear advectiondiffusion equation, which can be solved as discussed in Chapter 6. Then one computes the new pressure 7r n+1 and the new velocity u " + 1 as follows: 1 — (u"+1 - u n + 1 ^ 2 ) +V(wn+1 At (7.3.5)

= 0

div u n + 1 = 0 ,n+1 • n = 0

(7.3.6)

dnn+1

dn?

dn

dn

aw dt

divF(W) = divG(W)

inftx(0,T),

where ft C R d , d = 2,3. The array W contains the conserved variables, W = (p, pu, pE), p being the density, u the velocity vector, and E the total energy per unit mass (E is the sum of the internal thermodynamic energy per unit mass e and the kinetic energy per unit mass |u|2/2). The convective and diffusive terms F ( W ) and G ( W ) are the (d + 2) x d matrices defined as / F ( W ) :=

pu \ p u ® u + pl , G ( W ) :=

\ (pE+p)u J

/

0 r \ r u - q ,

Here, p is the pressure, u ® u denotes the tensor whose components are U[Uj, I is the unit tensor 5[j, q is the heat flux, and finally (r • u); := . TijUj, where r is the viscous stress tensor, the components of which are defined as rij : = p(DiUj + DjUi) + ^C - ^

divu<Sy,

in ft

with p > 0 and ( > 0 being the shear and bulk viscosity coefficients, respectively. The heat flux q is related to the absolute temperature 9 by the standard Fourier law (7.3.10)

aft,

un+1 = un+1/2 _ AiV(7rn+1 - 7

Thus we have a Poisson equation for the pressure (with a Neumann boundary condition), which can be treated as indicated in Chapter 1 (see in particular Section 1.4.1).

q = -KVO,

where K > 0 is the heat conductivity coefficient. In (7.3.8), the divergence of F ( W ) is the (d -I- 2)-vector d i v F ( W ) = (div(pu), d i v (pu ® u + pi), div[(pS + p)uj) (and similarly for d i v G ( W ) ) , and the divergence of a tensor T is the vector with components

together with the final updating for the velocity field (7.3.7)

flows

In the case of a compressible fluid, the Navier-Stokes equations, which express the conservation of mass, momentum and energy, can be written as (see, for example, Landau and Lifshitz 1959)

in ft on

Navier—Stokes e q u a t i o n s for c o m p r e s s i b l e

273

(7.3.9)

on 3ft.

= ATT? + -J- d i v u n + 1 / 2 At

NON-LINEAR TIME-DEPENDENT PROBLEMS

in ft

The latter is called the projection step, because it amounts to projecting the intermediate velocity field u n + 1 ' / 2 onto the space of divergence-free functions. This step can be reformulated as a boundary value problem for the pressure ^n+l. Ann+1

7.3.2

(7.3.8)

C A U N + 1 ' 2 + VJT?

+ (u n • V ) < + | (div un)u™ = f n + 1 / 2 u

T

§7.3

(divT); := £

DjTu. j

The entropy per unit mass s can be introduced via the second law of thermodynamics (7.3.11)

9ds = de

P

-dp.

277

TIME-DEPENDENT PROBLEMS

Ch. 7

For ideal polytropic gases, the pressure p and the internal energy e are related to the other thermodynamic quantities p and 9 through the equations of state p = Rp9,

T

§7.3

NON-LINEAR TIME-DEPENDENT PROBLEMS

The interface condition (7.3.12) 3 states the continuity of the normal fluxes, and can be rewritten as PiUi • n = p2u2 • n

e = cy9,

where R > 0 is the difference between the specific heat at constant pressure cP > 0 and the specific heat at constant volume cv > 0. From these relations it follows that P={ 7 - 1 )pe,

d PiUi,;Ui • n + pirn - Y i=i

8t (7.3.12)

+ divF(Wi) = d i v G ( W i )

ui = u 2 , Ei - E2 [F(Wi) - G ( W i ) ] • n = [F(W2) - G ( W 2 ) ] • n

in fii x (0, T) on r x (0, T) on Y x (0,T),

for i = 1,2, where F ( W ) • n denotes the (d+ 2)-vector

(piEi + p i ) u i • n -

n3>

' =

d Y Ti,ijUi,jni - kV9i • n j,i=i d

= (p2E2 +p2)u2

• n - Y T2,ijU2tjm - kV92 • n. 3,1=1

In particular, since u is continuous across F, from the first relation it follows that (7.3.14)

pi = p2

at all points on T x ( 0 , T ) where u • n ^ 0.

Finally, the flux matching property (7.3.12) 3 is a natural consequence of the fact that the variable W is a distributional solution to (7.3.8) in fi. R e m a r k 7.3.1 Within the frame of iterative substructuring methods, equations (7.3.12)2 can provide the Dirichlet conditions for u and 9 on T for one subdomain, while (7.3.13)2 and (7.3.13)3 can yield the Neumann conditions on T for the other subdomain. Concerning the interface condition (7.3.13)i (or, equivalently, (7.3.14)), it must be split into a Dirichlet condition for pi on T fl { u • n < 0 } , and a Dirichlet condition for p2 on T n { u • n > 0}. • 7.3.3

Euler e q u a t i o n s for c o m p r e s s i b l e

flows

As already noted, the Euler equations are obtained from the Navier-Stokes equations by taking the viscosity coefficients pt = ( = 0 and the heat conductivity coefficient K = 0; that is, disregarding all the diffusive terms G ( W ) . The equations read

(pu • n, pu(u • n) -I- pn, (pE + p)u • n) (and similarly for G ( W ) -n), and n is the unit normal vector on T, directed from fii to fi2. In other words, the subdomain restrictions W i and W 2 satisfy the NavierStokes equations separately in fii and fi2, together with suitable interface conditions. Clearly, equations (7.3.12) inherit the same boundary and initial conditions prescribed for W on <9fi and at t = 0, respectively. The interface conditions (7.3.12) 2 are a consequence of the following fact: the unknowns u and 9 appear in (7.3.8) through their second-order derivatives, hence they must be continuous across T. Since E = e + |u|2/2 = cy9 + |u|2/2, continuity holds also for the total energy E.

d j=1

p = kp1 exp (s/cy) for a suitable constant k > 0. Equations (7.3.8) provide an incomplete parabolic system (the continuity equation for the density p is hyperbolic). The hyperbolic Euler equations for inviscid and non-conductive flows can be obtained by dropping all the diffusive terms; that is, by taking p = ( = k = 0, or, equivalently, r = 0 and q = 0. Since, for compressible flows, explicit as well as implicit methods are used, our analysis of the multi-domain formulation will be carried out without explicitly indicating the discretisation of the time derivative. Let the domain fi be partitioned into fix and fi2, as in Section 1.1. Denoting by W i the restriction of W on fii, z = 1,2, equations (7.3.8) can be reformulated in the following split form

n,ij Tij

= P2U2>IU2 • n + p2ni - YT2'l3

(7.3.13)

with 7 > 1 being the ratio of specific heats. Moreover, from (7.3.11) we find that

dWlL

273

dW

(7.3.15)

~df

divF(W) = 0

in fi x (0, T).

Still denoting by W j the restriction of W to fii, we have the equivalent formulation dWj dt

(7.3.16) >

divF(Wi) = 0

F(Wi)-n = F(W2)-n

infiix(0,T),

i = l,2

onrx(0,T).

This time the only matching conditions on T are those prescribing the continuity °f the normal inviscid flux, which can be rewritten as

TIME-DEPENDENT PROBLEMS

278

Ch. 7T§7.3NON-LINEARTIME-D

PIVLI • n = p2u2 • n (7.3.17)

piui^ui

• n + pini = p2u2iiu2

• n +p2ni,

l—

l,...,d

{piEi + p i ) m • n = {p2E2 + p 2 ) u 2 • n. As we have noted already, the first condition is equivalent to (7.3.14). This latter condition is in agreement with the physical properties of compressible fluid flows, which admit two types of discontinuities: shock waves or contact discontinuities. In particular, on contact discontinuities the normal velocity is zero, the pressure is continuous, but density, as well as tangential velocity and temperature, may have a non-zero jump. If the interface F is not kept fixed but moves along in time, then denoting by cr(t) its velocity at time t along the normal direction n = n ( t ) , the matching condition (7.3.16) 2 has to be replaced by (7.3.18)

[ W i (t) - W 2 ( t ) ] a ( t ) = [ F ( W i ( f ) ) - F ( W 2 ( t ) ) ] • n(t)

on r ( f ) .

If r ( t ) intercepts (or coincides with) a shock front S(t), then (7.3.18) can be easily recognised as the Rankine-Hugoniot jump condition across (see, for example, Hirsch 1990, p. 136). A characteristic analysis of Euler equations enlightens the role that the interface conditions could play in the framework of a substructuring iterative method. To simplify our analysis, first we consider one-dimensional flows, in which case (7.3.15) becomes (7

.3,9)

^

+

in Si x (0, T ) ,

0 being an interval. This time, the unknowns quantities are W = ( p , p u , p E ) , while the inviscid flux can be written as a three-dimensional vector F ( W ) = (pu, pu2 + p, (pE +

p)u).

Let us recall some known facts about the characteristic form of the onedimensional Euler equations. Using (7.3.11) to express the derivatives of e in terms of s and p, the quasi-linear form of (7.3.19) in terms of the vector of primitive variables U = (p, u, s) is given by (7.3.20)

= 0

in fi x ( 0 , T ) ,

where (7.3.21)

and we have set c : =

(the speed of sound) and ps :=

r

jjy.d

iiuii-unvnftn ± ±iviEj-L>rj.r r/iNJUcjiN ± rxwj-DijiMVio

The eigenvalues of A are A I = u + c,

(7.3.22) a nd

A

2

=U

- c,

A3 = U,

the matrix L of left eigenvectors is /

(7.3.23)

c/p -cjp V 0

L :=

1 1 0

Ps/(PC) \ -ps/(pc) . 1 J

For subsonic flows in which the left end of the interval fl is the upstream boundary, i.e. 0 < u < c, the two eigenvalues A I and A 3 are positive, while A 2 is negative. If the flow is supersonic (u > c > 0), all the eigenvalues are positive. In principle, system (7.3.20) can be diagonalised by using the eigenvectors of A, leading to a fully decoupled problem. A suitable set of variables, the characteristic variables U , is introduced by means of the following differential form (see, for example, Hirsch 1990, p. 162) (7.3.24)

dU :=LdU

= (-dp \p

+ du + —ds, --dp pc p

+ du -

pc

—ds,ds]. J

The introduction of the above relations based on variations, and not upon the quantities themselves, is motivated by the fact that the coefficients of the system; that is, the elements of A, are not constant and depend on the solution itself. Since LAL_1 = A = diag ( A I , A 2 , A 3 ) , owing to (7.3.24) equations (7.3.20) become (7.3.25)

lir

+

A

l ^

=

0

infix

(°>T)>

which splits into three scalar independent equations

(7.3.26)

^ at

+

ox

in ft x (0, T ) ,

r = l,2,3.

It follows that, for each r = 1,2,3, the component Ur is constant along the characteristic curve {(xr(t),t)\x'r(t)

= A R}.

Note the formal analogy between system (7.3.26) and the characteristic system (7.2.28). Assuming that we know U , we can then proceed as for problem (7.2.28), and enforce the continuity of the incoming characteristic variables on each subdomain at every iteration. This procedure can be illustrated on the isentropic Euler equations, which read (7.3.27)

0

in Q x ( 0 , T ) ,

TlMtt-JJliPJiiNJJUiN I m u C l i B M S

280

on. y

where V = (p, u) and B :=

(7.3.28)

u c2 / p

p u

Here, the entropy s is assumed to be constant, so that the pressure p depends only on the density p; precisely, p = Kp7 for a suitable constant K > 0. The eigenvalues of B are Ai = u + c, where c =

X2 = u — c,

and the matrix L is given by r . _ ( c/p - [ - c / p

1 1

L

The characteristic variables V are defined as (7.3.29)

dV

LdV = I -dp + du, --dp P P

+ du );

hence, by direct integration one finds that VL = u + J / -dp -dp == uu++ J J P 2 = uH c + const, 7 - 1

^p^2-V2dp

T

Here, Xy is the interface point separating the subintervals Cli and and we have assumed that fii is on the left side of Therefore, C_ is incoming in fti at xr, and C+ is incoming in at x r , and we are imposing a boundary condition at xr for any characteristic variable associated with an incoming characteristic curve. Following this procedure, for a supersonic flow both interface conditions have to be attributed to the domain Furthermore, there are the initial condition and two additional boundary conditions, one assigned at the left end of the interval Oi (where the characteristic C_|_ is incoming) and the other at the right end of the interval Cl2 (where the characteristic C_ is incoming). It is easily seen that, at convergence of the iterative scheme, the continuity of R+ and R- at xr implies that (7.3.17)i and (7.3.17)2 are satisfied. Unfortunately, for the non-isentropic system (7.3.20) the functions R± are no longer constant along the characteristic curves C±, and in general the characteristic variables U are not explicitly known (see, for example, Hirsch 1990, pp. 155-6), except for the entropy s, which is constant along the characteristic curve C0 = {{x(t),t) j cc7(t) = u}. To compensate for the lack of knowledge of the characteristic variables, we consider the following iteration procedure. The Euler system is solved in fti and O2 5 with given initial data at t — 0 and boundary data on dfl (these latter conditions have to be determined by means of a suitable eigenvalue analysis). Concerning the interface conditions, we can proceed as for the isentropic Euler equations, and, assuming that each eigenvalue of A is different from zero, enforce the continuity of the quantities L U and iterate accordingly, even though L U are not characteristic variables. For the Euler system, since LTIT U = (\ u ,+ c , P s, u — c \ PC =: (Q+, Q-, Qo), g

and similarly for V2 . In conclusion we have (7.3.30)

V = {R+,R-),

R±:=u

+

7 - 1

The Riemann invariant R+ is constant on C+ = {(x(t),t) \ x'(t) = u+c}, whereas R- is constant on C _ = {(x(t),t) \ x'(t) = u — c}. In this case, assuming for instance that the flow is subsonic (0 < u < c), the iterative method would read as follows:

P*

s, s

pc

when all the eigenvalues are non-zero we enforce the continuity of the three functions Q+, and Qo, that are associated with the eigenvalues u + c, u — c and u, respectively. For instance, in the subsonic case 0 < u < c, the interface conditions in the iterative process become Qk+l{xr,t)=Qk_^{xr,t),

(7.3.31) at i r x ( 0 , T )

Rkj[i=R-,2

2

(7.3.32)

dt

+ B ^ — = dx

jyk+l _ jyk K +,2 — -"-+,!

0

in f?2 x ( 0 , T ) at x r x (0, T).

because the characteristic curve C _ , associated with the eigenvalue u — c, is incoming in fli, and Qk+21(xr,t) = Qk+d(xr,t) Qk+21(xr,t)

=

Qk0il(xr,t),

because the characteristic curves C+ and Co, associated with the eigenvalues u+c and u, are incoming in fl 2 . For each subdomain we are therefore prescribing a

boundary condition at the interface for any variable associated with an incoming characteristic line. Another possibility is to enforce the continuity of the Riemann 'invariants' R- and RQ — s. In both cases, it is straightforward to verify that the continuity of L U = ( Q + , Q - , Q o ) or that of (R+,R^,R0) at z r implies that (7.3.17) is satisfied. When considering discretisation, at the interface point xp one has to enforce three additional conditions (besides the other three related to Q+, Q- and Q 0 , or R+, R- and RQ), in order to recover all the six interface variables. This can be accomplished through the equations aui

r

, dUi + A: dx

dt

(7.3.33)

(xp,t)

dU2

(xr,t)

dx

=0

for r = 1,3

=0,

where l1

I 2 : = | — —, 1, —— | , l 3 : = (0, 0,1) \ p pcj

pcj

are the left eigenvectors of A. In each subdomain, equations (7.3.33) are obtained from the Euler system (7.3.20) by taking the scalar product with l r , r = 1,2,3, at the interface point, but only for those values of r whose corresponding eigenvalue Ar identifies a characteristic line that is outgoing at xp. These equations are sometimes called the compatibility equations. Let us return now to the three-dimensional Euler equations, which, with respect to the primitive variables U = (p, ui,u2,u3,s), can be written as dV

(7.3.34)

+

dt

A^DtU = 0

in fi x (0, T ) ,

i=i

where /

AW

:=

"1 c2/p 0 0 V o

p Ui

0 0 0

0 0 111 0 0

0 0 0

0

Ui

0

/ 4(3)

=

\ As usual, we have set c

U3 0 0 c2/p 0

/ U2 0

^

Ps/P 0 0 UI /

4(2)

0 u3 0 0 0

\/ jj^ and ps :=

0 0

: =

P 0 0

113 0 u3 0 0

c2/p 0 V 0 0

0 \ 0 Ps/P u3 /

0 u2 0 0 0

p 0 u2 0 0

0 0 0 u2 0

0

0

\

Ps/P 0 u2 /

7.3

NON-LINEAR TIME-DEPENDENT PROBLEMS

283

For any point x G T and any time t G (0,T), denote by C = C(n) the ni A O . The eigenvalues of C are given by characteristic matrix C = (7.3.35)

Ai = u • n + c,

A2 = u • n - c,

A3j4,5 = u • n.

Finally, denote by L the matrix of the left eigenvectors of C, which is given by c

/

p c

ni

n2

n3

Ei pc

ni

n2

n3

Ps

T3( 1 ) r (2)

1

p 0

^

0

r{2)

r 2( 1 ) r (2) 2 T (l) T(2) 2 2

0

__(1) _ _(2) 3 3

>

pc 1 1

)

where r' 1 ' and r ( 2 ' are two unit orthogonal vectors, spanning the plane orthogonal to n. Assuming that at time t the interface F is not characteristic at point x; namely, that no eigenvalue is zero at x, the matching conditions are naturally extrapolated from the one-dimensional case, and yield 5 (7.3.37) ?=1

5 =J2LrqU2,q 9=1

at x G T,

r = 1 , . . . , 5.

Again, we are enforcing the continuity of the 'characteristic' variables LTJ. It is easily verified that, as a consequence of (7.3.37), the interface conditions (7.3.17) are satisfied. The iteration-by-subdomain algorithm used for solving the multi-domain problem alternates the solution of the Euler equations (7.3.34) in fii and in with the Dirichlet boundary condition (7.3.37) imposed at a point x on F for all indices r corresponding to incoming characteristic lines. For instance, if we assume that at time t the interface point x is an outflow point for fti and that the flow is subsonic (namely, 0 < u • n < c, with n directed from f2i to f ^ ) , we have to impose at the (k + l)th iteration

E ^ f t and

1

=

5 5 J^LrgU^1 = Y,LrqU^q 9—1 9=1

atxer,

at x G r ,

r = 1,3,4,5.

When a numerical discretisation is being applied, other equations, i.e. the compatibility equations, have to be imposed at x. Proceeding as in (7.3.33), in

284

TIME-DEPENDENT PROBLEMS

Ch. 7

each subdomain fij, i = 1,2, they are obtained by taking the scalar product of (7.3.34) (stated for instead of U ) with the rth left eigenvector l r , r ~ 1 , . . . , 5, but only for those values of r for which the eigenvalue Ar is associated with a characteristic line that is directed outward from fi; at x.

r 282 HETEROGENEOUS DOMAIN DECOMPOSITION METHODS In this chapter we address the case of heterogeneous domain decomposition, arising whenever, in the approximation of certain physical phenomena, two different kinds of (initial-) boundary value problems hold within two disjoined subregions of the computational domain. This is a generalisation of the classical, homogeneous domain decomposition approach, in which the same kind of problem occurs in each subdomain. The abstract mathematical setting that we have in mind can be described as follows. W e denote by LI and L2 two different operators and consider the problem: find Uj : fij —> R , i = 1, 2, such that (8.1)

LiU\ = /

infi1;

L2u2

= f

in fi2,

together with suitable boundary conditions for ui on <9fii \ T and u2 on dfl2 \ F. (As usual, fii and fi2 denote two subdomains that provide a non-overlapping partition of the computational domain fi, and F is their interface.) Moreover, the unknown functions u\ and u2 should satisfy proper matching conditions on T, which we can formulate in an abstract form as (8.2)

$(uj) = $(u2)

(8.3)

¥(ui) = $(u2)

on T'

on T " ,

where the functions $ and , as well as T' C T and T " C T, will depend upon the nature of the problem at hand. In some problems we shall consider some terms in the modelling equations are dimensionally irrelevant on a certain portion (say, fii) of the computational domain fi. For such cases, if L2 denotes the operator associated with the complete model, LI will denote the operator obtained from L2 after omitting the 'irrelevant' terms. In such a way, the split problem (8.1)-(8.3) can be regarded as a reduced version of the global problem (8.4)

L2W = f

in fi.

From a mathematical viewpoint, the crucial issue is the set-up of the matching equations on T' and T". The derivation of such interface conditions ought to be carried out so that we model as closely as possible to the physics of the underlying problem. Most often, the leading criterion is that the solution to the

286

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch.

8^

coupled problem (8.1)-(8.3) is also the limit of a sequence of solutions to global variational problems set in the whole domain fi, following a singular perturbation approach. We will illustrate how these general principles can be adopted to analyse several kinds of heterogeneous models, remarkably: advection-diffusion equations, the coupling of viscous and inviscid model equations for compressible and incompressible flows, and the coupling of Maxwell equations in heterogeneous media (conductor and insulator). Some physical motivations that justify the adoption of heterogeneous decomposition models are given hereafter. A family of problems that are heterogeneous 'in nature' is supplied by the interaction between fluids and solids. For instance, this is the case for off-shore mechanics, undersea pipelines, underwater acoustic phenomena in shallow waters (see Lie 1992), or the mutual action of blood flow and compliant vessel walls (see Perktold and Rappitsch 1995; Taylor et al. 1998; and Quarteroni et al. 1998). Typically, these problems are modelled by the coupling between the NavierStokes equations (or other flow equations) and the system of (linear or non-linear) structural analysis (e.g. Navier equations, the system of elasticity). Another example is provided by the coexistence of different flow regimes. An instance arises from the simultaneous simulation of surface and subsurface hydraulics, where shallow water equations are coupled to equations describing flow in porous media. At the opposite extreme, for the simulation of the reentry of a vehicle from the upper atmosphere, the molecular and continuous flow regimes are modelled by coupling the Boltzmann kinetic equations and the Navier-Stokes (respectively, Euler) equations for viscous (respectively, inviscid) compressible flows (see Bourgat et al. 1992, 1994). The propagation of electromagnetic waves in heterogeneous media can be modelled through the Maxwell equations, whose conductivity coefficient degenerates upon a subregion of the computational domain (see Alonso and Valli 1997, and Section 8.5). This kind of approach is nowadays known as multi-physics or multi-field analysis. Another class is provided by those problems that, although 'homogeneous' in nature, can be solved in an heterogeneous fashion after reducing the given problem to a simplified one in a subregion of fi. We encounter situations of this type in fluid dynamics whenever convection-dominated viscous flows yield internal and/or boundary layers. In aerodynamic simulations, ignoring the viscous effects far from sharp layers leads to the coupling of Navier-Stokes and Euler equations or Euler and full potential equations for irrotational isentropic flows. We address these models in Section 8.3. An analogous coupling between Navier-Stokes equations for incompressible flows and their linearised form (Oseen or Stokes equations) can be suitably adopted to enforce far field boundary conditions more efficiently, see Section 8.2.2.

§8.1

HETEROGENEOUS ADVECTION-DIFFUSION EQUATIONS 284

Similarly, ignoring the thermal conductivity coefficient in a thermodynamic problem, or else the viscous diffusivity in convection—diffusion—reaction equations, leads to the coupling of hyperbolic and parabolic equations. The potential interest behind the heterogeneous approach is manifold. Using two rather than one model problem allows higher flexibility in devising the numerical method that fits better the nature of the physical phenomenon within each subregion. Most often, far from sharp layers, the expected solution is smooth and exhibits slow variations; hence, a very inexpensive numerical approximation of the reduced problem suffices to produce accurate results. Besides, the set-up of sound numerical algorithms allows the solution to the coupled problem to be obtained through a sequence of independent solves upon each subdomain (see, for a general presentation, Quarteroni et al. 1992).

8.1

Heterogeneous models for advection-diffusion equations

The first example that we consider concerns an advection-diffusion problem with dominating advection. As described in Chapter 6, this boundary value problem reads ' Leu := —eAu + div(bu) -I- a0u = f in fl on r D

U = ipu

£

du d^

on V N ,

= < P N

where fl is a bounded domain in R d , d = 2,3, with a Lipschitz boundary dfl; the subsets r D and Tjv provide a partition of dfl, and n* is the unit outward normal vector on dfl. Here f , < p D and (fiN are prescribed functions, e > 0 is a parameter, ao G L°°(fl), b G ( L ° ° ( f l ) ) d with Djh G ( L 0 0 ( f l ) ) d for every j = l,...,d, and satisfy - divb(x) -I- ao(x) > fJ-o > 0 b(x) • n*(x) > 0

for almost every x £ fl,

for almost every x G Tpj.

These conditions ensure solvability of the problem, because the latter can be associated with a continuous and coercive bilinear form. We are interested in the case in which the convective field is dimensionally dominating over the diffusion, i.e. e
288

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

FIG. 8.1.1. The computational domain, fi, and the convective field at the subdomains' interface. ' L\U\

in fii

=
on r 1 : D

u2 = ip2to

on r 2 , D

UI

(8.1.1)

div(bui) -I- aoUi = /

du2 £d^

=

on r 2 ,jv

^

. L2,eU2 •= - e A u 2 + div(bu 2 ) + aou 2 = /

in fi2,

together with suitable interface conditions on T to be determined in a correct way. Here, we have denoted by := difl F^, T^AT •'= <9fi, n tpi^ : = i = 1, 2, : = yAr|r2,jv> a n d w e suppose that b ( x ) • n*(x) < 0

for almost every x G T i ^ .

Having defined

(8.1.2)

r°

= { x G r I b ( x ) • n ( x ) = 0}

pin

= { x G r I b(x) • n(x) < 0}

pout

= { x G r | b ( x ) • n ( x ) > 0},

where n is the unit normal vector on T directed towards fi2, it turns out that Ti.d U P n is the inflow boundary for fii relative to the convection field b . An illustrative description is provided in Fig. 8.1.1, where arrows indicate the local direction of the vector b. The operator L x is formally the limit of as e —>• 0 + , while L 2iE coincides with LEJFJ2. Therefore, the conditions at the interface can be found by the following procedure. We add an elliptic perturbation —5A (5 being a small positive

§8.1

HETEROGENEOUS ADVECTION-DIFFUSION EQUATIONS

289

coefficient) to the hyperbolic operator Li and impose the associated interface conditions, which, in the current case, express the continuity of the perturbed a s the continuity of the fluxes solutions and u2,s as

across T. It has been proved (see F. Gastaldi et al. 1990) that the limit solution, as 5 —> 0 + , satisfies (8.1.1) as well as the interface relationships on r i n

ui = u2 (8.1.3)

du2 • n u i = e— an

,

b • 11112

on r .

The flux is therefore continuous across the whole interface, whereas the solution is continuous only across r i n . Equations (8.1.3) will therefore be added to (8.1.1) to complete the coupled advection-diffusion boundary value problem. The coupled formulation (8.1.1), (8.1.3) allows the independent solution of a sequence of hyperbolic problems in fii and elliptic problems in f) 2 , in the framework of iterative processes between subdomains. The different possible treatments of the interface relations is what distinguishes one iterative method from another. In this respect, a very natural approach is defined as follows. Let us summarise the boundary conditions on <902 \ T in the form B2U2 = ip2, with obvious meaning of notation and let A0 be a given initial function on T. For k > 0 we define and u 2 + 1 to be the solutions to the problems =f

in fli

< u k + 1 = \k

on r

' Lxuk+1 (8.1.4)

uk+1

L2^uk+1

=

on r i i j D

= f — b • n u 2 + 1 = —b • n u k + 1

in Jl2 on F o u t U T 0

(8.1.5) on r i n B2uk+1

=
where (8.1.6)

A fc+1 : = 6u k + l + (1 - B)\k

and 9 > 0 is an acceleration parameter.

on dn2 \ r ,

290

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

The convergence properties of this method are analysed in F. Gastaldi et al. (1990), while several numerical results can be found in Frati et al. (1993). Other research papers connected with this approach are those by F. Gastaldi and Quarteroni (1989), Scroggs (1990), Chan and Mathew (19946) and Aguilar and Lisbona (1994). 8.1.1

T h e Steklov-Poincare reformulation

Also in this heterogeneous situation, the transmission conditions (8.1.3) can be enforced through an equation on T (that can be solved autonomously from (8.1.1)), which is written in terms of the Steklov-Poincare operator and reads (8.1.7)

SA =

on r .

X

The right-hand side x depends on the data of the problem / ,
= 0

Kir] = 77|rin , Kit] = 0

infix on T i n on rx,£>.

On the other hand, the function E\ is defined in Cl2 as the solution to ' L2,£(Eh) (8.1.9)

E\T) = T] ^B2{Eb2r])=0

= 0

mU2 on T ondft2\r

(see also (6.4.12)). For the sake of brevity we shall not introduce the rigorous variational setting for the above problems, but rather we stick with their differential form. The variational formulation can be easily recovered by following the guidelines of Sections 5.6 and 6.4.

§8.1

HETEROGENEOUS ADVECTION-DIFFUSION EQUATIONS

291

The solution Kir] can be shown to satisfy the a priori estimate (8.1.10)

kUKuWIk+U

Z YROUTURI

\h- n\IKXR,)2 < i

F

L JRIN

N

|b • n\R,2

(see, for example, F. Gastaldi et al. 1990). Define, moreover, the functions Vj* '. —y R , i — 1)2, th&t axe solutions to the problems in fii

' Liu\ = f (8.1.11)

(8.1.12)

ul = 0

on T i n

u\ = ipi,D

on

VltD

L2,eU2 = /

in fi2

u\ = 0

on T

= tp2

. B2u2

on dO,2 \ r .

We are now in a position to define the function % and the operator S appearing in (8.1.7). It is clear that the solution (1/1,^2) to the coupled problem (8.1.1), (8.1.3) is given by ui = Ki\ + it*, u2 = E\\-\-UI. Therefore, the function A we are looking for on T is precisely the solution to (8.1.7) with x and S defined as *

3.1.13)

X

3.1.14)

St] : = —b • nKirj

-

b

nti!

9Un

on r

dElrt + h • nr] — e dn

on r ,

which can also be written as —b • nKif] Sv

dEkr, + h • xit] — s- 2 dn

dE\rj - £

on

p o u t |j p d

on P n .

dn

If we define the two operators Si and S2 through —b • n Kir/ (8.1.15)

onroutur°

Si?7 : = 0

on T"

292

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

b

(8.1.16)

.

I

n

o n r

o

u

rtbZn

t

Ch.8^

u r °

on r m ,

on we obviously have (8.1.17)

Sr] = Sn] + S2r].

The operator S turns out to be continuous and coercive in A, where, as usual, we have defined A : = {q G H1/2(T)

\r] = v{r for a suitable v £ V},

having set V = (ft) := { v £ Hl(Sl) | W|rD = 0}. In fact, denoting by <•, •) the duality pairing between A and its dual, we have

and

= s [ VElV • VEln + f (£AEh) Jn2 J&2 = s [ VEb2r]-VE^+ Jn2

[

E\

[&v(bE%ri) + aoE%rj\E%ii.

Moreover [ [div(bE%ri)]E%ti = - [ E \ n b - V E \ i i Jn2 Jn2 - f b n Jr Lnypout

/ h • n* E\r) E\pi. JV2,N

In conclusion, we have obtained (S2ri,n)=e (8.1.18)

+ f

f J n

-

'

f

J r in

a ^ E ^ -

, b-n?7M+ / h • n* Ev2r] E^nJr2,N

On the other hand, (8.1.19)

f

(Siri, !*) = - [

JT

out

b - n (K^)

p.

E^b-VE^

§8.1

HETEROGENEOUS ADVECTION-DIFFUSION EQUATIONS

293

Therefore, continuity of S2 and Si in A is a straightforward consequence of the a priori estimates (6.4.13) and (8.1.10). The coerciveness is proved as follows: at first - f E^h-VE^^Jn2 =

1

-

f b-V(i^)2 Jn2

z

i JQ2

(divb)(Elr,f+1-[ b-nr?2 Z ,/pinupout b • n* (E^rj)2,

- I f hence

+

(S2ri,v)=efa

nr]2 — ~ j b Z Jpin

+1 I b Z J pout

(8.1.20)

+ l f

0divb +

a o

) ( ^ )

2

n??2

b • n* (E^rj)2.

On the other hand, (Siv,v)

= - f b-n(KlV J pout

)n

= - [ b n Kin J pout = -

to-KM)

-

[ b-n J pout

b • n Kir] (rj - Km) r ° J n

f J

+ f bnr)2+ J r in

(Km)2

f div[b 1

[ b-n* Jri,N

(Km)2]

(Kir,)2,

and -

[ div[b (Km)2} Jn1

= -

[ div(b Km) Km-I Jn 1

= Ja

1

[ b • Jn 1

Q d i v b + oo)

(Km)2

(Km)2-\l^b-nrj2

- \ f b - n (Km)2-If & Jrout

z

^ri.jv

h-n*

(Km)2-

Therefore, (Sm,r])= (8.1.21)

f

(Uivb

j Q l

V

+a0)

(Km)2+ J

- \ f b -nKm(2v J r° ut

1 f

b • nV2

Jrin

- Km) + I f 1 Jr 1

b-n* N

(Km)2-

294

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

Adding (8.1.20) and (8.1.21) we finally have (Sr1,r]}=ef

\VEb2r,\2 + [

(8.1.22)

( i d i v b + <*>)

Q d i v b + ao) b n *(Ebrj)2

+ \f

(E^)2

+ + l [

b - n ( r j - Ki V)2 b n *

(Kir,)2.

Now the coerciveness of 5 and S2 follows from (8.1.22) and (8.1.20), respectively, using (6.4.14). By analogy with the homogeneous elliptic case considered in Chapter 1 (see, in particular, Section 1.3), it is not difficult to see that (8.1.4)-(8.1.6) is equivalent to the following iterative method for the Steklov-Poincare equation (8.1.7): (8.1.23)

S2(Xk+l

on T,

-Xk)=9(X-SXk)

k > 0.

This is a preconditioned Richardson method for the interface problem (8.1.7), with S2 as a preconditioner. A convergence result on the whole P for the iterates Xk defined in (8.1.23) could be obtained as in Section 5.1.2 by means of Theorem 4.2.2. But this would require the convective field b to be small in comparison with e, which is not the case we are interested in. However, proving convergence of Xk to A on the whole F is not strictly necessary for proving that the iterates uk+1 and uk+1 of (8.1.4)-(8.1.6) converge to ui and u 2 . As a matter of fact, if one succeeds in showing that A|*rin converges to a value Ain on r i n , then a further step (8.1.4), (8.1.5) (taking A in instead of Xk) will provide the desired solution m and u2, and therefore the solution A = u2\r t o (8.1.7). To this end, let us restrict (8.1.7) to a problem set solely on r m . This is easily accomplished by eliminating from (8.1.7) the restriction of A to r o u t U T 0 ; say, A°" 4 ' 0 . This yields a reduced Steklov-Poincare problem of the form (8.1.24)

SinAin =

xin

on P i n .

A characterisation of S i n in terms of S2 is as follows. Let us refer for simplicity to Si, S2 and A with a 'matrix' notation of the type t,o 5i = (S

.0

0

0

2 S2=(S21

-

M

v^23 s 2 4 ; '

*

Then

(8.1.25)

Sin = (S24 - 52352-11522) - 52352-115i2 =: T™ + T~

l i u i m L w u u i i u v u u n u v u v j i i u i i u i i i u Jiun

urt.J. JLV_1 /N D

The iterative method (8.1.4)-(8.1.6), for the variables Xln'k shown to be equivalent to the following iterations: (8.1.26)

r in( A in,fc+l

_ Ai„,fc)

=

^ i n _ ginyn.kj

Qn

zyo

: = A*rin, can be

pin^

> ^

which is nothing but a Richardson method for the reduced Steklov-Poincare problem (8.1.24) where acts as a preconditioner. At the algebraic level, the matrix associated with the operator T"1 is the Schur complement of the matrix associated with S2, taken with respect to the variables Ain (i.e. after elimination of A o u t '°). In F. Gastaldi et al. (1990) it is proved that the iteration operator I — 9(Tln)~~1Sm is a contraction in the Hilbert space

when 9 ranges in a suitable neighbourhood of 9 = 1. This property yields in particular that the iterative procedure (8.1.26) converges if we take 9 = 1. Furthermore, it ensures that (T}n)~1Sin is a coercive operator in L ^ ( r m ) and therefore its eigenvalues ( satisfy Re C > p > 0 for a suitable p. The same properties are enjoyed by the finite dimensional counterparts of the above operators. This enables one to use other iterative methods to solve (8.1.24), such as, for example, non-symmetric conjugate gradient methods or the GMRESt method. In all cases, since the range of the eigenvalues of ( T l n ) ~ 1 S m is bounded uniformly with respect to the number N of the numerical degrees of freedom, convergence is achieved at a rate independent of N (i.e. T"1 is an optimal preconditioner for S m ) . In the numerical tests the convergence rate looks indeed independent of all critical parameters involved in the equation, including £ and ||b||L=o(n). 8.1.2

The coupling for non-linear convection—diffusion equations

Let us consider the viscous Burgers equation, which reads (8.1.27)

—

+ divf(w) = divg(w)

in ft x (0, T)

where f(-) and g(-) are two (a priori non-linear) vector functions, playing the role of inviscid and viscous fluxes, respectively, whilst w = w(x,t) is the scalar unknown function. Without loss of generality we can assume that g(w) = uVw, where v > 0 is a given viscosity. In such a case (8.1.27) is a parabolic advectiondiffusion Burgers equation. Referring to the domain splitting introduced before, the heterogeneous problem associated with (8.1.27) is the one in which the right-hand side is neglected in the subregion fii of the given computational domain. Denoting by Wi the unknown solution in the subdomain fij, i = 1,2, the reduced coupled problem reads, for each 0 < t < T:

Zi J7U

i i u 1 m i V / U U i l JJW U U J_/ \_/ IVi ITl JLJ. > i^UVV/iVll V U l l l V i i

— - +divf(tui) = 0

(8.1.28)

— - + divf(iu 2 ) = dt

(8.1.29)

luunivi/u

in fix x (0, T)

in

x (0,T),

with the interface conditions vVw2 • n - {(w 2 ) • n = — f(iui) • n on T x

(8.1.30)

W\ — w2

(0,T)

on rip.

The continuity of normal fluxes on T is enforced in equation (8.1.30)i. More precisely, the normal component of the inviscid flux associated with wi must be equal to the normal component of the whole flux (inviscid plus viscous) associated with w 2 . In equation (8.1.30)2 we have made use of the following notation: (8.1.31)

i

T

:

j ( x , i ) £ T x (0,T) | ^ ( M ) . n ( x ) < 0 }

The intersection of Fif with the time level t is the inflow interface; that is, the subset of T from which the flow at time t is entering into fli. In conclusion, the interface conditions for the coupled hyperbolic-parabolic problem (8.1.28), (8.1.29) impose the continuity of the fluxes on the whole F, as well as of the solution itself, but limited to the portion of F through which the flow is entering into the 'inviscid' domain Qi. These conditions generalise to the non-linear scalar case those introduced in (8.1.3) for the linear advectiondiffusion equation. The coupled problem (8.1.28)-(8.1.30) can still be solved by a splitting method that alternates the solution of the second-order boundary value problem in Vi2 with the Neumann boundary condition induced from (8.1.30)i on T x (0,T), and a first-order problem in Hi with a Dirichlet condition obtained from (8.1.30)2 on the inflow boundary . Following the discussion in Chapter 7, if the time derivative is discretised by an explicit method, then at each time step the two boundary value problems are independent of one another. Otherwise, they are coupled through the interface conditions and an iterative procedure between subdomains can be devised by splitting the interface conditions as outlined above. Note that the inflow boundary Fj? depends in turn on the solution and is therefore unknown; it might, however, be determined on the basis of the latest information available on w\, such as u>i at the previous time level, or, even better, w± at the previous subdomain iteration in the case of implicit methods.

8.2

Heterogeneous models for incompressible flows

Incompressible flow problems in exterior domains are encountered, for example, when investigating flow past bodies or obstacles, or flow through ducts or pipes

XXXI* X XJXVWVjIIilN Xl/W U O 1V1WX-»X^XjO JT V^XV XiN X^WiVlX X\,XJOOXXJXjXK r X)U vv o zy <

with artificial boundaries representing their inlets or outlets. In this process, it is sometimes convenient to resort to a mechanism of coupling between the full Navier-Stokes system (7.3.2) and a reduced model. The latter can be obtained by applying different kinds of strategies; for example, omitting the viscous stresses from the momentum equations (7.3.2)i, or linearising the convective terms, or even assuming an irrotational flow regime in a subregion of the computational domain. We analyse some of these models in what follows, after having considered a heterogeneous model for the linear Stokes equations. 8.2.1

T h e coupling for the linearised Stokes problem

Let us refer to an idealised geometrical situation as depicted in Fig. 8.2.1.

r.

FIG. 8.2.1. The geometrical configuration. Here, fi is a bounded domain of R d , d = 2,3, external to a body whose boundary is r&, and is partitioned into a subdomain next to the body and a far field subdomain fii; the interface between fii and 0 2 is denoted by F, and we set Too : = dCl \ Finally, let n indicate the unit normal vector on T directed from fii to O2, and n* the unit outward normal vector on dfl. The problem we are considering reads: a u — z/Au + Vp = f

in 0

div u = 0

in fi

u = 0

on r 6

Bu =
on r ^

(8.2.1)

where we have denoted by B the boundary operator on Too- This problem may arise in the process of solving the full Navier-Stokes equations, when the discretisation of the time derivative is performed by means of a scheme that is explicit in the non-linear convective term. The unknowns u and p represent velocity and

298

MJli J. JiltUGfiilNJl/UUS IJUiViAHN

iiUIN IViHi X ilWJJQ

Vjil. O

pressure field, respectively. The boundary conditions on Too have to be prescribed in a suitable way for assuring well-posedness. In this respect, on a portion F1^ of F ^ an onset flow u = is given. However, assigning conditions on the outflow section may not be simple. It is also clear that all interesting flow features occur in the vicinity of the body due to the role of viscosity in this area. For these reasons, Schenk and Hebeker (1993) have proposed the replacement of (8.2.1) with the following coupled problem, where the viscosity v is set to 0 in fii: aui

+

Vpi

=f

in fii

div ui = 0

in fii

ui - u£

on

rz

=o

on

r™*

Pi

u2 =

0

div u 2 — 0

on r& in fi2

au2 — i^Au2 + Vp2 = f in fi2 . This problem is incomplete, because the matching conditions that have to be fulfilled on T are missing. In Schenk and Hebeker (1993) these conditions are recovered through a singular perturbation analysis similar to that carried out for the advection-diffusion problem in Section 8.1. The limiting coupled problem is proved to be: find PI £ H1(fii), P2 € L2(Q2) and u 2 £ H1^) such that

§8.2

HETEROGENEOUS MODELS FOR INCOMPRESSIBLE FLOWS f Api = div f d

Pi

-— = dn*

in fii

,r

in x

v

00

*

f — QU„.

on F1"

•n

Pi = 0 dpi dn

(8.2.3)

299

on r - 4

= (f - Q U 2 ) • N

on r

+ DjU2,i)nj

-p2ni = -pint

on T,

1=

1,..

j u2 = 0

on

divu2 = 0

in fi2

_ au2

-

I>AU2 + V p 2 = f

in

fi2.

This coupled problem can now be solved in the framework of an iterative process as follows. Assume that A is given, satisfying Jr A • n = 0, and for any k > 0 solve ' AP*+1 =

(8.2.4)

in fii

divf

dPki+1 dn

= (f-au£).n*

onTS

Pki+1=

0

on pout

dPki+1 dn

= (f — aA ) • n

on r ,

then - I/AU 2 + 1 + V p 2 + 1 = F

auk+1

divu*+1 (8.2.5)

in fi2

=0

in fi2

,fc+i _

on r 6 fc+1 ni =

-Pki+1nt

on T,

and finally set (8.2.6)

A

: = flu**1 + (1 - 9)A

on T,

I = 1 , . . . , d,

300

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch.

8^

where 9 > 0 is a relaxation parameter. Since d i v u 2 + 1 = 0 in ft2, the trace u ^ 1 satisfies

L r

<

• n = 0;

hence, J r A • n = 0 for each k > 0. The same holds for u2|p. The analysis of the coupled problem and the proof of convergence of the above iterative process are reported in Schenk and Hebeker (1993). They can be performed also by writing the problem in terms of the associated Steklov-Poincare operators, and then proving convergence by applying the abstract Theorem 4.2.2. A similar approach for the coupling of the compressible Stokes equations will be presented in Section 8.4. 8.2.2

The coupling for the Navier-Stokes equations in exterior domains

We consider now the case of the Navier-Stokes equations in an unbounded exterior domain Q = R d \ fit,, where fib is the region of R d , d = 2,3, occupied by a body (or an obstacle), and I1/, denotes the boundary of The problem reads: find u : Q R d and p : fl R satisfying -VAVL

(8.2.7)

+ (u • V ) u + V p = f

in fi

div u = 0

in fi

u = 0

on r&

lim u = u•oo*

|x|—>-oo

Typically, the unbounded domain 0 is replaced by a smaller bounded computational region, say fi2, with an artificial boundary T (see Fig. 8.2.2). At this stage, we should prescribe suitable far field boundary conditions on F to feed the 'interior' Navier-Stokes problem in fi2; however, this choice can be quite critical. A way to circumvent this difficulty is to describe the flow process in the far region fii beyond T by a reduced (linear) model of the Navier-Stokes equations. Clearly, the resulting coupled problem consisting of the Navier-Stokes system in fi2 and the reduced linear problem in fij has to be provided with matching conditions on the interface T. Remark 8.2.1 A slightly different situation is that in which the physical domain fi is actually bounded; however, the flow field has to be entirely computed only in a subregion of fi. This is, for instance, the case illustrated in Fig. 8.2.3, where fi may represent a long channel or a blood vessel with a bifurcation that very likely induces relevant modification of the local flow pattern.

§8.2

HETEROGENEOUS MODELS FOR INCOMPRESSIBLE FLOWS

301

FIG. 8.2.2. External flow problem.

In this case, the 'far field' fii (where the reduced problem is solved) obviates the need to prescribe boundary conditions on T, which could be difficult to devise with sufficient accuracy. On the other hand, the treatment of defective boundary conditions at the far field inflow F1^ and outflow r™ 4 is less critical for the reduced model in fii than it would be for the complete Navier-Stokes system in fi2. Although our discussion will be focused on the situation depicted in Fig. 8.2.2, adapting the results to the case of Fig. 8.2.3 is straightforward. • Suitable models for describing the flow in the far field fii are: the incompressible Euler (or linearised Euler) equations, the inviscid full potential equations, the Stokes equations, and the Oseen equations. The last three cases are treated in Feistauer and Schwab (1996, 1998). For the coupling of the Navier-Stokes equations in fi2 with the Stokes equations in fii they propose the following problem, which has to be used if the velocity field

302

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8

u is assumed to be rather small on the interface T: for i = 1, 2 find u» : ftj —• R d , Pi : ft; —>• R such that ' -VAU.2 + (u2 • V)u 2 + Vp 2 = f

in ft2

divu 2 = 0

in ft2

u2 = 0

on

u 2 = ui

on T

3.2.J _

lim Ui = iioo |x| —foo

U 2

12 ni

on T,

I — 1,...

,d

in Oi

div Ui = 0

in fti,

^ —i/Aui + Vpi = f

where Uoo is a prescribed constant vector and T;j(w, OJ) are the components of the stress tensor Tij(w,ui) := v(DtWj + DjWi) - toSij. The second interface condition derives from having written the non-linear convective term as (u2 • V)u 2 = ^ V (|u2|2) + (rotu 2 ) x u 2 . A different way of formulating it is ^{Diuij (8.2.9)

+ DjUij)rij

- pin;

3

=

+ DjU2,i)nj - ( p 2 + -|u2|2 j ni

on T,

for I = 1 , . . . ,d, which is perhaps more immediately understandable since the kinematic static pressure pi for the Stokes problem plays the same role as the dynamic pressure p2 + ||u2|2 for the Navier-Stokes problem. Note that this choice of the flux interface condition is not guided by the strategy proposed in Remark 1.1.1. An iterative process can be devised solving at each new step k +1 the exterior Stokes problem in fti with a Dirichlet condition (8.2.10)

u1 + 1 = ul;

on r ,

and the interior Navier-Stokes problem in ft2 with the interface condition

^

§8.2

HETEROGENEOUS MODELS FOR INCOMPRESSIBLE FLOWS

(8.2.11)

i

onT,

= J2Tij(uk1+1,pk1+1)nj

l=

303

l,...,d,

with a possible relaxation procedure to update the velocity u 2 + 1 on I\ The role of the interface conditions can also be reversed: in this case, the Stokes problem in fti is solved with the Neumann condition Y,Tlj(u1+\pk+1)nj = -l\nk2\2nl+YTij(
j

then the Navier-Stokes problem in

on T,

l=

l,...,d,

is solved with the Dirichlet condition

= uk+1

on T.

The proof of the convergence of the preceding iterative schemes is still an open problem. Another heterogeneous model is obtained by coupling Navier-Stokes equations in CI2 and Oseen equations (see Section 5.3) in fti. The model is suitable if the velocity field u is supposed to be close to the far field Uqo on the interface I\ The coupled problem can still be formulated as in (8.2.8) upon replacing (8.2.8)g with the Oseen momentum equation -z^Aui + (uqo • V ) u i + Vpi = f

in fti,

and the interface equation (8.2.8)5 by -^(Uoo - n ^ i , ; +

j

=--(u

YTlj(Ul>Pl)nj 2

• n)u 2 ,j + ^ T / j ( u 2 , P 2 ) " j

on r ,

l = l , . . . , d ,

j

which comes from having written the convective term in the skew-symmetric form 1 d 1 + 2 (UQO • V)ui,i = ' 3=1

for I = 1 , . . . , d (see also the treatment of the convective term in the case of linear advection-diffusion equations in (5. 1.3) and Section 6.4). Again, we note that the choice of this interface condition does not fit the guidelines presented in Remark 1.1.1. The iterative procedure for the solution at each step of Navier-Stokes equations in ft2 and Oseen equations in fti can be defined similarly to what was carried out for problem (8.2.8).

304

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

A further heterogeneous model advocated in Feistauer and Schwab (1996) can be adopted under the assumption that the flow is inviscid and irrotational in the exterior domain fii. Assuming that fii is simply connected, we can therefore postulate the existence of a velocity potential i such that V(px = ui in fii. In view of the continuity equation we obtain A fa = 0

(8.2.12)

in fix.

Correspondingly, assuming that the external force field is irrotational in fii, so that fjnj = V'^i, the pressure px is determined from
(8.2.13)

=Coo,

where the constant C ^ is given by Coo

Poo + 2 l U °°l

—

^1,00)

Poo, Uoo and Vi,oo being the constant pressure, velocity and force potential of the homogeneous flow at infinity (for |x| - > oo). A first interface condition expresses the continuity across T of the normal component of the velocity fields ui and u 2 . Besides, since the dynamic pressure plays the same role for both the Euler and Navier-Stokes equations, the following equality holds on T:

(8.2.14)

(Diu2-j

J

+

DjU2,i)r

P2 + 2 l u 2l

ni

= - \ P i + ^4>i\)ni=-(Cao+ipx)ni

onT,

1 = 1,.

where the last equality follows from the Bernoulli equation. The resulting problem reads: -fAu

2

in fl2

+ (u 2 • V ) u 2 + Vp2 = f

in f l 2

divu2 = 0 3.2.15)

on T b

< u2 = 0 +

DjU2,i)n

P2+

2 lu2|

m

j = -(Coo

+rpi)ni

on r ,

1 = 1,.

,d,

HETEROGENEOUS MODELS FORINCOMPRESSIBLEFLOWS

§8.2

A0i = 0

(8.2.16)

dn

305

in fii on r

= u2 • n

lim (i - 0i iOO ) = 0, where the velocity potential 01 > 0 0 is defined as i,oo( x ) : = x • u o o , x £ fti. Although well posed from the mathematical point of view, the physical significance of the heterogeneous problem (8.2.15), (8.2.16) is questionable. Actually, in this form, the Navier-Stokes problem (8.2.15) is independent of the exterior problem and can be solved first. Then the associated velocity u 2 is used to provide the Neumann data for the exterior potential problem (8.2.16). The Navier-Stokes problem in the near field domain is influenced by the external problem solely through the potential ipi and the constant Coo, which accounts for the far field quantities p ^ , u ^ and Vi,oo (the homogeneous flow at infinity). The coupling of the Navier-Stokes and the Euler equations for incompressible flows has also been considered by C. Xu (1996), using the spectral approximation method.

8.3

Heterogeneous models for compressible flows

Compressible flows are modelled through the Navier-Stokes equations, that express the conservation of mass, momentum, and energy of a fluid. They provide a complete description of all flow phenomena, but need empirical laws to relate the viscous coefficients and the heat conductivity coefficient to the thermodynamic flow variables. Different types of formulations can be used for these equations, depending upon the set of variables that are chosen to represent the flow field. Here, we consider the conservative form already presented in Section 7.3.2 (and use the same notation introduced therein): (8.3.1)

dW —

+divF(W) = divG(W)

inftx(0,T).

Omitting the diffusive term G ( W ) , we obtain the Euler equations (8.3.2)

5W —-+divF(W)=0 dt

inftx(0,T),

which describe the motion of inviscid and non-conductive compressible fluid flows. The Euler system (8.3.2) shares with the Navier-Stokes system the same number of equations d + 2. However, the viscous stress and the heat-conduction terms are no longer considered; neither are those equations that might supplement (8.3.1) when the Navier-Stokes equations are Reynolds averaged through suitable turbulence models (e.g. Lesieur 1997).

306

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

NS

FIG. 8.3.1. The three-subdomain decomposition around an airfoil. When the flow field is potential and isentropic, the Euler equations can be further simplified by resorting to the so-called full potential equation, which is a single, scalar equation. In this case, the conservation of mass reads (8.3.3)

^

+ div(pV<^) = 0

in ft x (0, T ) ,

where

The coupling between the Navier-Stokes and Euler equations

This approach is motivated by the fact that experimental evidence has shown that at high Reynolds numbers (that is, when the viscosity coefficients are small, or, equivalently, the convection is dominant), apart from a thin region near the

§8.2

HETEROGENEOUS MODELS FORINCOMPRESSIBLEFLOWS

307

b o d y surface, the velocity of a fluid flow around a streamlined b o d y is of the same order of magnitude as the free-stream velocity and the inertial forces are dominant over the viscous stresses. On the other hand, near the b o d y surface the viscous stresses cannot be ignored even for very small viscosities, due to the presence of boundary layers. Consequently, the flow-field can be split into two regions: the near-body domain fi2, that includes the boundary layer, and the outer region fii (see Fig. 8.2.1). In fii we can omit the viscous stresses from the governing equations. The resulting problem is a heterogeneous model that entails the coupling between the Navier-Stokes equations in fi2 and the Euler equations in fii; that is, (8.3.4)

(8.3.5)

+ divF(Wi) = 0

in fii x ( 0 , T )

aw? - ^ + divF(W2) = d i v G ( W 2 )

infi2x(0,T).

Finding the matching at interface T is a more difficult mathematical problem. Let us start by considering the following one-dimensional parabolic model problem: find W satisfying dW _

(8.3.6)

<9F(W) +

=

<9G(W)

. ^ . m\ in fi x ( 0 , T ) ,

where W = (W\, W2, W3), fi = (A, /3) is a one-dimensional interval that is supposed to be partitioned into the subdomains fii = (a, 7 ) , fi2 = (7,/?) for a suitable 7 . We assume that F ( W ) is a hyperbolic flux, namely, its Jacobian ^ ( W ) : = <9F/<9W is a matrix with real eigenvalues, and that the right-hand side is a second-order elliptic term. Therefore, F(-) and G ( - ) play the roles of convective and diffusive fluxes, respectively. The coupled hyperbolic-parabolic problem associated with equation (8.3.6) is, for all 0 < t < T, 9W1

+

« F ^

at

1

)

= 0

f o r Q < I < 7

ox

Denoting by L and A, respectively, the matrices of the left eigenvectors and associated eigenvalues of ^4(W) (A is therefore a diagonal matrix), the interface matching conditions to be satisfied at the point x = 7 are 3

(8.3.9)

Y , V ^ V r=1

3

=

LirW2,r

r=1

at a; = 7 ,

t £ (0,T)

308

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

for all indices q associated with the negative eigenvalues (see Section 7.3.3), and (8.3.10)

G(W2) - F(W2) = - F ( W i )

at x = 7 ,

t e (0, T).

Condition (8.3.10) expresses the continuity of fluxes at x = 7 : the inviscid flux associated with W i is equal to the complete (inviscid plus viscous) flux associated with W 2 . This is precisely the one-dimensional vector analogue of the multi-dimensional scalar condition expressed by equation (8.1.30)i, which was established for the multi-dimensional scalar equation. On the other hand, (8.3.9) states the continuity of the 'characteristic' variables ( I W ) , for all indices q that are associated with a characteristic line entering into the hyperbolic subdomain Oi. Again, this property can be regarded as the one-dimensional vector analogue of the condition expressed by equation (8.1.30) 2 . This analysis was carried out in F. Gastaldi and Quarteroni (1989) for the case of linear systems with constant coefficients. We can now turn back to the non-linear, multi-dimensional case in which the Euler equations in fli are coupled with the Navier-Stokes equations in see (8.3.4), (8.3.5). As an immediate generalisation of the case considered in the previous subsections, following (8.1.30)i and (8.3.10) we can guess that the normal fluxes are continuous across interface P at all time t, i.e. (8.3.11)

[G(W2) - F ( W 2 ) ]

n = -F(Wi)

n

onrx(0,T).

An a priori less obvious task is how to generalise the matching conditions given by (8.1.30) 2 and (8.3.9) for the problem at hand. One possibility is to proceed as in Section 7.3.3, first rewriting equation (8.3.4) in non-conservative form, like (7.3.34). Recalling that the hyperbolic flux F ( W ) is a (d + 2) x d matrix whose entries will be denoted by Fri(W), we can write (8.3.4) as q-«*T (8.3.12)

d +

=0

in Oi x ( 0 , T ) ,

/=1 where the (d + 2) x (d + 2) matrices A^ are defined as A\.'] : = dFrl/dWs. For each time level t and each point x 6 P, denoting by C = C(n) the niA^l\ characteristic matrix C = and by L the matrix whose rows are the left eigenvectors of C, the matching conditions at time t are 5

(8.3.13)

^ V ^ l . r r=1

5

=

at X £ r, r=1

for all indices q associated with negative eigenvalues of C . The iteration-by-subdomain algorithm used for solving (8.3.7), (8.3.8) alternates the solution of the second-order problem (8.3.8) with the Neumann boundary condition on T x ( 0 , T ) given by (8.3.11), and the solution of the first-order

§8.2

HETEROGENEOUS MODELS FORINCOMPRESSIBLEFLOWS

309

X FIG. 8.3.2. Converging-diverging nozzle. problem (8.3.12) with the Dirichlet boundary condition (8.3.13), which, at each time level t and at each point x of T, has to be imposed for all indices q that correspond to a negative eigenvalue of C. Numerical evidence about the effectiveness of this coupling procedure is given in Quarteroni and Stolcis (1995). 8.3.2

T h e coupling between the Euler equations and the full potential equation

As previously pointed out, the Euler equations provide an acceptable mathematical description of compressible flows whenever the viscous stresses and the heat conducting terms can be ignored. This is the case for flows around an obstacle (e.g. an airfoil) far from the obstacle (and from the boundary layer induced in its vicinity), or flows in ducts or pipes (e.g. the converging-diverging nozzle of Fig. 8.3.2). If there are regions of irrotational and isentropic flow, the full potential equation (8.3.3) can be used instead of the Euler equations, yielding two scalar equations rather than the d + 1 equations of the isentropic Euler system. Before facing the issue of the heterogeneous coupling between the Euler and full potential equations, it is worthwhile to address the full potential equation alone in order to discuss the interface conditions to be used in the event of adopting a homogeneous coupling involving solely the full potential. Assume, therefore, that the full potential equation is considered in the whole domain ft; we aim to consider it separately in fti and ft2 and couple the corresponding solutions through suitable matching conditions on interface F between fti and ft2. Since the flow is potential and isentropic, the conservation of momentum now reads (8.3.14)

dVw

where, as usual, c

+

c2 V)u+-Vp = 0

in ft x ( 0 , T ) ,

y V ( / ° ) is the sound speed. On the other hand,

310

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

(u • V ) u = ^V|u|2 + r o t u x u = ^V|u|2 and c2 f c2 — V/9 = V / — dp. P J P Since the gas is ideal and polytropic, namely, p = RpO and e = cyO, where R = cp — cy is the difference between the specific heats, for an isentropic flow we also have p = Kp1, where K > 0 is a suitable constant and 7 is the ratio of specific heats. Hence f —dp =

= - K — T T P 7 - 1 + const

f Kipi-2dp

7 7 7 V = — - — - + const = — — R 0 + const 7 — 1 p 7 — 1

= cpd + const. Introducing the enthalpy per unit mass, h = e + h = cve

+ R9 =

one has

cp6.

Thus, the equation of conservation of momentum finally reads (8.3.15)

+

+h

in fl x ( 0 , T ) ,

= H0

where HQ is the (constant) stagnation total enthalpy. In the case of isentropic flows, the second law of thermodynamics (7.3.11), expressed in terms of the enthalpy h, reads dh

c2 P

dp = 0,

because the left-hand side equals dds. Hence, differentiating in t equation (8.3.15) and using (8.3.3) we find d2tp

dh

dVip

_ d2ip =

W d2(p

„

c2 dp +

~pdt c2

dS/ip +

,

~df

' ,

* dVtp

By a direct computation, differentiating equation (8.3.14) with respect to x one also has

HETEROGENEOUS MODELS FORINCOMPRESSIBLEFLOWS

§8.2

c2 P

c2

div(pV) = —c2Aip

V

= -c2A(p -

—

311

c2A(p + \7(p •

dV
1 + -V|Vvf

dt

We have thus found the single equation for the potential tp: (8.3.16) ^

dVip - c2Aip + V(p • [ ( V p • V ) V
0

=

i n f i x (0, T).

Upon introducing the vector unknown Z : = (Dpp,Dnp,..

.,Dd
we also obtain the following quasi-linear vector form of the full potential equation dz

(8.3.17)

at

Y^A^DIZ i=i

in fi x (0, T),

= 0

where, for d — 3, the Jacobian matrices A^

AW

=

'2u\ -1 0 0

u

- c

2

UIU2

/2 u2

W1U3 \

0

0

0

0

0

0

0

0

0

A(2)

/

/2 u3 U1U3 ,4(3) =

are given by

0

0 0 0

0

=

- 1

V

0

U2U3 u2

0 0 0

-

0

0

0

0

0

0

0

V - 1

0

0

0

If d = 2, they reduce to A ^ and A ^ upon omitting the last row and column. The hyperbolic form (8.3.17) suggests the coupling mechanism at the interface. As a matter of fact, following the guidelines of Section 7.2.2 we deduce that the whole vector C ( n ) Z is continuous across the interface, where niA^ C(n) := is the characteristic matrix. To analyse what consequences can be drawn about the continuity of the unknown variables Z, let us investigate the spectral properties of C ( n ) . If d = 3, its eigenvalues are Ai = u • n + c,

A2 = u • n - c,

A3 = A4 = 0.

The first two of them coincide with those of the Euler system (see (7.3.35)), whereas the third and fourth are zero. The reason is that, in contrast with the Euler case, the two characteristic curves along which vorticity and entropy

312

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

propagate are missing (remember that vorticity is zero and entropy is constant for irrotational and isentropic flows). In the one-dimensional case, the full potential equation (8.3.16) becomes (8.3.18)

=

where u —

is the flow velocity. The eigenvalues of the characteristic matrix 2u

u2 - c 2

-1

0

are Ai = u + c and A2 = u — c, and the matrix of left eigenvectors is given by

x

c

c

The characteristic variables are defined through the relations dZ = LdZ, i.e. 1 /. u\ . „ dZi = — d Z i + ( l - - ) dZ2, C n C'

(8.3.19)

1 dZ2 = -dZx C

+ ( l + - ) dZ-2, ^ C'

where Z\ = ^ and Z2 = = u. For isentropic flows the sound speed can be written as c2 =p'(p) (8.3.20)

= 7— = "fR@ 9

= (7 - l)cP6

( Z2 = (7 - l)/i = (7 - 1) ( Ho - Z, -f

having used (8.3.15). Then we find that 2 -cdc 7 - 1


— Z2 dZ2 =

2 7C dc — u du, 7 - 1

and therefore u / w\ 2 - d c -I—du - M l ) du = dc + du 7 — 1 c \ c/ 7 — 1 2 - d c u du + (1—1— / u\ 2 dc + du. = ) du = 7 — 1 c V c) 7 — 1

dZi =

2

Integrating these relations we obtain (8.3.21)

Zi = U + - A r c , 7 - 1

= u

~C, 7 - 1

which coincides with the first two Riemann invariants of the one-dimensional isentropic Euler system (see (7.3.30)).

HETEROGENEOUS MODELS FORINCOMPRESSIBLEFLOWS

§8.2

313

To simplify the notation, let us denote for velocity u and sound speed c the functional dependence on the potential ip as follows: u = u(tp) :=

c = c(ip) :=

dip dx (7-1)

H0-

d
dip

~dt

dx

Repeating the analysis of Section 7.3.3, we obtain the following two-domain formulation for the one-dimensional full potential equation: dVi

, ,

\2

,

+2u{ipi)^£L

d2
(8.3.22)

^ d V i

=

dtdx

-(C(ip2)2-U(ip2)2)

in fti x (0, T)

0

d 2 (p 2 dx2 in n2 x (0, T)

u(
-c(i) = U {V2) H 7 — 1

7c{
at x r x ( 0 , T ) at xT x ( 0 , T ) ,

where a'r is the interface point separating the subintervals fii and 0. 2 . As usual, this system has to be supplemented by the initial condition and the additional boundary conditions for each characteristic variable whose associated characteristic line is incoming. An iterative procedure like (7.3.31), (7.3.32) can be defined in order to prescribing the values of the incoming Riemann invariants for the interval at hand. Assuming that Oi is on the left side of Q 2 , that u is positive at xp and that the flow is subsonic, at the fcth step we prescribe -

7 — i

= ^ 2 ) -

7 —I

pk2)

< V k 2 + 1 ) + — ~ r c ( v 3 2 + 1 ) = " ( V i ) + —-_rc(
at

x (0,T)

at x r x ( 0 , T ) ,

whereas for supersonic flows we assign both values to the downstream domain that is, = u(ip$) 7 — - 11 " " •• 2 , £>_L 1 V - 1.V k + l + 1
-^—ciifil) 77 — I 2 -^-c(rf) 7—1

at xr x (0, T ) at x r x ( 0 , T )

314

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

Observe that the continuity of Z\ and Z2 at the interface ensures that of u and c, hence that of (the potential flux) and ^ (through (8.3.20)), or else, equivalently, that of ^ and
MM

fc+1

dt2 2.-fc+1 + 2 u{ip1fc+i 5 M = 0 dtdx

(8.3.23)

in fii x (0, T)

(fc+i dx

=

BVk+1

at z r x (0, T)

U0

dVh+1

(8.3.24) „*+i + J L . c « + i 7 — 1

=

„(vf)

+

-i-c(rf) 7 — 1

at xT x ( 0 , T ) ,

together with initial conditions at t = 0 and boundary conditions at a and f3. The interface condition in fii can be substituted by 3io k + 1

=H0-

1 -(uk2)2

I - —Y(ck)2

at xr x (0,T),

or else by the Dirichlet condition for
§8.2

HETEROGENEOUS MODELS FORINCOMPRESSIBLEFLOWS

315

When the flow is supersonic, we have to distinguish between the inflow and the outflow case. Let us first assume that u > c > 0. The iterative procedure is defined as follows: k+1 d2
k+l\2

dt2

2, Js+1

av?

dx2 dtdx

dVk+1

„dVk+1

dt (8.3.26)

n

in ft2 x (0, T)

— = 0 dx

2

u k2+1 + uk2+1 -

= «(¥>{) + ^ c ( ^ ) = u{ipk)

7 - J-

in fti x (0, T)

0

=

7

^yc(^f)

at z r x ( 0 , T ) at xp X (0, T).

Instead, when u < — c < 0 we consider the iterative scheme

dx2 2,„k+l

dt2 +2u(fk+1) (8.3.27)

k+1 d
dt

(8.3.28)

dtdx

—0

in fti x (0, T) at z r x (0, T )

l Ho-^(uk2)2-^rj(ck)2

=

dVk+1 d t ~

+

B

dVk+1 d ^ = °

at xr x (0, T)

in ft2 x (0, T).

The above procedures can also be extended to the case of a flow that is not isentropic in ft2. As we have already noted in Section 7.3.3, in this case the Riemann 'invariants' u ± are no longer constant along the corresponding characteristic curves. However, we can decide to impose their continuity at the interface. Furthermore, the continuity of the Riemann invariant s, the entropy, that is constant along the characteristic curve Co = {(x(t),t) \ x'(t) = u } , has to be imposed when the curve Co is incoming in ft2. Let us denote by Si the constant entropy in fti, and consider first the subsonic case 0 < u < c. The first step of the iterative scheme remains the same as the isentropic case (see (8.3.23)), while the second step reads (for notation see (7.3.20), (7.3.21)):

316

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS f d\Jk+1 2 dt

(8.3.29)

,k+1

+

k+1 „ dU 2 + Adx

in n 2 x ( 0 , T ) u(Vk)

7-1

Ch. 8 ^

+

7-1

c(^)

ata;rx(0,T) at zp x ( 0 , T ) ,

together with initial conditions at t = 0 and boundary conditions at (3. If the direction of the flow is reversed (—c < u < 0), the eigenvalue u is negative and consequently the characteristic line associated with the entropy s is no longer incoming in fi2; the Dirichlet condition for s 2 + 1 at the interface is therefore omitted in (8.3.29). On the other hand, u is not an eigenvalue for the full potential equation, and only one interface condition for the potential
„ dUk+1 n + A—^— = 0 dx

2

dt (8.3.30)

uk+1

+ ^—ck+1 —I

7-1 ck+1

_

ck+1

in

= u{Vk) = u(cpk)

x (0, T)

+ -A-cOrf) 7 —I

at xr x (0, T )

-c(pf) 7 - 1

at xr x (0, T) at xr x (0, T).

Si

Instead, when u < — c < 0 no characteristic curve is incoming in 0 2 , hence all the three interface conditions in (8.3.30) must be omitted. Since u is not an eigenvalue of the full potential equation, we are led to consider the iterative scheme given by (8.3.27) and (8.3.31)

dTJk+1 at

Ad\Jk+1

dx

=

0

in fi2 x (0, T).

The well-posedness of all boundary value problems we have considered for the full potential equation can be analysed by the classical energy method or the Kreiss' normal mode analysis as in Oliger and Sundstrom (1978). In the two- or three-dimensional case the eigenvalues of the characteristic matrix CE of the isentropic Euler equations and those of the characteristic matrix CFP of the full potential equation are different, and so are the corresponding 'characteristic' variables defined by means of the matrix of left eigenvectors of C. Indeed, we have already seen that the eigenvalues for the Euler equations are u - n + c, u n — c and u - n (with a multiplicity equal to d— 1), whereas for the full potential equation the characteristic matrix Cpp is given by (for d = 3)

COUPLING FUK COMPRESSIBLE STUKES EQUATIUJNS

§8.4

Cpp =

' 2u • n -n i -n2 -n3

317

w i u - n —c 2 ni u2 u • n — c n2 U3U • n — c n3 ' 0 0 0 0 0 0 0 0 0

with eigenvalues u - n + c, u • n — c and 0 (with a multiplicity equal to d — 1). Besides, a straightforward computation shows that the matrix of left eigenvectors for the Euler equations is ( Le

c/p -c/p

•=

0 V

0

n1 ni rP J2)

n2 n2 (i) T.2 T,

(2)

"3 \ n3 (i)

T,3

.(2) 3

)

(see (7.3.36)), while that associated with the full potential equation is given by

- 1 / c

1/c

Lpp :—

0

\

0

-

UiU • n — c n 1 c(u • n + c)

itiu • n — c 2 ni c(u • n — c)

TP J2)

u 2 u • n — c n2

U3 u • n c 2 n 3 \

c(u • n + c) u2u • n c2n2

c(u • n + c) u 3 u • n c2n3

c(u • n — c)

c(u • n — c)

r 2( 1 )

,(2)

/

Hence, the extension of the above algorithms to these multi-dimensional cases is not straightforward and needs some further investigation.

8.4

The coupling for the compressible Stokes equations

The next example we shall consider is the so-called compressible Stokes problem, a stage of the linearisation process of the solution to the Navier-Stokes equations for compressible viscous flows through a fractional step approach (see Bristeau et al. 1987). We have presented this problem in Section 5.4, as well as its inviscid counterpart related to the Euler equations in Section 5.5. We are now interested in a problem in which the viscous terms do not give significant contributions in a subregion, say fii, of the whole domain fl (fii could be, for instance, the far field region in a flow problem). Therefore, we propose to omit them from the equations, and consider an inviscid problem in fii, to be coupled with the viscous problem in fi2. The viscous-inviscid coupled problem reads as follows: find u t , a,, i = 1,2, such that

318

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

(8.4.1)

aui + fiVai = f

in

aai + d i v u i = g

in Oi

ui • n* = ip1>D

on r i ) D

Ol = ipi,N

on F l t N

v'^2(Diu2tj

+ DjU2ti)n*

j

-0cr2n* u

+ (7 -

= (¥>2,JV);, l =

Ch. 8 ^

divu2n*

2v/d)

on r2lAT

l,...,d,

on r 2 , D

2 =
a a 2 + div u 2 = g

in £l2

a u 2 — i / A u2 — 7 * V d i v u 2 + p \ r a 2 — f

in

together with suitable interface conditions on T. Here, a > 0, v > 0, 7 > 0, 7* : = 7 + (d — 2)v/d and /3 > 0 are constants, f, (p2ia a n d i,d and
uf = u 2 5

Djui,inj

-

P
j

+

DJU2J)RIJ

i

+ (7 - 2vjd) d i v u 2 n ; - f3a2ni

on T,

l =

l,...,d,

we find for the limit solution (see Quarteroni et al. 1991) Ui • n = u 2 • n (8.4.2)

- P v i n i

=

on T +

v

Dj

u

2,i)

n

j

j

+ (7 - 2v/d) d i v u 2 n ; - 0cr2ni

on T,

l=l,...,d.

Hence, the flux is continuous across the interface, while only the normal c o m p o nent of the velocity is continuous across P. We can formulate the interface condition in terms of the Steklov-Poincare operator. If we summarise the operators acting in (8.4.1) in the form I a ( u i , a i )

§8.4

COUPLING FUK COMPRESSIBLE STUKES EQUATIUJNS

319

and Z/ 2 (u 2 , o"2), and take for the sake of simplicity PAT = 0,
•n

(8.4.4)

=

infix on r

ip

U f ^ - n* = 0

on a f i i \ r

L 2 (F 2 rj) = ( 0 , 0 )

in fi2

u

on r

=

n

on dfi 2 \ T. These operators have already been introduced in (5.5.8) and (5.4.8), respectively. Introduce, moreover, the functions ( U ^ , / ^ ) , solutions of the problems [ (8.4.5)

M

U

^

P

^

M

infix

Ug.n = 0

on r

(0) U i,* n* = 0

on 9 f i i \ T

and the functions ( U 2 ^ , P 2 ^ ) , solutions to [L2{V(£,P$) (8.4.6)

= {i,g)

in fi2

U ^ = 0

on r

u

on dft 2 \ T,

0

already defined in (5.5.9) and (5.4.9), respectively. The Steklov-Poincare equation is now given by SX = X

(8.4.7)

on r ,

where

(Si/), := (8.4.8)

(17 •

-PP™

N ) n l + /3P2("VI

j - ( 7 - 2u/d) divU { 2 ] r)ni

on V

320

(8.4.9)

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

j + (7 — 2v/d) d i v U ^ ] ni

Ch. 8 ^

on P,

for I = 1 , . . . , d. The solution A to (8.4.7) represents the value of u 2 on T and at the same time A • n is the value of ui • n on T. In other words, we get the solution to the coupled problem (8.4.1) by setting (8.4.10)

(8.4.11)

K , ^ ) = ( u ( ° ) ( A - n ) , J P 1 ( 0 ) ( A - n ) ) + (u(°l,P 1 ( ° ) )

( u

2

, a

2

)

=

(") p O h

+

Let us now focus on the iteration-by-subdomain procedure. Let A0 be a given initial vector on T. For k > 0 we define ( u j + 1 , cr* +1 ) and (u£ + 1 , o-£ +1 ) as the solution of f (8.4.12)

( u i + 1 > a i + 1 ) = (f, g)

in fii on r

u* + 1 • n = A* n ,k+1 n

on

L2(uk2+1,ak+1)

=

(f,g)

\T

in ft2

•YiDiu^+D^n, j (8.4.13)

+ (7 - 2v/d) d i v u k + 1 n i -/3ak+1ni

=

Pak+1nt

on F,

I = 1 , . . . , d,

on dfl2 \ r,

uk2+1=0 where (8.4.14)

ik+l

:= (1 - 9)\k +

du^1

and 9 > 0 is again an acceleration parameter. 8.4.1

Variational formulation and finite element approximation

Referring for notation to Sections 5.4 and 5.5, the variational formulation of the Steklov-Poincare operator S is given by (8.4.15)

(SR,, Ai) : = 4

0 )

(FI (V • n), FX (JI • n)) +

(F2T,,

F2H),

§8.4

COUPLING FUK COMPRESSIBLE STUKES EQUATIUJNS

321

where (•, •) denotes the duality pairing between the trace space (A) d and its dual and, as in (5.4.10), we define -421/)[(W2,P2),

(V2,?2)]

/

J n.

/3ap2q2 + f3 div w 2 g2 + a w 2 • V2 — (3p2 d i v v 2

(8.4.16)

v d + - Y

(Diwi,j

+ D3w2,I) (DIV2J + DjV2tl

1,3 = 1

+ ( 7 — — J div w 2 div v 2 and (o)

(8.4.17)

[(wi,/9i),(vi,gi)] := / {(3apiqi + / 3 d i v w i qx Jn i + a w i • v i - (3pi d i v v i ) .

We can split the Steklov-Poincare operator as S = Si + S 2 , where (8.4.18)

(Si V ,fi) = 4 0 ) ( F 1 ( I 7 - n ) , F i ( / i - n ) ) ,

which corresponds to (5i»j); : = —/3P[°\r) • n) ni, I = 1 , . . . , d, and (8.4.19)

{S 2 V,H) = 4' / ) (F2V,F2H),

which corresponds to (S2ri)i := pp^rim

- v ^ D ^ f , ) ;

+ Dj(U^ij),)

nj

3

— (7 —

2v/d)

divU^tyn/,

l =

l,...,d.

Both of the operators Si and S2 is symmetric. Their continuity in (A) d is an easy consequence of elliptic regularity theorems applied to the extension operators Fi (see (5.5.10)) and F2. Moreover, Si is non-negative, because there exists a positive constant a* such that (Siij.ij) >a*||fi(i7-n)||g ifll > 0

V n e (A) d .

Finally, S2 is coercive in (A) d , owing to estimate (5.4.5) and the trace inequality (1.2.5). It is easily verified that the iterative method (8.4.12)-(8.4.14) is equivalent to

322

(8.4.20)

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS S 2 (A f c + 1 - A f c ) =6{x-S\k)

Ch. 8 ^

on r ,

that is, a Richardson procedure for (8.4.7) with preconditioner S2• Owing to the properties of Si and S 2 , the convergence of this scheme can be established by means of Theorem 4.2.2. In Quarteroni et al. (1991) the finite element approximation of the multidomain problem (8.4.1) and the iterative procedure (8.4.12)-(8.4.14) are considered. The finite dimensional space is constructed by taking (continuous) piecewiseP r elements for u 2 , piecewise-P,—i elements for cri and <j2, whereas the RaviartThomas (in two dimensions) or div-conforming Nedelec elements (in three dimensions) are used for ui (see Section 5.5.3). The coupling between the finite dimensional spaces related to Cli and fi2 occurs through the additional condition (8.4.21)

Ph(v2,h

•

n) =

vi

t h

•n

on T,

where ph is the L 2 (T)-orthogonal projection from L2 (T) onto ¥h:=tyeL2(r)|V|i€Pr_i(/)

vies*}

(here Eh is the decomposition of F induced by the triangulation Th of 0 , and we make the assumption that no triangle of Th crosses T). All the procedures introduced before can be extended to the finite dimensional case. In particular, we can introduce in the natural way the operators Sith, i = 1,2 (the finite dimensional analogue of the operators Si and S2), defined on the space (Ah)d where (8.4.22)

Ah := {Xh G C°(T) | Ah\i G P r ( J )

V I G Eh} C A.

The operators Sith and S2,h are symmetric. Owing to the continuity of the bilinear forms and by proceeding as in Sections 5.4 and 5.5 we can apply Theorems 4.1.3 and 4.1.9 and obtain that they are continuous, uniformly with respect to h. Moreover, Sirh is non-negative and S2>h is coercive, uniformly with respect to h. Therefore, the Richardson method with S2,h as a preconditioner converges; in view of the above properties of Si,/, and S2jn we could also use the preconditioned conjugate gradient iterations. In both cases, the convergence occurs at a rate independent of h. 8.4.2

A n alternative formulation: variational setting and finite element approximation

An alternative approach has been proposed by Alonso and Valli (1995). The novelty resides in a different coupling, which permits us to consider two elliptic problems, one for the pressure a± in fii, and another for the velocity u 2 in fi2 (whereas <j2 and Ui are directly computed from u 2 and <7i, respectively). At the

§8.4

COUPLING FUK COMPRESSIBLE STUKES EQUATIUJNS

323

finite dimensional level, this permits us to use standard piecewise-polynomial finite element spaces; moreover, no matching condition between these elements is necessary on T, and the meshes in fti and ft2 can be chosen in a completely independent fashion. T h e starting point is t o transform the coupled problem into one that is equivalent. Combining (8.4.1)i and (8.4.2) 2 we find that (8.4.23)

ckti - Q _ 1 /3Acri = g - a - 1 d i v f .

Similarly, we obtain (8.4.24)

a u 2 - I/AU 2 - (7* + a _ 1 / 3 ) V d i v u 2 = f -

a^/SVg.

For deriving the new variational formulation, let us start by setting V! = {
I
Vi = {v2 G H1^)

| v2 = 0 on r 2 , D }

A

= {q G i f 1 / 2 (R) | r) = ¥>i|r FOR a suitable ipi G V i } .

Taking into consideration the equation (8.4.1)i and the boundary condition ( 8 . 4 . 1 ) 3 , and integrating by parts we obtain from (8.4.23) / [aanpi + a _ 1 (/3V<7i - f ) • V<£i] + /
/ 9Vi Jn 1

V

IPI G VI.

Similarly, using ( 8 . 4 . 1 ) 7 and (8.4.1) 5 , we have from (8.4.24)

/

Jn;

cm 2 • v 2 + -

+ (7 -

+

r

d

Y

( A « 2 j + DjU 2 ,i) ( D i v 2 j + £>jU2,i

] d i v u 2 div v 2 + a _ 1 / ? ( d i v u 2 - g) d i v v 2

d

L V ,

[ f • v2 Jn2

Diuu)n)

j=1 - (7 -

=

+

- vY(DlU2>i

f2,N • v 2 J 1 2 ,N• L

V v2 G

2v ~d

)

d i v U2nf

+

Pa2nl

v2l

(V2)d,

where n 2 is the unit outward normal vector on 3 f t 2 , therefore n 2 = —n on T. B y means of the interface conditions (8.4.2), we can finally write

324

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

I

JQi

(aaytpi

=

JOi

1 /3V
+ a

(m

• V^i)

u 2 • n ipi

-

/
V
CtU2 • V2 + - Y ( D l u 2 , j + DjU2,l) ( A « 2 j + j Jn-. 1,3 =1 2v + ( 7 —— + a 1f3 ) div u 2 div v 2 =

(f • v 2 + a + f

8 ^

• V
+ a~lf

—J

Ch.

1(3gdivv2)

0(Tin-v2

+

/

DjV2,l)

¥>2,at • v 2

Vv2e(V2)d.

Let us define in H1 ( f i i ) the bilinear form

:

= / (aV'iV'i+a~1/3VV'i • V ^ i ) Jfl!

and the linear functional JCI(VI)

/

( m + «

- l f

• V<^i) -

(vi,z),Vi|riD)ri,D,

where, here and in what follows, we use the same notation (•, •) for the duality pairing between any couple of dual spaces. Similarly, we define in ( H x ( Q 2 ) ) d the form

S(w2,v2) :=

aw2 • v 2 + - £ (DlW2>j / Jn-. 1,3=1

+

D3wi,i)

iDiv2,j

+ DjV2,i

( 2z/ , \ + I 7 — — + a P J div w 2 div v 2 and the functional

£2 ( v 2 ) : = /

J n2

(f • v 2 + a

1/3ffdivv2)

+ (v»2>jv>v2|r2

The variational formulation we are dealing with thus reads: find (<TI,U2) G tf^fti) x [Hl{Q,2))d such that

§8.4

COUPLING FUK COMPRESSIBLE STUKES EQUATIUJNS

Vi) = £ i ( V i ) - J u 2 • n
(8.4.25)

V
i 2 ( u 2 , v 2 ) = £ 2 ( v 2 ) + J Per in. • v 2 ci =

on rliiV

U 2 = V>2,D

on r 2 n.

325

V v 2 G (V 2 )'

Note that the unknowns <7\ and u 2 are only coupled through the boundary integrals that appear in (8.4.25)i and (8.4.25) 2 . For the sake of simplicity, let us assume in what follows that • ( V 2 ) d in the following way: E*rj is the solution to the mixed boundary value problem (with a Dirichlet boundary condition on T) :

(EZrjeVi ai(EiVi

(8.4.26)

^l)

. E{r) = r)

=

0

V i <= V?

on T,

whereas M 2 r) is the solution to the mixed boundary value problem (with a Neumann boundary condition on T) (8.4.27)

A/"27? G (V 2 ) d :

S ;(A/ 2 t?,v 2 )

= J pVn-v2

Vv2G(V2)c

The Steklov-Poincare operators Si, i = 1,2, are defined as follows: (8.4.28)

(S^ti^aKEfaEln)

Vrj^eA

and (8.4.29)

(S2V,p)

=(3-1s*2{M2r],tf2lA

V r),(j,£ A.

Moreover, we set Sr] := Si77 + S2?7. We also introduce the functions crJ G Vf and u 2 G ( V 2 ) , as solutions to (8.4.30)

(8.4.31)

a? G V? : a l i a l ^ i ) = £1(^1)

u 2 G (V 2 ) d : s ^ ( u 2 , v 2 ) = £ 2 ( v 2 )

V i G V?

V v 2 G (V 2 ) d .

If we define x G A' (the dual space of the trace space A) as

326

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

(8.4.32)

Ch. 8 ^

V p G A,

we can easily conclude that the couple <Ji = A + af, u 2 = M2 A + u 2 is a solution to (8.4.25) if and only if A € A is the solution of the Steklov-Poincare equation (8.4.33)

SX = x-

To solve (8.4.25) we propose the following iterative algorithm: given A0 G A, for each k > 0 solve (8.4.34) u * + 1 G (V2)d

:

S;(u2fc+1,v2)

= £ 2 ( v 2 ) + ^/3Afcn-v2

Vv2 G (V2)d,

then (8.4.35)

(J k + 1 G Vi : a j ( a k + 1 ,

) = A

) - J uk+1 • n

V

G Vi,

and finally set (8.4.36)

Xk+1

+ (1 -

=

9)Xk,

where 0 > 0 is an acceleration parameter. It must be noted that at each step we need t o solve two mixed (Dirichlet-Neumann) boundary value problems, one in fii and the other in fi2, associated with linear elliptic operators (a scalar Helmholtz operator in fii, and an elasticity-like operator in fi2). The boundary condition on the interface T is always of a Neumann type. The proof of the convergence of this algorithm is still based on Theorem 4.2.2 (interchanging the role of Si and S 2 ) . In fact, one can verify that it is equivalent t o the preconditioned Richardson method: xk+1

= xk

+eSi1{x-sxk).

Let us show that the operators Si and S 2 satisfy the assumptions of Theorem 4.2.2. For brevity, we will consider directly their finite element approximations. Define for r > 1 and s > 1 the following spaces of finite elements: V U = {
G C°(FI 2 ) I V2MT


ThN}

G P S V T G 7 ^ , v2ih = 0 on r 2 v D } .

Note that the polynomial degrees r > 1 and s > 1 are not related t o each other, and that the triangulations T^ of fii and T 2 of fi2 are not required to match on the interface T. Finally, denoting by

the decomposition of F induced by T f l, we define

Afc : = Wh\r I
= {Vh

G C°(T)

I

T)h]I

G P r ( J ) V I G E^}.

§8.4

COUPLING FUK COMPRESSIBLE STUKES EQUATIUJNS

327

Note that Ah is clearly a finite dimensional subspace of the trace space A. We consider this approximation of (8.4.25): find (<TI,/,, u 2> /,) € Vi,/, x {V2,h)d such that 01(01,/,, fi,h)

= Ci{
V tpi,h G V\th

(8.4.37) s${u2,h, v 2 ,h) = C2{\2th)

+ J (3ai,hn • v 2 ,/,

V v2,h

e

(V2,h)d.

We can then define the finite dimensional extension operators Ej* h and A/2,ft (the discrete counterpart of (8.4.26), (8.4.27)), the finite dimensional SteklovPoincare operators Si,/,, S 2 ,h and S/, = Si,/, + S 2 ,/, (the discrete counterpart of (8.4.28), (8.4.29)), and finally the functions a \ h G V® h , u 2 , A G (V 2 , h ) d and Xh e A'h (the discrete counterpart of (8.4.30)-(8.4.32)). The discrete Steklov-Poincare operators are symmetric. The coerciveness of the form a\ and the trace inequality (1.2.5) yield that Si,/, is coercive, uniformly with respect to h; moreover, S2th is non-negative. The uniform continuity of Si,/, is a straightforward consequence of the continuity of the form in H 1 ( f i i ) and the uniform continuity of the extension operator E^ h , proved in Section 4.1.1. Finally, the uniform continuity of S 2t h follows from the continuity of the form s*2 in (H1(Q2))d and the following estimate: (8.4.38)

\\M2,hVh\\i,n2 < C £ | M | a

V % G Ah,

for a suitable constant C3 > 0 independent of h. This last inequality can be verified as follows:

< ^|Hlo,r||(//'2,fc»7)|r||oJ 11^2,^111,11,

V/?gA,

where a 2 > 0 is the coerciveness constant of the bilinear form s ^ ' ; •) (see (5.4.5)), and C q 2 > 0 is the constant of the trace inequality from H 1 ^ ^ into H l / 2 ( d U 2 ) (see (1.2.3)). The convergence of the finite dimensional Dirichlet-Neumann iterative scheme corresponding to (8.4.34)-(8.4.36) is therefore a consequence of Theorem 4.2.2, because it again corresponds to a preconditioned Richardson iterative method for solving the discrete Steklov-Poincare equation (8.4.39)

S „ A,, =

Xh,

with Si,h as a preconditioner. The rate of convergence is independent of h.

328

8.5

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS

Ch. 8 ^

The coupling for the time-harmonic Maxwell equations

The derivation of the time-harmonic Maxwell equations has been carried out in Section 5.7. Omitting from the complete set of equations (5.7.2) the terms —ieaE (which corresponds to considering a low-frequency problem), we are left with r rot H - oE = 0

in fi

I rot E + ifj.aH = 0

in fi,

(8.5.1)

where \i is the magnetic permeability coefficient and a is the electric conductivity. In a conductive medium, \i and a are assumed to be symmetric matrices, depending on the space variable x , and uniformly positive definite. On the other hand, in an insulator, n is a constant /j.q > 0 and a is vanishing. On the boundary, we impose the Dirichlet boundary condition for the electric field: (8.5.2)

n x E = $

on dfi,

where $ is a given tangential vector field defined on dfi. We are concerned here with a medium that is heterogeneous-, namely, composed of two parts, one that is a non-homogeneous, non-isotropic conductor, and another that is a perfect insulator. In other words, this means that we are assuming that the conductivity a is a symmetric, uniformly positive definite matrix a in a subset, say fi2, of fi, while it is equal to 0 in the complementary part, fii. We are thus naturally led to considering an appropriate equivalent twodomain formulation. Its main feature will be that in each subdomain we have to solve an equation of a simple type, the interaction between the two subdomains being governed by two interface conditions. This formulation yields a solver based on a domain decomposition iterative approach: one interface condition is assigned to a subdomain, the other to the complementary domain, and then an iterative procedure is employed, solving at each step the 'simple' problems in each subdomain, just by modifying step by step on the interface the value of the data that are related to the solution on the other domain. A first remark is now in order: once we have chosen a as we described above, for problem (8.5.1), (8.5.2) uniqueness clearly does not hold, because we can add to E the gradient of any function tp having compact support in fii. Therefore, we need to modify (8.5.1) by adding another equation to it. A perturbation argument suggests we add the constraint div(E|^ 1 ) = 0 in fii (which corresponds to the natural physical condition that no charge is present in the insulator fii). Furthermore, it is possible to find a vector field E* such that n x E ' = $ o n <9fi, and, moreover, divE* = 0 and rot rot E* = 0. Writing u : = E - E*, taking the rotation of (8.5.1) 2 and using (8.5.1)i, we can finally rewrite the problem as

§8.5

COUPLING FOR TIME-HARMONIC MAXWELL EQUATIONS

I

rot(/i

(8.5.3)

1

r o t u ) + iaau = — vot(p

1

rotE*) — iaaW

329

in fi

div(u|Qj) = 0

in fii

n x u= 0

on dfl.

The construction of such a vector field E* relies on suitable geometric assumptions on the domain fl (see Alonso and Valli 1996); for instance, it is true if fl is a convex polyhedron in R 3 . In Alonso and Valli (1997) it has been proved that this problem has a unique solution in a suitable Hilbert space, provided that the subdomain fii is also a convex polyhedron. We will not give the proof of this rather technical result here; we want only to show that (8.5.3) can be rewritten in terms of an equivalent two-domain formulation, which leads in a natural way to an iteration-bysubdomain procedure. It can also be shown that this iteration-by-subdomain procedure is equivalent to a preconditioned Richardson method applied to the Steklov-Poincare equation on the interface. Let us turn to the equivalent two-domain formulation. First, this requires us to determine the correct interface conditions on F for the solution u defined in fl. As we have already noted in Remark 1.1.1, these conditions are often determined by requiring that the solution u belongs to a space of functions defined over the whole of fi (this space embodies the regularity properties of u ^ in fii and U|q2 in fi2, together with a suitable matching on T); moreover, the solution u satisfies the equation in the sense of distributions in the whole fi; namely, through the interface F and not only in fii and fi2 separately. For the problem at hand, denoting by Uj the restriction of u to fii, i = 1,2, this means that we have to impose the following interface conditions on T:

(8.5.4)

n x ui = n x u 2 n x

(/j,0 1

r o t u i ) = n x (p

on F 1rotu2)

on T.

The first condition ensures that u G i?(rot; fi), and the second that u is a solution of (8.5.3)i in fi in the sense of distributions. The equivalent two-domain formulation is therefore

330

HETEROGENEOUS DOMAIN DECOMPOSITION METHODS ( rot rot ui = 0

(8.5.5)

in fii

divui = 0

in fii

n x ui = 0

on dfii \ T

< n x ui = n x u 2 n x (/i 0

1

r o t u i ) = n x (/i

Ch. 8 ^

on 1rotu2)

r

on F

n x u2 = 0

on dfi 2 \ T

„ rot(/u _1 r o t u 2 ) + iaau2 = F

in fi2,

where F := - r o t ^ - 1 rot E*) - ia&E*. The equation in fii can be rewritten in a simpler way. In fact, it can be seen that ui is the solution of rot Ui = Vwi,r((n x u 2)|r)

in

div Ui = 0

in fii

n x ui = (n x u2)|r

on dfii.

!

Here, Wi,r(7) is the solution of the Neumann problem:

(8.5.7)

' Awi ) r(7) = 0 o»wi,r(7) = —div r 7 dn

/

JQ.

in fii on <9fii

wi,r(7) = 0

where 7 is a tangential vector on T, 7 is its extension by 0 on dfii \ I1, and d i v r 7 is the tangential divergence of 7 (for a precise definition, see, for example, Begue et al. 1988; Alonso and Valli 1996). Therefore, we can rewrite the two-domain formulation (8.5.5) as

T

§8.5

COUPLING FOR TIME-HARMONIC MAXWELL EQUATIONS ' At-pi = 0

in fti

A • T (n ( x u 2 )|r ^ = —div

(8.5.8)

/

JQ!

331

on 5f2i

ipi = 0

n x u2 = 0

on d 0 2 \ T

n x

on T

r o t u 2 ) = / i ^ n x V
. r o t ( p _ 1 r o t u 2 ) + iao\i2 = F

in fi2,

and the iteration-by-subdomain procedure is easily derived as follows: given a tangential vector field A0 on T, for each k > 0 solve in Qi

f A
M+1

(8.5.9)

dn

h JQi

(8.5.10)

I

= —div r A

on dQ±

= 0

r o t ( / i _ 1 r o t u 2 + 1 ) + ia&u.2 +1 = F

in fi2

n x (fj,- 1 rot u 2 + 1 ) =

on F

1n

x V<^+1

on 80,2 \ T,

n x u2+1 = 0 where (8.5.11)

A f c + 1 : = 9{nr

+ 1 ) | r + (1 x u *k+l\

9)AK.

The convergence of this iterative procedure has been proved for the corresponding finite dimensional problem, using suitable finite elements for space discretisation. More precisely, standard piecewise-polynomials for the approximation in i ) and the rot-conforming Nedelec finite elements for the approximation in .ff(rot; fi2) : = { v G ( L 2 ( 0 2 ) ) 3 | r o t v G ( L 2 ( 0 2 ) ) 3 } are employed. We refer to Alonso and Valli (1997) for a precise variational formulation and the corresponding finite element approximation. There it is proved that the assumptions of Theorem 4.2.10 are satisfied, uniformly with respect to the grid size h.

330 APPENDIX 9.1

Function spaces

In this last section we recall the definitions of some function spaces which have been often used in the book. A complete presentation of this subject can be found for instance in Yosida (1974), Brezis (1983), J.-L. Lions and Magenes (1972), and Adams (1975). 1. Hilbert and Banach spaces Let V be a complex linear space. A scalar product on V is a map (•, •) : V x V —>• C , linear in the first argument and such that (w,v) = (v,w) for each w,v G V (symmetry; the bar denotes complex conjugation); (v,v) > 0 for each v 6 V (positivity); and = 0 if, and only if, v = 0. In the case of a real linear space, the map (•, •) takes values in R . A seminorm is a map || • || : V —> R such that ||w|| > 0 for each v £ V; ||cu|| = |c| ||u|| for each c G C and v G V; and ||w + < ||IU|l + IMI for each w,v G V (triangular inequality). A norm on V is a seminorm satisfying the additional property that ||t>|| = 0 if, and only if, v = 0. Two norms || • || and ||| • ||| on V are equivalent if there exist two positive constants M i and M2 such that M1\\v\\ < \\\v\\\<M2\\v\\ for each v E V. It is readily verified that any scalar product defines a norm by setting: := (v^)1/2. Moreover, any norm defines a distance: d(w,v) := ||u; — A linear space V endowed with a scalar product (respectively, a norm) is called pre-hilbertian (respectively, normed) space. A sequence vn is a Cauchy sequence in a normed space V if it is a Cauchy sequence with respect to the distance d(w,v) = - i>||. If any Cauchy sequence in a pre-hilbertian (respectively, normed) space V is convergent, the space V is called a Hilbert Hilbert space (respectively, Banach spaceBanach) space. In a Hilbert space the Schwarz inequality holds: (9.1.1)

1(^,^)1 < IMI |M| for each w,v G V.

2. Dual spaces If (V, || • ||y) and ( W , || • 11w) a r e normed spaces, we denote by C(V; W) the set of linear, continuous functionals from V into W, and for T G £ ( V ; W) we define the norm

Ch. 9

APPENDIX

334

(9-1.2) 0 Thus C(V\ W) is a normed space; if W is a Banach space, then C(V\ W) is a Banach space, too. If W = C (respectively, W = R , if V is a real normed space), the space C(V~, C ) (respectively, C(V\ R ) ) is called the dual space of V and is denoted by V'. The norm in V is indicated by || • ||y. The bilinear form (•,•) from V' x V into C (respectively, R ) defined by (F,v) := J-{v) is called the duality pairing between V and V. 3. Weak convergence In a normed space V it is possible to introduce another type of convergence, which is called weak convergence. It is defined as follows: a sequence vn is called weakly convergent to v £ V if !F(vn) converges to !F(v) for each T € V'. Clearly, if the sequence vn converges to v in V , it is also weakly convergent. The converse is not true, unless V is finite dimensional. It can be proved that the weak limit v, if it exists, is unique. Moreover, if vn is weakly convergent to v £ V, one has |M| < l i m i n f | K | | . n—> oo 4. The Riesz theorem and the Lax-Milgram lemma An important result which holds in Hilbert spaces is the following one. T h e o r e m 9 . 1 . 1 (Riesz representation theorem) Let V be a (real or complex) Hilbert space, endowed with the scalar product (•, -)v • If J- £ V', then there exists a unique w? £ V such that (9.1.3)

T{v)

V v £ V.

= (wr,v)v

As a consequence of the Riesz representation theorem, if V is a Hilbert space, then the dual V' is a Hilbert space which can be canonically identified to V. Another consequence of the Riesz theorem is this result. T h e o r e m 9.1.2 endowed with the is a bilinear form, T £ V. Assume,

(Lax-Milgram lemma) Assume that V is a real Hilbert space, scalar product (•, -)y and the norm || • \\y, that A : V x V —> R and that J7 : V H is a linear continuous functional, namely, moreover, that A is continuous, namely,

(9.1.4)

3 7 > 0 : |^l(u;,v)| < 7||w|k I M k

and coercive,

namely,

(9.1.5)

3 a > 0 : A{v,v)

> a\\v\\2v

Then there exists a unique u £ V solution

to

V w,v £ V,

VveV.

FUNCTION SPACES

§9.1

(9.1.6)

A(u,v)=F(

v)

335

VvGV,

and, moreover, (9.1.7)

\\u\\v
If V is a complex Hilbert space, the Lax-Milgram lemma holds provided that (9.1.5) is substituted by the assumption (9.1.8)

3 a > 0 :\A{v,v)\ > a\\v\\2v

VugK

5. Lp spaces Let ft be an open set contained in R d , d > 1, and consider in ft the Lebesgue measure. A very important family of Banach spaces is the following one. Let 1 < p < oo, and consider the set of measurable functions v such that (9.1.9)

I \v{x)\pdx < oo, Jn

l
or, when p = oo, (9.1.10)

s u p { K x ) | |x £ ft} < oo.

These spaces are usually denoted by L p (ft) and the associated norm is

'-IP (9.1.11)

| M U , ( n ) := QT Kx)|*dx

1 < p < oo,

or, when p = oo,

(9.1.12)

IM|i~(ti) := sup{|u(x)| | x 6 ft}.

More precisely, Lp(Vt) is indeed the space of classes of equivalence of measurable functions, satisfying (9.1.9) or (9.1.10), with respect to the equivalence relation: w = v if w and v are different on a subset having zero-measure. In other words, in the space L p (ft) two functions, different on a subset which has zero-measure, are identified to each other. Thus the definition of the space L°°(ft) in (9.1.10) and of its norm in (9.1.12) should be modified in the following way: v 6 L ° ° ( f t ) if i n f { M > 0 | |w(x)| < M almost everywhere in ft} < oo, and IMIz,°°(fi)

:=

i n f { M > 0| |u(x)| < M almost everywhere in ft},

where 'almost everywhere in ft' means 'everywhere except on a subset of ft having zero-measure'.

APPENDIX

336

Ch.

9

The space L 2 ( f t ) is a Hilbert space, endowed with the scalar product (w,v)L2{n) often indicated by (w,v)otn

:=

/ w(x)v(x)dx, Jn

or simply (vu,v).

If 1 < p < oo, the dual space of Lp(Cl) is given by Lp (ft), where (1 jp) (1 / p <) = l (and p' = oo if p = 1).

+

6. Distributions Let us recall that Cq°(Q) (or £>(ft)) denotes the space of infinitely differentiable functions having compact support; that is, vanishing outside a bounded open set ft' c ft which has a positive distance from the boundary <9ft of ft. It is useful to define the concept of convergence for sequences of X>(ft). W e say that vn e X>(ft) converges to v £ £>(ft) if there exists a closed bounded subset K £ ft such that vn vanishes outside K for each n, and for every non-negative multi-index a the derivative Davn converges to Dav uniformly in ft. We recall that if a = ( a i , . . . , ad), non-negative integers, then

where |a| : = a\ + ... + is the length of a . The space of linear functionals on T>(Q) which are continuous with respect to the convergence introduced above is denoted by D ' ( f t ) and its elements are called distributions. If L G I?'(ft) and v 6 D(Q.), the action of the functional L on v is usually denoted by the duality pairing (L,v). It is easily seen that each function w £ Lp(fl), 1 < p < oo, can be associated with the following distribution: v^-

Jn

w(x)v{x)dx,

vET>(Q).

However, setting for instance ft = (—1,1), the Dirac functional v -»• (6,v) :=v(0),

v£V{Q),

is a distribution which cannot be represented through any function belonging to L p ( f t ) , 1

:=(-l)W(L,Dav)

\/v£V(tt).

Note that, from this definition, a distribution turns out to be infinitely differentiable. On the other hand, when L is a smooth function, it is easily verified

§9.1

FUNCTION SPACES

337

by integrating by parts that its derivative in the sense of distributions coincides with the usual derivative. Let us also recall that the Dirac distribution 6 is the distributional derivative of the Heaviside function: H{x)

1

:= {

0

x > 0 x < 0

Finally, we say that the a-derivative of a distribution L is a function belonging to Lp(fl), 1 < p < oo, if there exists a function ga G LP(Q) such that {DaL,v)=

[ Jn

V v e 2>(n).

ga(x)v(x)dx

7. Sobolev spaces W e finally introduce another class of functions, which furnish the natural environment for the variational theory of partial differential equations. The Sobolev space Wk'p(Cl), k a non-negative integer and 1 < p < oo, is the space of functions v E D'ifi) such that all the distributional derivatives of v of order up to k belong to Lp(fl). In short Wk'p(n)

{ v G Lp(0)

I Dav

G L p ( 0 ) for each non-negative multi-index a such that |a| < fc}.

Clearly, for each p, 1 < p < oo, VF 0,p (r2) = and Wk2'p(fl) C Wkl'p{0.) when fci < For 1 < p < oo, is a Banach space with respect to the norm I/P

IMU,p,n : =

£

p

I I D a v LP( n)

\a\
Moreover, its seminorm is defined as follows: I/P

M*, P ,n ==

On the other hand, Wk,°°(fl)

E \a\=k

p

L?(n)

is a Banach space with respect t o the norm

I M k o o . n : = max \\Dav\\L^iQ), while the corresponding seminorm is denoted by M/o,oo,n : = max H - D ^ H ^ m ). y ' \a\=k In particular, when p = 2 we write Hk(0)

instead of Wk<2(tt),

instead of || • ||jt,2,ft and | • \k,2,n, respectively.

|| • ||jt,n and | • |fc)o

APPENDIX

338

Note that Hk(Q)

Ch. 9

is a Hilbert space with respect to the scalar product

\a\
Finally, for 1 < p < oo we denote by W 0 fc ' p (ft) the closure of respect to the norm || • ||jfc,p,n, and with As before, when p -

2 we write

# 0 f c (ft)

W~k'p and

(ft) with

(ft) the dual space of

H~k(Q)

instead of

WQ'p(£1).

W0M(ft)

and

W~k'2(£l), respectively. It can also be proved that WQ ' p ( f t ) = LP(Q) and that, if ft has a Lipschitz continuous boundary and 1 < p < oo, Wk'p(fl) is indeed the closure of C ° ° ( f t ) with respect to the norm || • In other words, C ° ° ( f t ) is dense in Wk'p(£l) for 1 < p < oo. It is sometimes useful to consider the Sobolev space W s ' p ( f t ) , where s £ R and 1 < p < oo. For the general definition, we refer to Adams (1975). In particular, we only recall that, if ft = R d and p = 2, W s - 2 ( R d ) = Hs(Rd) can be characterised as follows by means of the Fourier transform 0(£): Hs(Rd)

= {v £ L 2 ( R d ) | (1 + |£| 2 ) s/2 6(f) € L 2 ( R d ) } .

When considering vector-valued functions v : ft —>• R d , the Hilbert space i f (div; ft) : = { v £ ( L 2 ( f t ) ) d | d i v v £ i 2 ( f t ) } is also often used. It is endowed with the graph norm l|v|k(div;n) : = (l|v||o,n + l l d i v v l l 2 ^ ) 1 / 2 . Similarly to the preceding cases, if ft has a Lipschitz continuous boundary, it can be proved that i?(div; ft) is the closure of ( C o c ( f l ) ) d with respect to the norm II ' l l i f ( d i v ; f i ) -

For three-dimensional vector-valued functions we also introduce the Hilbert space H(rot; ft) : = { v £ ( i 2 ( f t ) ) 3 | r o t v € ( L 2 ( f t ) ) 3 } , endowed with the graph norm IM|H(rot;fi) == (I v| |g,n + 11 rot v| |2_n f ' 2 . Again, if ft has a Lipschitz continuous boundary, then H(rot; ft) is the closure of ( C ° ° ( f t ) ) 3 with respect to the norm || • ||jj(rot;f2)Another important class of Sobolev spaces is given by Ws'p(S), where s > 0, 1 < p < co and E is a suitable subset of the boundary dfl (again, we write i F ( E ) instead of VF S ' 2 (£)). Their definition needs the introduction of some technical tools, especially if E is a non-smooth hypersurface (for instance, the boundary

§9.2

SOME PROPERTIES OF THE SOBOLEV SPACES

339

of a polygonal domain). For this we refer to Adams (1975) or Brezzi and Gilardi (1987); however, we return on a characterisation of these spaces in the following section. When £ = 80, the dual space of HS(80) is denoted by H~S(80).

9.2

Some properties of the Sobolev spaces

In this section we present without proofs some relevant properties enjoyed by functions belonging to Sobolev spaces. We mainly limit ourselves to the Hilbert spaces Hs(0), referring the reader to J.-L. Lions and Magenes (1972) or Adams (1975) for the general case and all the proofs. Let us start with the so-called trace theorems. The trace on the boundary 80 of a function v £ Hs(0) is, in a sense to make precise, the value of v restricted to 80. Indeed, the latter statement has not even a meaning, as a function in Hs(0) is not univocally defined on subsets having measure equal to zero. If we denote by C'°(0) the space of continuous functions on 0, the precise result reads as follows. Theorem 9.2.1 (Trace theorem) Let 0 be a bounded open set o / R d . that the boundary 80 is smooth and that s > 1/2. (a) There exists a unique linear continuous map 70 : HS(0) such that 7o« = V\QQ for each v £ HS(0) fl C°(ft). (b) There exists a linear continuous map TZQ : HS~L/2(80) joRof = V> for each

—y

Assume

HS"1/2(80)

—^ HS(0)

such that

Analogous results hold true if we consider the trace jsv over a smooth subset £ of the boundary 80. In particular, for 1/2 < s < 1, it is enough to assume that the boundary 80 or the set £ are Lipschitz continuous. Thus, we have seen that any function belonging to HS~" ^ ( E ) , 0 > 1/2 and £ smooth, is the trace on £ of a function in HS(0). This provides a useful characterisation of the space For vector functions belonging to i f (div; fi) the following trace result can be proved. Theorem 9.2.2 (Normal trace theorem) with a Lipschitz continuous boundary 80.

Let 0 be a bounded open set of R d

(a) There exists a unique linear continuous map 7 „ : //"(div; 0) -»• H~xl2(8O) such that 7 „ v = (v • n * ) ^ for each v € H(div; fi) n (C°(f2)) d . (b) There exists a linear continuous map Hn : H~ll2(d0) that 7 n T l n f = (p for each

2{dO).

->• H(div; Q) such

Here we have denoted by n* the unit outward normal vector on 80. Let us note, moreover, that the normal trace of a vector function v e # ( d i v ; 0) over a Lipschitz continuous subset £ of 80 different from the whole boundary 80 does not belong in general to 7 J - 1 / , 2 ( £ ) , but to a larger space, which is usually denoted by i 7 ~ 1 / 2 ( £ ) (see, for instance, J.-L. Lions and Magenes 1972).

APPENDIX

340

Ch.

9

Now, let us introduce the space XQQ : = { £ { H - ^ 2 { d n ) ) 3 \ $ • n* = 0, divrV> £ i T " 1 / 2 ^ ) } , where div r tf> denotes the tangential divergence of ip (see, for example, Begue et al. 1988). For three-dimensional vector functions belonging to i f (rot; ft) the following trace result can be proved (see Alonso and Valli 1996). Theorem 9.2.3 (Tangential trace theorem) Let ft be a bounded open set of R 3 with a Lipschitz continuous boundary 3ft. (a) There exists a unique linear continuous map yT : H{rot; ft) XQQ such that 7 r v = (n* x v)| 9n for each v £ H(rot; ft) n (C°(ft)) 3 . (b) If either the boundary dfl is smooth or ft is a convex polyhedron, then there exists a linear continuous map 1Z.T : XQQ, —> H(rot; ft) such that frUr^ = V" for each £ Xdn. By means of these trace operators it is possible to characterise the spaces ff^ft), Ho (div; ft) := (C 0 oo (ft)) d and # 0 ( r o t ; f t ) : = (C 0 °°(ft)) 3 (here the closure has to be intended with respect to the norms || • ||/f(div;n) a n d II • ||j?(rot;fi)> respectively). As a matter of fact, if the boundary dft is Lipschitz continuous, we have: = {v £ H1 (ft) 170U = 0 } Ho (div; ft) = { v £ ff (div; ft) | 7 « v = 0 } H0(rot; ft) = { v £ H(rot; ft) 17 r v = 0 } . A similar characterisation holds for the space

An important result is the so-called Poincare inequality (see (1.2.2)). Theorem 9.2.4 (Poincare inequality) Assume that ft is a bounded connected open set of R d and that S is a (non-empty) Lipschitz continuous subset of the boundary dfl. Then there exists a constant CQ > 0 such that (9.2.1)

f v2{x) dx
for each v £

[ |Vw(x)|2 dx Jn

H^(fl).

As a consequence of the density of C°°(ft) in i / 1 ( f t ) (under the assumption that 5ft is Lipschitz continuous), it is easily proved that for each w,v £ f f 1 ( f t ) the following Green formula holds: (9.2.2)

/ (Djw)vdx Jn

= -

/ wDjvdx+ Jn

/ (7ou>) (70 w) n* dy, Jan

j =

l,...,d,

§9.2

SOME PROPERTIES OF THE SOBOLEV SPACES

341

where we have denoted by Dj the partial derivative gfr and by d j the surface measure on <9fi. Similarly, if w G i?(div; fi) and v G Hl(fl), (9.2.3)

we find that

/ (div w ) v dx = - / w - V i > d x + / ( 7 „ w ) (-y0v) dy. Jn Jn Jan

Finally, if w G # ( r o t ; f i ) and v G ( f l ^ f i ) ) 3 , we find that (9.2.4)

/ (rot w ) • v dx = / w - r o t v d x - | - / ( 7 r w ) • (7 0 v) d-y. Jn Jn Jan

As we have already noted, the functions belonging to the Sobolev spaces l F s ' p ( f i ) are not univocally defined over subsets having measure equal to zero. However, if suitable restrictions on the indices s and p are assumed, these functions indeed turn out to be regular functions. This is made clear by the following theorem. T h e o r e m 9 . 2 . 5 (Sobolev embedding theorem) Assume that fl is a (bounded or unbounded) open set of Rd with a Lipschitz continuous boundary, and that 1 < p < oo. Then the following embeddings are continuous. (a) If 0 < sp < d, then Ws'p{fl) (b) If sp = d, then (c) Ifsp

Ws'p(fi)

> d, then Ws'p{fi)

C Lp* (fi) for p* =

C L9(fl)

dp/(d-sp).

for any q such that p < q < oo.

C C°(fi).

In the one-dimensional case, we have in particular that H1 (fi) C C° (fi), with continuous embedding.

REFERENCES 1. Achdou, Y., Kuznetsov, Yu.A. and Pironneau, 0 . (1995). Substructuring preconditioners for the Qi mortar element method. Numer. Math., 71, 419-449. 2. Achdou, Y., Maday, Y . and Widlund, O.B. (1996). Methode iterative de sousstructuration pour les elements avec joints. C. R. Acad. Sci. Paris Ser. I Math., 322,185-190. 3. Agoshkov, V.I. (1988). Poincare-Steklov's operators and domain decomposition methods in finite dimensional spaces. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 73-112. 4. Agoshkov, V.I. and Lebedev, V.I. (1985). Poincare-Steklov operators and the methods of partition of the domain in variational problems. In Computational Processes and Systems, No. 2, G.I. Marchuk, ed., Nauka, Moscow, pp. 173-227 [Russian]. 5. Agoshkov, V.I. and Ovtchinnikov, E. (1994). Projection decomposition method. Math. Models Methods Appl. Sci., 4, 773-794. 6. Aguilar, G. and Lisbona, F. (1994). Interface conditions for a kind of non linear elliptic hyperbolic problems. In Domain Decomposition Methods in Science and Engineering, A. Quarteroni et al. eds., American Mathematical Society, Providence, pp. 89-95. 7. Alonso, A. and Valli, A. (1995). A new approach to the coupling of viscous and inviscid Stokes equations. East-West J. Numer. Math., 3, 29-41. 8. Alonso, A. and Valli, A. (1996). Some remarks on the characterization of the space of tangential traces of i?(rot; ft) and the construction of an extension operator. Manuscr. Math., 89, 159-178. 9. Alonso, A. and Valli, A. (1997). A domain decomposition approach for heterogeneous time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Eng., 143, 97-112. 10. Alonso, A., and Valli, A. (1999). An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput., to appear. 11. Alonso, A., Trotta, R.L. and Valli, A. (1998). Coercive domain decomposition algorithms for advection-diffusion equations and systems. J. Comput. Appl. Math., 96, 51-76. 12. Auge, A., Lube, G. and Otto, F.C. (1997). A non-overlapping decomposition method with adaptive interface conditions for elliptic problems. In Proceedings of XIII GAMM-seminar, Friedr. Vieweg & Sohn, Braunschweig, to appear. 13. Auge, A., Kapurkin, A., Lube, G. and Otto, F.C. (1996). A note on domain decomposition of singularly perturbed elliptic problems. In Proceedings of IX Domain Decomposition Conference, P. Bj0rstad et al.. eds., John Wiley &

344

REFERENCES

Sons, New York, to appear. 14. Babuska, I. (1973). The finite element method with Lagrangian multipliers. Numer. Math., 20, 179-192. 15. Bank, R.E. and Xu, J. (1995). The hierarchical basis multigrid method and incomplete LU decomposition. In Domain Decomposition Methods in Scientific and Engineering Computing, D.E. Keyes and J. Xu, eds., American Mathematical Society, Providence, pp. 163-173. 16. Begue, C., Conca, C., Murat, F. and Pironneau, 0 . (1988). Les equations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression. In Nonlinear Partial Differential Equations and Their Applications. College de France Seminar, Vol. IX, H. Brezis and J.-L. Lions, eds., Longman, Harlow, pp. 179-264. 17. Ben Belgacem, F. and Maday, Y. (1997). The mortar element method for three-dimensional finite elements. R.A.I.R.O. Model. Math. Anal. Numer., 31, 289-302. 18. Bernardi, C. and Girault, V. (1998). A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal., 35, 1893-1916. 19. Bernardi, C., Debit, N. and Maday, Y . (1990). Coupling finite element and spectral methods: first results. Math. Comput., 54, 21-39. 20. Bernardi, C., Maday, Y. and Patera, A. T. (1994). A new nonconforming approach to domain decomposition: the mortar element method. In Nonlinear Partial Differential Equations and Their Applications. College de France Seminar, Vol. XI, H. Brezis and J.-L. Lions, eds., Longman, Harlow, pp. 13-51. 21. Berselli, L.C. (1997). Neumann-Neumann Domain Decomposition Preconditioners for Non-symmetric Problems. Ph.D. Thesis, Dipartimento di Matematica, Universita di Pisa, in preparation. 22. Bj0rhus, M. (1995). On Domain Decomposition, Subdomain Iteration, and Waveform Relaxation. Dr Ing. Thesis, Department of Mathematical Sciences, University of Trondheim. 23. Bj0rstad, P.E., and Widlund, O.B. (1986). Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal., 23, 1097-1120. 24. Blum, H., Lisky, S. and Rannacher, R. (1992). A domain splitting algorithm for parabolic problems. Computing, 49, 11-23. 25. Boffi, D. (1994). Stability of higher-order triangular Hood-Taylor methods for the stationary Stokes equations. Math. Models Methods Appl. Sci., 4, 223236. 26. Borgers, C. and Widlund, O.B. (1990). On finite element domain imbedding methods. SIAM J. Numer. Anal., 27, 963-978. 27. Bossavit, A. (1993). Electromagnetisme, en Vue de la Modelisation. SpringerVerlag, Paris. 28. Bourgat, J.-F., Glowinski, R., Le Tallec, P. and Vidrascu, M. (1989). Variational formulation and algorithm for trace operator in domain decomposition calculations. In Domain Decomposition Methods, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 3-16.

REFERENCES

345

29. Bourgat, J.-F., Le Tallec, P., Perthame, B. and Qiu, Y. (1994). Coupling Boltzmann and Euler equations without overlapping. In Domain Decomposition Methods in Science and Engineering, A. Quarteroni et al. eds., American Mathematical Society, Providence, pp. 377-398. 30. Bourgat, J.-F., Le Tallec, P., Tidriri, D and Qiu, Y. (1992). Numerical coupling of nonconservative or kinetic models with the conservative compressible Navier-Stokes equations. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, D.E. Keyes et al. eds., SIAM, Philadelphia, pp. 420-440. 31. Bramble, J.H. and Pasciak, J.E. (1989). A domain decomposition technique for Stokes problems. Appl. Numer. Math., 6, 251-261. 32. Bramble, J.H., Pasciak, J.E. and Schatz, A.H. (1986a). An iterative method for elliptic problems on regions partitioned into substructures. Math. Comput., 46, 361-369. 33. Bramble, J.H., Pasciak, J.E. and Schatz, A.H. (19866). The construction of preconditioners for elliptic problems by substructuring. I. Math. Comput., 47, 103-134. 34. Bramble, J.H., Pasciak, J.E. and Schatz, A.H. (1989). The construction of preconditioners for elliptic problems by substructuring. IV. Math. Comput., 53, 1-24. 35. Bramble, J.H., Pasciak, J.E. and Vasiliev, A.E. (1997). Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J Numer. Anal., 34, 1072-1092. 36. Bramble, J.H., Pasciak, J.E. and Xu, J. (1990). Parallel multilevel preconditioners. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 341-357. 37. Bramble, J.H., Pasciak, J.E. and Zhang, X. (1996). Two-level preconditioners for 2m'th order elliptic finite element problems. East-West J. Numer. Math., 4, 99-120. 38. Bramble, J.H., Pasciak, J.E., Wang, J. and Xu, J. (1991). Convergence estimates for product iterative methods with application to domain decomposition. Math. Comput., 57, 1-21. 39. Brenner, S.C. (1998a). The condition number of the Schur complement in domain decomposition. Preprint. 40. Brenner, S.C. (19986). Personal communication. 41. Brezzi, F. (1974). On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrange multipliers. R.A.I.R.O. Anal. Numer., 8, 129-151. 42. Brezzi, F. (1998). Personal communication. 43. Brezzi, F. and Fortin, M. (1991). Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin. 44. Brezzi, F. and Marini, L. D. (1994). A three-field domain decomposition method. In Domain Decomposition Methods in Science and Engineering, A. Quarteroni et al. eds., American Mathematical Society, Providence, pp. 27-34.

346

REFERENCES

45. Brezzi, F., Douglas, J., Jr and Marini, L.D. (1985). Two families of mixed finite elements for second order elliptic problems. Numer. Math., 47, 217-235. 46. Bristeau, M.-O., Glowinski, R. and Periaux, J. (1987). Numerical methods for the Navier-Stokes equations. Application to the simulation of compressible and incompressible viscous flows. Comput. Phys. Rep., 6, 73-187. 47. Buzbee, B.L., Dorr, F.W., George, J.A. and Golub, G.H. (1971). The direct solution of the discrete Poisson equation on irregular regions. SIAM J. Numer. Anal., 8, 722-736. 48. Cai, X.-C. (1990). An additive Schwarz algorithm for nonselfadjoint elliptic equations. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 232-244. 49. Cai, X.-C. (1991). Additive Schwarz algorithms for parabolic convectiondiffusion equations. Numer. Math., 60, 41-61. 50. Cai, X.-C. (1995). A family of overlapping Schwarz algorithms for nonsymmetric and indefinite elliptic problems. In Domain-based Parallelism and Problem Decomposition Methods in Computational Science and Engineering, D.E. Keyes et al. eds., SIAM, Philadelphia, pp. 1-21. 51. Cai, X.-C. and Sarkis, M. (1997). A restricted additive Schwarz preconditioner for general sparse linear systems. Technical report CU-CS-843-97, University of Colorado at Boulder. 52. Carlenzoli, C. and Quarteroni, A. (1995). Adaptive domain decomposition methods for advection-diffusion problems. In Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, I. Babuska et al. eds., IMA Volumes in Mathematics and its Applications, 75, SpringerVerlag, New York, pp. 165-199. 53. Carpenter, M.H., Nordstrom, J. and Gottlieb, D. (1997). A stable and conservative interface treatment of arbitrarily spatial accuracy. Technical report, Division of Applied Mathematics, Brown University. 54. Casarin, M. (1996). Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University. 55. Cazabeau, L., Lacour, C. and Maday, Y. (1997). Numerical quadratures and mortar methods. In Computational Science for the 21st Century, M.-O. Bristeau et al. eds., John Wiley & Sons, Chichester, pp. 119-128. 56. Chan, T.F. and Mathew, T.P. (1992). The interface probing technique in domain decomposition. SIAM J. Matrix Anal. Appl., 13, 212-238. 57. Chan, T.F. and Mathew, T.P. (1994a). Domain decomposition algorithms. Acta Numer., 161-143. 58. Chan, T.F. and Mathew, T.P. (19946). Domain decomposition preconditioners for convection diffusion problems. In Domain Decomposition Methods in Science and Engineering, A. Quarteroni et al. eds., American Mathematical Society, Providence, pp. 157-175. 59. Chan, T.F. and Resasco, D.C. (1985). A survey of preconditioners for domain decomposition. Technical report / D C S / R R - 4 1 4 , Yale University.

344 REFERENCES 60. Chan, T.F. and Smith, B.F. (1995). Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes. In Domain Decomposition Methods in Scientific and Engineering Computing, D.E. Keyes and J. Xu, eds., American Mathematical Society, Providence, pp. 175-189. 61. Ciarlet, Ph.G. (1978). The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam. 62. Ciccoli, M.C. (1995). Adaptive domain decomposition algorithms and finite volume/finite element approximation for advection-diffusion equations. Technical report, CRS4, Cagliari. 63. Collino, F. (1993). High order absorbing boundary conditions for wave propagation models: straight line boundary and corner case. In Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman et al. eds., SIAM, Philadelphia, pp. 161-171. 64. Dauge, M. (1988). Elliptic Boundary Value Problems on Corner Domains. Springer-Verlag, Berlin. 65. Dawson, C.N. and Du, Q. (1991). A domain decomposition method for parabolic equations based on finite elements. In Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 255-263. 66. Dawson, C.N., Du, Q. and Dupont, T.F. (1991). A finite difference domain decomposition algorithm for numerical solution of the heat equation. Math. Comput., 57, 63-71. 67. De Boeck, Y.-H. and Le Tallec, P. (1991). Analysis and test of a local domaindecomposition preconditioner. In Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 112-128. 68. Dryja, M. (1982). A capacitance matrix method for Dirichlet problem on polygon region. Numer. Math., 39, 51-64. 69. Dryja, M. (1988). A method of domain decomposition for three-dimensional finite element elliptic problems. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 43-61. 70. Dryja, M. (1991). Substructuring methods for parabolic problems. In Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 264-271. 71. Dryja, M. and Widlund, O.B. (1989). Some domain decomposition algorithms for elliptic problems. In Iterative Methods for Large Linear Systems, L. Hayes and D. Kincaid, eds., Academic Press, San Diego, pp. 273-291. 72. Dryja, M. and Widlund, O.B. (1990). Towards a unified theory of domain decomposition algorithms for elliptic problems. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 3-21. 73. Dryja, M. and Widlund, O.B. (1992). Additive Schwarz methods for elliptic finite element problems in three dimensions. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, D.E.

348

REFERENCES

Keyes et al. eds., SIAM, Philadelphia, pp. 3-18. 74. Dryja, M. and Widlund, O.B. (1994). Some recent results on Schwarz type domain decomposition algorithms. In Domain Decomposition Methods in Science and Engineering, A. Quarteroni et al. eds., American Mathematical Society, Providence, pp. 53-61. 75. Dryja, M. and Widlund, O.B. (1995). Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Commun. Pure Appl. Math., 48, 121-155. 76. Dryja, M., Smith, B.F. and Widlund, O.B. (1994). Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal., 31, 1662-1694. 77. Duvaut, G. and Lions, J.-L. (1976). Inequalities in Mechanics and Physics. Springer-Verlag, Berlin. 78. Elman, H.C. and Golub, G.H. (1994). Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal., 31, 1645-1661. 79. Engquist, B. and Majda, A. (1977). Absorbing boundary conditions for the numerical simulation of waves. Math. Comput., 31, 629-651. 80. Faccioli, E., Maggio, F., Paolucci, R. and Quarteroni, A. (1997). 2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method. J. Seismology, 1, 237-251. 81. Faccioli, E., Maggio, F., Quarteroni, A. and Tagliani, A. (1996). Spectral domain decomposition methods for the solution of acoustic and elastic wave equations. Geophysics, 61, 1160-1174. 82. Farhat, C. and Roux, F.-X. (1991). A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng., 32, 1205-1227. 83. Farhat, C. and Roux, F.-X. (1994). Implicit parallel processing in structural mechanics. Comput. Mech. Adv., 2, 1-124. 84. Feistauer, M. and Schwab, C. (1996). On coupled problems for viscous flow in exterior domains. Research report No. 96-07, Seminar fur Angewandte Mathmatik, Eidgenossische Technische Hochshule Zurich. 85. Feistauer, M. and Schwab, C. (1998). Coupling of an interior Navier-Stokes problem with an exterior Oseen problem. Research report No. 98-01, Seminar fur Angewandte Mathmatik, Eidgenossische Technische Hochshule Zurich. 86. Fichera, G. (1972). Existence theorems in elasticity. In Handbuch der Physik, VIa/2, S. Flugge and C. Truesdell, eds., Springer-Verlag, Berlin, pp. 347-424. 87. Fischer, P.F. and R0nquist, E.M. (1994). Spectral element methods for large scale parallel Navier-Stokes calculations. Comput. Methods Appl. Mech. Eng., 116, 69-76. 88. Franca, L.P., Frey, S.L. and Hughes, T.J.R. (1992). Stabilized finite element methods, I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Eng., 95, 253-276. 89. Frati, A., Pasquarelli, F. and Quarteroni, A. (1993). Spectral approximation to advection-diffusion problems by the fictitious interface method. J. Comput. Phys., 107, 201-212.

REFERENCES

349

90. Funaro, D., Quarteroni, A. and Zanolli, P. (1988). An iterative procedure with interface relaxation for domain decomposition methods. SIAM J. Numer. Anal., 25, 1213-1236. 91. Garbey, M. (1996). A Schur alternating procedure for singular perturbation problems. SIAM J. Sci. Comput., 17, 1175-1201. 92. Garbey, M. and Kaper, H.G. (1997). Heterogeneous domain decomposition for singularly perturbed elliptic boundary value problems. SIAM J. Numer. Anal., 34, 1513-1544. 93. Gastaldi, F. and Gastaldi, L. (1994). On a domain decomposition for the transport equation: theory and finite element approximation. IMA J. Numer. Anal., 14, 111-136. 94. Gastaldi, F. and Gastaldi, L. (1995). Convergence of subdomain iterations for the transport equation. Boll. U.M.I., 9 - B , 175-202. 95. Gastaldi, F. and Quarteroni, A. (1989). On the coupling of hyperbolic and parabolic systems: analytical and numerical approach. Appl. Numer. Math., 6, 3-31. 96. Gastaldi, F., Gastaldi, L. and Quarteroni, A. (1996). Adaptive domain decomposition methods for advection dominated equations. East-West J. Numer. Math., 4, 165-206. 97. Gastaldi, F., Quarteroni, A. and Sacchi Landriani, G. (1990). On the coupling of two dimensional hyperbolic and elliptic equations: analytical and numerical approach. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 22-63. 98. Gear, C.W. (1971). Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs. 99. Gervasio, P., Ovtchinnikov, E. and Quarteroni, A. (1997). The spectral projection decomposition method for elliptic equations. SIAM J. Numer. Anal., 34, 1616-1639. 100. Girault, V. and Glowinski, R. (1995). Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Ind. Appl. Math., 12, 487514. 101. Girault, V. and Raviart, P.-A. (1986). Finite Element Methods for NavierStokes Equations. Springer-Verlag, Berlin. 102. Girault, V., Glowinski, R. and Lopez, H. (1997). Error analysis of a finite element realization of a fictitious domain/domain decomposition method for elliptic problems. East-West J. Numer. Math., 5, 35-56. 103. Glowinski, R. and Pan, T.-W. (1992). Error estimates for fictitious domain/penalty/finite element methods. Calcolo, 29, 125-141. 104. Glowinski, R., Pan, T.-W. and Periaux, J. (1994). A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng., I l l , 283-303. 105. Golub, G.H. and Mayers D. (1984). The use of pre-conditioning over irregular regions. In Computing Methods in Applied Sciences and Engineering, VI, R. Glowinski and J.-L. Lions, eds., North-Holland, Amsterdam, pp. 3-14.

350

REFERENCES

106. Golub, G.H. and Van Loan, C.F. (1989). Matrix Computations (2nd edn). The Johns Hopkins University Press, Baltimore. 107. Griebel, M. (1994). Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM J. Comput., 15, 547-565. 108. Hirsch, C. (1990). Numerical Computation of Internal and External Flows. Volume 2: Computational Methods for Inviscid and Viscous Flows. John Wiley & Sons, Chicester. 109. Israeli, M., Vozovoi, L. and Averbuch, A. (1993). Parallelizing implicit algorithms for time-dependent problems by parabolic domain decomposition. J. Sci. Comput., 8, 151-166. 110. Johnson, C. (1987). Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge. 111. Kantorovich, L.V. and Krylov, V.I. (1964). Approximate Methods of Higher Analysis. P. Noordhoff, Groningen. 112. Keyes, D.E. and Gropp, W.D. (1987). A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation. SIAM J. Sci. Stat. Comput., 8, sl66-s202. 113. Klawonn, A. (1996). Preconditioners for Indefinite Problems. Ph.D. Thesis, Westfalische Wilhelms-Universitat Miinster. 114. Klawonn, A. (1998a). Block-triangular preconditioners for saddle point problems with a penalty term. SIAM J. Sci. Comput., 19, 172-184. 115. Klawonn, A. (19986). An optimal preconditioner for a class of saddle point problems with a penalty term. SIAM J. Sci. Comput., 19, 540-552. 116. Klawonn, A. and Pavarino, L. (1998a). Overlapping Schwarz methods for mixed linear elasticity and Stokes problems. Comput. Methods Appl. Mech. Eng., 165, 233-245. 117. Klawonn, A. and Pavarino, L. (19986). A comparison of overlapping Schwarz methods and block preconditioners for saddle point problems. Technical report 15/98-N, Schriftenreihe Angewandte Mathematik und Informatik, Westfalische Wilhelms-Universitat Miinster. 118. Klawonn, A. and Starke, G. (1999). Block-triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis. Numer. Math., to appear. 119. Kuznetsov, Yu.A. (1988). New algorithms for approximate realization of implicit difference schemes. Sov. J. Numer. Anal. Math. Modelling, 3, 99-114. 120. Kuznetsov, Yu.A. (1989). Matrix iterative methods in subspaces. In Proceedings of the International Congress of Mathematics, Warsaw 1983, NorthHolland, Amsterdam, pp. 1509-1521. 121. Kuznetsov, Yu.A. (1991). Overlapping domain decomposition methods for FE-problems with elliptic singular perturbed operators. In Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 223-241. 122. Kuznetsov, Yu.A. (1995). Efficient iterative solvers for elliptic finite element problems on nonmatching grids. Russ. J. Numer. Anal. Math. Modelling, 10, 187-211.

\

REFERENCES

351

123. Kuznetsov, Yu.A. (1997). Iterative analysis of finite element problems with Lagrangian multipliers. In Computational Science for the 21st Century, M.-O. Bristeau et al. eds., John Wiley & Sons, Chichester, pp. 170-178. 124. Ladyzhenskaya, O.A. (1969). The Mathematical Theory of Viscous Incompressible Flow (2nd edn). Gordon and Breach, New York. Russian edn: Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961. 125. Landau, L.D. and Lifshitz, E.M. (1959). Fluid Mechanics. Pergamon Press, Oxford. Russian 2nd edn: Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953. 126. Lesaint, P. and Raviart, P.-A. (1974). On a finite element method for solving the neutron transport equation. In Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, pp. 89-123. 127. Lesieur, M. (1997). Turbulence in Fluids (3rd edn). Kluwer, Dordrecht. 128. Le Tallec, P. (1993). Neumann-Neumann domain decomposition algorithms for solving 2D elliptic problems with nonmatching grids. East- West J. Numer. Math., 1, 129-146. 129. Le Tallec, P. (1994). Domain decomposition methods in computational mechanics. Comput. Mech. Adv., 1, 121-220. 130. Le Tallec, P. and Patra, A. (1997). Non-overlapping domain decomposition methods for adaptive hp approximations of the Stokes problem with discontinuous pressure fields. Comput. Methods Appl. Mech. Eng., 145, 361-379. 131. Lie, I. (1992). Heterogeneous domain decomposition for acoustic bottom interaction problems. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, D.E. Keyes et al. eds., SIAM, Philadelphia, pp. 506-517. 132. Lions, J.-L. and Magenes, E. (1972). Non-homogeneous Boundary Value Problems and Applications 1, Springer-Verlag, Berlin. 133. Lions, P.-L. (1988). On the Schwarz alternating method I. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 1-42. 134. Lions, P.-L. (1989). On the Schwarz alternating method II: stochastic interpretation and order properties. In Domain Decomposition Methods, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 47-70. 135. Lions, P.-L. (1990). On the Schwarz alternating method III: a variant for non-overlapping subdomains. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 202-231. 136. Lube, G., Otto, F.C. and Miiller, H. (1998). A non-overlapping domain decomposition method for parabolic initial-boundary value problems. Appl. Numer. Math., to appear. 137. Maday, Y., Meiron, D., Patera, A.T. and R0nquist, E.M. (1993). Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations. SIAM J. Sci. Comput., 14, 310-337. 138. Mandel, J. (1993). Balancing domain decomposition. Commun. Appl. Numer. Meth., 9, 233-241.

352

REFERENCES

139. Marchuk, G.I., Kuznetsov, Yu.A. and Matsokin, A.M. (1986). Fictitious domain and domain decomposition methods. Sov. J. Numer. Anal. Math. Modelling, 1, 3-35. 140. Marini, L.D. and Quarteroni, A. (1988). An iterative procedure for domain decomposition methods: a finite element approach. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 129-143. 141. Marini, L.D. and Quarteroni, A. (1989). A relaxation procedure for domain decomposition methods using finite elements. Numer. Math., 55, 575-598. 142. Matsokin, A.M. (1972). Fictitious components and subdomain alternating methods. In Vychisl. Algoritmy v Zadachakh Mat. Fiz., V.V. Penenko, ed., Vychisl. Tsentr Sib. Otdel. Acad. Nauk USSR, Novosibirsk, pp. 76-88 [Russian]. English transl.: Sov. J. Numer. Anal. Math. Modelling, 5 (1990), 53-68. 143. Matsokin, A.M. and Nepomnyaschikh, S.V. (1985). A Schwarz alternating method in a subspace. Izv. VUZ Mat., 29 (10), 61-66 [Russian], English transl.: Sov. Math. (Iz. VUZ), 29 (10) (1985), 78-84. 144. Meurant, G.A. (1991). A domain decomposition method for parabolic problems. Appl. Numer. Math., 8, 427-441. 145. Mikhlin, S.G. (1951). On the Schwarz algorithm. Dokl. Akad. Nauk S.S.S.R., 77, 569-571 [Russian]. 146. Nataf, F. and Rogier, F. (1995). Factorization of the convection-diffusion operator and the Schwarz algorithm. Math. Models Methods Appl. Sci., 5, 67-93 147. Nedelec, J.-C. (1980). Mixed finite elements in R 3 , Numer. Math., 35, 3 1 5 341. 148. Nedelec, J.-C. (1986). A new family of mixed finite elements in R 3 , Numer. Math., 50, 57-81. 149. Nepomnyaschikh, S.V. (1984). On the application of the bordering method to the mixed boundary value problem for elliptic equations and on the mesh 1 /2 norms in W2 (S). Technical report 106, Computing Center of the Siberian Branch of the USSR Academy of Sciences, Novosibirsk [Russian]. English transl.: Sov. J. Numer. Anal. Math. Modelling, 4 (1989), 493-506. 150. Nepomnyaschikh, S.V. (1986). Domain Decomposition and Schwarz Methods in a Subspace for the Approximate Solution of Elliptic Boundary Value Problems. Ph.D. Thesis, Computing Center of the Siberian Branch of the USSR Academy of Sciences, Novosibirsk. 151. Oliger, J. and Sundstrom, A. (1978). Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math., 35, 419-446. 152. Ovtchinnikov, E. (1993). The construction of a well-conditioned basis for the Projection Decomposition method. Calcolo, 30, 255-271. 153. Ovtchinnikov, E. (1995). Projection decomposition method with a wellconditioned basis for the generalised Stokes problem. In Numerical Methods for Fluid Dynamics V, K.W. Morton and M.J. Baines, eds., Clarendon Press,

1

REFERENCES

«J«_>U

O x f o r d , pp. 515-521.

154. Pasciak, J.E. (1989). Two domain decomposition techniques for Stokes problems. In Domain Decomposition Methods, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 419-430. 155. Pavarino, L. (1998). Preconditioned mixed spectral element methods for elasticity and Stokes problems. SIAM J. Sci. Comput., 19, 1941-1957. 156. Pavarino, L. and Widlund, O.B. (1997). Iterative substructuring methods for spectral element discretizations of elliptic systems. II: mixed methods for linear elasticity and Stokes flow. Technical report n. 755, Courant Institute of Mathematical Sciences, New York University. SIAM J. Numer. Anal., to appear. 157. Perktold, K. and Rappitsch, G. (1995). Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. J. Biomech., 28, 845-856. 158. Pironneau, 0 . (1988). Methodes des Elements Finis pour les Fluides. Masson, Paris. 159. Prohl, A. (1997). Projection and Quasi-compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Teubner, Stuttgart. 160. Proskurowski, W. and Widlund, 0 . (1976). On the numerical solution of Helmholtz's equation by the capacitance matrix method. Math. Comput., 30, 433-468. 161. Przemieniecki, J.S. (1985). Theory of Matrix Structural Analysis. Dover, New York. 162. Quarteroni, A. (1989). Domain decomposition algorithms for the Stokes equations. In Domain Decomposition Methods, T. Chan et al. eds., SIAM, Philadelphia, pp. 431-442. 163. Quarteroni, A. (1990). Domain decomposition methods for systems of conservation laws: spectral collocation approximations. SIAM J. Sci. Stat. Comput., 11,1029-1052. 164. Quarteroni, A. and Sacchi Landriani, G. (1989). Domain decomposition preconditioners for the spectral collocation method. J. Sci. Comput., 3, 4 5 75. 165. Quarteroni, A. and Stolcis, L. (1995). Heterogeneous domain decomposition for compressible flows. In Proceedings of the ICFD Conference on Numerical Methods for Fluid Dynamics, M. Baines and W . K . Morton, eds., Oxford University Press, Oxford, pp. 113-128. 166. Quarteroni, A. and Valli, A. (1990). Domain decomposition for a generalized Stokes problem. In Proceedings of the Third European Conference on Mathematics in Industry, J. Manley et al. eds., Teubner, Stuttgart, pp. 59-74. 167. Quarteroni, A. and Valli, A. (1991a). Theory and applications of SteklovPoincare operators for boundary-value problems. In Applied and Industrial Mathematics, R. Spigler, ed., Kluwer, Dordrecht, pp. 179-203. 168. Quarteroni, A. and Valli, A. (19916). Theory and applications of SteklovPoincare operators for boundary-value problems: the heterogeneous operator case. In Fourth International Symposium on Domain Decomposition Methods

354

REFERENCES

for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 58-81. 169. Quarteroni, A. and Valli, A. (1994), Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin. 170. Quarteroni, A., Pasquarelli, F. and Valli, A. (1992). Heterogeneous domain decomposition: principles, algorithms, applications. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, D.E. Keyes et al. eds., SIAM, Philadelphia, pp. 129-150. 171. Quarteroni, A., Sacchi Landriani, G. and Valli, A. (1991). Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements. Numer. Math., 59, 831-859. 172. Quarteroni, A., Tuveri, M. and Veneziani, A. (1998). Computational vascular fluid dynamics: problems, models, methods. In preparation. 173. Raviart, P.-A. and Thomas, J.-M. (1977). A mixed finite element method for second order elliptic problems. In Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes, eds., Springer-Verlag, Berlin, pp. 2 9 2 315. 174. Raviart, P.-A. and Thomas, J.-M. (1983). Introduction a I' Analyse Numerique des Equations aux Derivees Partielles. Masson, Paris. 175. Rieder, A. (1997). On embedding techniques for 2nd-order elliptic problems. In Computational Science for the 21st Century, M.-O. Bristeau et al. eds., John Wiley & Sons, Chichester, pp. 179-188. 176. R0nquist, E.M. (1996). A domain decomposition solver for the steady Navier-Stokes equations. In ICOSAHOM.95, A.V. Ilin and R.L. Scott, eds., Houston Journal of Mathematics, Houston. 177. Saad, Y. (1996). Iterative Methods for Sparse Linear Systems. P W S Publishing Company, Boston. 178. Schenk, K. and Hebeker, F.K. (1993). Coupling of two dimensional viscous and inviscid incompressible Stekes equations. Preprint 93-68, Sonderforschungsbereich 359, Universitat Heidelberg. 179. Schwarz, H.A. (1869). Uber einige Abbildungsdufgaben. J. Reine Angew. Math., 70, 105-120. 180. Scroggs, J.S. (1990). A parallel algorithm for nonlinear convection-diffusion equations. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T.F. Chan et al. eds., SIAM, Philadelphia, pp. 373-384. 181. Secchi, P. and Valli, A. (1983). A free boundary problem for compressible viscous fluids. J. Reine. Angew. Math., 341, 1-31. 182. Seriani, G. and Priolo, E. (1994). Spectral element method for acoustic wave simulation in heterogeneous media. Finite Elem. Anal. Des., 16, 337-348. 183. Seshaiyer, P. and Suri, M. (1997). Uniform hp convergence results for the mortar finite element method. Technical report, Department of Mathematics and Statistics, University of Maryland, Baltimore County. Math. Comput., to appear.

REFERENCES

355

184. Shakib, F. and Hughes, T.J.R. (1991). A new finite element formulation for computational fluid dynamics. IX. Fourier analysis of space-time Galerkin/least-squares algorithms. Comput. Methods Appl. Mech. Eng., 87, 35-58. 185. Smith, B.F. (1990). Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University. 186. Smith, B.F. (1991). A domain decomposition algorithm for elliptic problems in three dimensions. Numer. Math., 60, 219-234. 187. Smith, B.F. (1992). An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems. SIAM J. Sci. Comput., 13, 364-378. 188. Smith, B.F., Bj0rstad, P.E. and Gropp, W.D. (1996). Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge. 189. Sobolev, S.L. (1936). Schwarz algorithm in the theory of elasticity. Dokl. Akad. Nauk S.S.S.R., 4, 235-238 [Russian]. 190. Stefanica, D. (1998). On the L 2 -stability of the 1-D mortar projection. Preprint, Courant Institute of Mathematical Sciences, New York University. 191. Taylor, C.A., Hughes, T.J.R. and Zarins, C.K. (1998). Finite element modeling of blood flow in arteries. Comput. Methods Appl. Mech. Eng., 158, 155— 196. 192. Temam, R. (1984). Navier-Stokes Equations. Theory and Numerical Analysis (3rd edn). North-Holland, Amsterdam. 193. Tong, P. (1970). New displacement hybrid finite element model for solid continua. Int. J. Numer. Methods Eng., 2, 73-83. 194. Toselli, A. (1999). Domain Decomposition Algorithms for the Finite Element Approximation of Maxwell's Equations. Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University. 195. Trotta, R.L. (1996). Multidomain finite elements for advection-diffusion equations. Appl. Numer. Math., 21, 91-118. 196. Widlund, O.B. (1988). Iterative substructuring methods: algorithms and theory for elliptic problems in the plane. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et al. eds., SIAM, Philadelphia, pp. 113-128. 197. Widlund, O.B. (1992). Some Schwarz methods for symmetric and nonsymmetric elliptic problems. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, D.E. Keyes et al. eds., SIAM, Philadelphia, pp. 19-36. 198. Wu, Y., Cai X.-C. and Keyes, D.E. (1998). Additive Schwarz methods for hyperbolic equations. In Domain Decomposition Methods 10, J. Mandel et al. eds., American Mathematical Society, Providence, pp. 468-476. 199. Xanthis, L. and Ovtchinnikov, E. (1994). Domain decomposition and the method of arbitrary lines for elliptic free boundary problems. In Proceedings of the Second Hellenic-European Conference on Mathematics and Informatics, Vol. I, E.A. Lipitakis, ed., Hellenic Mathematical Society, Athens, pp. 193-

356

REFERENCES

203. 200. Xu, C. (1996). A global algorithm in spectral methods for the coupled Navier-Stokes/Euler equations. In ICOSAHOM.95, A.V. Ilin and R.L. Scott, eds., Houston Journal of Mathematics, Houston, pp. 151-155. 201. Xu, J. (1992). Iterative methods by space decomposition and subspace correction. SIAM Review, 34, 581-613. 202. Xu, J. and Zou, J. (1998). Some nonoverlapping domain decomposition methods. SIAM Review, 40, 857-914. 203. Yosida, K. (1974). Functional Analysis (4th edn). Springer-Verlag, Berlin. 204. Zhang, X. (1992). Multilevel Schwarz methods. Numer. Math., 63, 521-539.

INDEX advection operator, 203, 268 advection-diffusion operator, 219, 239,287 agglomeration multi-grid, 100 Agoshkov-Lebedev iterative method, 16, 17 Banach space, 335, 337 Bernoulli equation, 304 (3- Robin-Neumann formulation, 223 iterative method adaptive, 228, 231 damped adaptive, 232 block-Gauss-Seidel iteration, 91, 93, 100 block-Jacobi iteration, 93, 100 Burgers equation, 295

Crank-Nicolson / Adams-Bashforth method, 273 Dirichlet-Neumann formulation, 221 iterative method, 5, 11, 13, 14, 17, 21, 65, 71, 74, 75, 77, 91, 118, 128, 133, 146, 147, 151, 153, 170-172, 177, 179, 184, 188, 194, 197, 201, 203, 214, 258, 302, 303, 314, 320, 327 adaptive, 228, 245 damped adaptive, 232, 246 for many subdomains, 24 preconditioner, 53, 54, 75, 79, 107, 110, 129, 178 for many subdomains, 86 distribution, 336 derivative of a, 336 dual space, 334 duality pairing, 334

Cauchy-Schwarz inequality, 115, 139 characteristic curve, 261, 269, 270, 279, 281, 308, 313, 315, 316 matrix, 269, 308, 311, 312, 316 variable, 268, 279, 281, 308, 312, 313

elasticity operator, 191 elliptic operator non-symmetric, 141 symmetric, 18, 28, 34, 133, 135, 252 "

Chorin-Temam method, 239, 274 coarse grid, 77, 86, 95, 257, 258, 268 compatibility condition, 20, 21, 149, 156, 159, 161 compatibility equations, 282 condition number, 51, 57, 107, 126, 257, 268 conjugate gradient method, 38, 51, 57, 66, 69, 76, 94, 99, 101, 118, 127, 178, 190, 322 conservation law, 261 Crank-Nicolson method, 260

equilibrium equations, 148 Euler equations for inviscid compressible flow, 273, 277, 286, 305-309, 311, 317 for inviscid incompressible flow, 305 for isentropic inviscid compressible flow, 279, 312, 314 extension harmonic, 3, 104

358

finite element, 46 matrix, 57, 88 operator, 7, 32, 44, 47, 71, 72, 104, 109, 112, 114, 115, 147, 237, 290, 325 uniform, 47, 105, 111, 113, 116, 247 Stokes, 162 finite element, 176 F E T I method, 45 fictitious domain method, 34, 36 finite element approximation, 41, 42, 46, 71, 101, 152, 174, 179, 181, 196, 210, 213, 244 full potential equation, 286, 301, 306, 309, 311-314, 317 G A L S stabilisation method, 249 7-Dirichlet-Robin iterative method, 234, 240, 246 7 - R o b i n - R o b i n iterative method, 240, 248 G M R E S method, 95, 99, 122-124, 249, 295 Green formula, 340 Green function, 257 Hilbert space, 61, 111, 114, 117, 138, 197, 204, 211, 295, 329, 334-336, 338 hyperbolic operator, 203, 261, 276 first-order, 268 inexact solver, 96 inf-sup condition, 67, 68, 156, 158, 159, 165, 173, 174, 177, 185, 189 integral matching conditions, 60 interface conditions, 2, 4, 19, 142, 148, 155, 191, 198, 203, 211, 217, 263-265, 276, 277, 285, 289, 296, 302-304, 307, 308, 318, 329

INDEX equation, 2, 5 interpolation error estimate, 106, 110 inverse inequality, 48, 106, 111 Korn inequality, 149, 151, 159, 163, 176 Krylov iterative method, 77, 95, 99, 272 Lagrange multiplier, 36, 38-40, 45, 66-68 Laplace operator, 1, 71, 104 Lax-Milgram lemma, 18, 43, 61, 108, 119, 192, 199, 211, 334, 335 leapfrog method, 271 linear elasticity operator, 147 Lp space, 335 LU decomposition, 52, 58 Maxwell equations, 114, 129, 210, 286, 328 mixed-type formulation, 111, 191, 198 mortar method, 60, 68 Nedelec finite elements div-conforming, 112, 202, 322 rot-conforming, 116, 213, 331 Navier-Stokes equations for compressible flow, 190, 191, 273, 275, 286, 305-308, 317 for incompressible flow, 155, 220, 239, 273, 300, 301, 303, 305 Neumann-Neumann iterative method, 5, 14, 22, 65, 72, 75, 77, 118, 128, 135, 146, 147, 151, 153, 172, 178, 180, 185, 195, 197, 201, 203, 216, 259 preconditioner, 75, 76, 79, 108, 110, 130, 178 for many subdomains, 84 Newton method, 272

INDEX n o r m , 333

orthogonal projection, 30, 89, 137, 168, 322 Oseen operator, 155, 286, 301, 303 parabolic operator, 252, 276 Poincare inequality, 6, 8, 18, 23, 47, 48, 61, 105, 149, 340 Poisson problem, 1, 2, 7, 41, 56, 59, 77, 274 polygonal domain, 41 preconditioner, 53, 57, 58, 74, 77, 98 additive Schwarz, 93, 257 balancing Neumann-Neumann, 85 block-Jacobi, 80 Bramble-Pasciak-Schatz, 82 Dirichlet-Neumann, 53, 54, 75, 79, 107, 110, 129, 178 for many subdomains, 86 Dryja, 78 Golub-Mayers, 78 interface, 79 multiplicative Schwarz, 95, 258 Neumann-Neumann, 75, 76, 79, 108, 110, 130, 178 for many subdomains, 84 optimal, 47, 51, 107, 108, 110 Probing, 79 restricted additive Schwarz, 96 scaleable, 79 Schwarz, 84, 99, 100 Vertex Space, 83 Wire-Basket, 84 predictor-corrector method, 259, 260, 272 pressure approximation continuous, 185 discontinuous, 173 Projection Decomposition method, 101 Raviart-Thomas finite elements, 112, 202, 322

restriction matrix, 57, 88 operator, 32 Reynolds number, 306 Richardson iterative method, 13-15, 31, 57, 74, 76, 77, 94, 118, 120-122, 124, 125, 127, 128, 130, 132, 146, 151, 153, 170, 172, 173, 178, 181, 184, 195, 197, 201, 203, 209, 214, 216, 237, 294, 295, 322, 326, 329 minimum-residual, 127 steepest-descent, 127 Riemann invariant, 280, 282, 312-315 Riesz theorem, 334 Robin iterative method, 5, 16, 17, 135 Robin-Neumann formulation, 222 iterative method adaptive, 228, 230, 244, 246 damped adaptive, 232, 246 saddle point problem, 65 scalar product, 333, 336, 338 Schur complement matrix, 50, 53, 56-58, 68, 77, 128, 258, 295 system, 51, 66, 68, 74, 77, 79 Schwarz inequality, 333 Schwarz method additive, 27, 30, 31, 90, 91, 93, 216, 267 for many subdomains, 33, 90 alternating, 26, 28, 31, 32, 76, 86, 137, 139 multiplicative, 27, 29-31, 86, 88, 90-92 for many subdomains, 33, 90 Schwarz preconditioner, 84, 99, 100 additive, 93, 257 multiplicative, 95, 258 restricted additive, 96

INDEX semi-discretisation continuous-in-space, 256 continuous-in-time, 254 seminorm, 333 Sobolev embedding theorem, 341 Sobolev space, 5, 337-339, 341 spectral approximation, 171, 305 spectral radius, 126, 130, 132, 134 splitting of operators, 117 static condensation, 51 Steklov-Poincare equation, 3, 9, 10, 14, 15, 22, 23, 74, 75, 101, 143, 150, 152, 167, 169, 170, 173, 176, 183, 194, 195, 197, 200, 201, 203, 206, 207, 214, 216, 290, 294, 295, 319, 326, 327, 329 operator, 2, 3, 5, 9, 22, 23, 38, 46-48, 50, 57, 66, 78, 104-107, 109-111, 118, 143, 144, 147, 150, 152, 161-164, 167, 169, 175, 182, 184, 194, 196, 200, 202, 207, 209, 237, 291, 292, 318, 320, 322, 325, 327 stiffness matrix, 45, 58 local, 57 Stokes operator, 153, 286, 297, 301 for compressible flow, 190, 317 for inviscid compressible flow, 197 subdomain colouring, 91 substructuring iterative method, 10, 71, 208, 224, 270, 280, 283, 289, 299, 308, 313-315, 326, 331 adaptive, 227 coercive, 234 SUPG stabilisation method, 249

three-field method, 38, 66, 68 time discretisation, 266 trace

inequality, 6, 8, 23, 47, 104, 106, 110, 112-115, 117, 144, 151, 163, 176, 194, 237, 321 of a function, 6, 23 space, 6 normal, 111 tangential, 114 theorem, 110, 339 normal, 339 tangential, 340 transmission conditions, see interface conditions two-level method, 96, 97, 99 preconditioner, 98 additive, 98 multiplicative, 98 Uzawa pressure operator, 189 wave acoustic, 262, 264, 267 elastic, 262, 265, 271 waveform relaxation, 255 weak convergence, 138, 334