Numerical Methods for Exterior Problems (Peking University Series in Mathematics) (Peking University Series in Mathematics)

Peking University liffiajll

Series in Mathematics — Vol. 2

NUMERICAL METHODS FOR EXTERIOR PROBLEMS Ying Lung-An

NUMERICAL METHODS FOR EXTERIOR PROBLEMS

PEKING UNIVERSITY SERIES IN MATHEMATICS Series Editor: Kung-Ching Chang (Peking University, China)

Vol. 1: An Introduction to Finsler Geometry by Xiaohuan Mo (Peking University, China) Vol. 2: Numerical Methods for Exterior Problems by Ying Lung-An (Peking University & Xiamen University, China)

Peking University fjqSSskjg?,ies T^y

in

Mathematics — Vol. 2

UMERICAL METHODS Ying Lung-An Peking University & Xiamen University, China

P World Scientific Hr

W H S

NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI

- HONG KONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

NUMERICAL METHODS FOR EXTERIOR PROBLEMS Peking University Series in Mathematics — Vol. 2 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-270-218-0 ISBN 981-270-526-0 (pbk)

Printed in Singapore by Mainland Press

Preface

Partial differential equations play an important role in mathematical physics. Usually some initial conditions or boundary conditions are imposed on the equations, and we are dealing with initial value problems, boundary value problems, or initial-boundary value problems. There are some different points of view to classify the problems: Linear or nonlinear, first order equations or higher order equations, steady problems or evolution problems, interior problems or exterior problems. The first three are according to the properties of the equations, and the last one is according to the properties of the underlying physical domains. For example, the flow problem in a chamber is an interior problem, and the flow problem around an aircraft is an exterior problem. All partial differential equations possess infinite number of degrees of freedom. To solve a problem numerically, one should approximate the problem so that only a finite number of degrees of freedom are solved in real computation. As a result a truncation error appears in every numerical scheme. This fact is the same for both interior and exterior problems. However when one intends to solve an exterior problem, one faces another kind of difficulty that the domain is infinitely large. Most approaches for the interior problems can not be applied to the exterior problems directly. The importance of exterior problems is obvious, and the challenge is serious. There are two kinds of approaches to deal with the exterior problems: to truncate the domain, or to solve the problem directly on the infinite domain. The former ones are: the introducing of artificial boundary conditions, and the introducing of perfectly matched layers. The later ones are: the infinite element method, and the spectral method. While in the boundary element method both approaches are applied, depending on the degree of complexity of the domains.

V

vi

Numerical Methods for Exterior

Problems

The aim of this monograph is to provide a comprehensive study of different approaches. At the beginning we investigate the mathematical theory of the exterior problems of some typical partial differential equations, which provides the foundation of all numerical methods. Restricted by the space we only state the basic results. The readers can refer to the literature for the details. The main body of the book is on some traditional and new methods. Each of them is effective, and certainly restricted to some particular classes of problems. One omission of the book is that no numerical example is included, because there are too many in the literature, and it is too difficult to choose a few, representing most of the problems and most of the methods. It is also due to the restriction of space. However I am not sure if it is appropriate to do so. This is an area growing rapidly. Limited by the author's knowledge, some important approaches must have been ignored. There is a proverb in China:" Cast away a brick and attract a jadestone". I will be pleased to listen to all comments. My friends, Professors Zhang Guan Quan, Guo Ben Yu, Yu De Hao, and Shen Jie kindly provided me their works on this subject. Without their help I could not finish this work. My students, Fang Nengsheng and Liao Caixiu, helped me to proof-read and illustrate the manuscript. I express my sincere thanks to them. Dr. Sim, the editor, helped me a lot for the style file, and Ms. Zhang Ji, the editor, helped me to prepare the final manuscript. I am grateful to them. Ying Lung-an

Contents

Preface

v

1.

1

Exterior Problems of Partial Differential Equations 1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8 1.9 1.10 1.11 1.12 2.

Harmonic equation-potential theory Poisson equations Poisson equations-variational formulation Helmholtz equations Linear elasticity Bi-harmonic equations Steady Navier-Stokes equations -linearized problems . . . 1.7.1 Navier-Stokes equations 1.7.2 Stokes equations 1.7.3 Behavior of solutions at the infinity 1.7.4 Stokes paradox 1.7.5 Oseen flow Steady Navier-Stokes equations Heat equation Wave equation Maxwell equations Darwin model

2 12 13 17 25 29 35 35 36 39 41 41 44 49 53 56 61

Boundary Element Method

71

2.1

71 71 75 77

Some typical domains 2.1.1 Harmonic equation 2.1.2 Bi-harmonic equation 2.1.3 Stokes equation vii

viii

Numerical

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.

Problems

2.1.4 Plane elasticity 2.1.5 Helmholtz equation General domains Subdivision of the domain Dirichlet to Neumann operator Finite part of divergent integrals Numerical approximation Error estimates Domain decomposition Boundary perturbation

80 82 85 93 96 98 103 108 113 114

Infinite Element Method

117

3.1

117 117 120 127 131 133 134 136 138 140 142 147 152 152 155 158 161 162

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

3.10 4.

Methods for Exterior

Harmonic equation-two dimensional problems 3.1.1 Infinite element formulation 3.1.2 Tranfer matrix 3.1.3 Further discussion for the transfer matrix 3.1.4 Combined stiffness matrix General elements Harmonic equation-three dimensional problems Inhomogeneous equations Plane elasticity Bi-harmonic equations Stokes equation Darwin model Elliptic equations with variable coefficients 3.9.1 A homogeneous equation 3.9.2 An inhomogeneous equation 3.9.3 General multiply connected domains 3.9.4 Transfer matrices Convergence

Artificial Boundary Conditions

167

4.1 4.2 4.3 4.4 4.5 4.6

167 172 175 176 178 178

Absorbing boundary conditions Some approximations Bayliss-Turkel radiation boundary conditions A lower order absorbing boundary condition Liao extrapolation in space and time Maxwell equations

Contents

4.7 4.8

5.

6.

Finite difference schemes Stationary Navier-Stokes equations 4.8.1 Homogeneous boundary condition at the infinity . 4.8.2 Inhomogeneous boundary conditions at the infinity 4.8.3 A linear boundary condition

ix

182 183 183 186 187

Perfectly Matched Layer Method

191

5.1 5.2 5.3 5.4 5.5 5.6

191 197 201 208 210 212

Wave equations Berenger's perfectly matched layers Stability analysis Uniaxial perfectly matched layers Maxwell equations Helmholtz equations

Spectral Method

217

6.1 6.2 6.3

217 225 230 230

6.4 6.5 6.6

Introduction Orthogonal systems of polynomials Laguerre spectral methods 6.3.1 Mixed Laguerre-Fourier spectral method 6.3.2 Spherical harmonic-generalized Laguerre spectral method 6.3.3 Generalized Laguerre pseudo-spectral method . . 6.3.4 Nonlinear equations Jacobi spectral methods Rational and irrational spectral methods Error estimates

235 237 239 241 243 245

Bibliography

251

Index

265

Chapter 1

Exterior Problems of Partial Differential Equations

This chapter is devoted to the theory on well posedness of the exterior problems of some important partial differential equations in mathematical physics. The emphasis is the boundary conditions at the infinity, which play an essential role in the mathematical theory, and differ from different equations and dimensions. An appropriate numerical method must reflect these boundary conditions correctly. In this respect this chapter is the foundation of the other chapters. On the other hand this chapter provides some tools for numerical methods. For example, the potential theory is the basic tool in the boundary element method. Another example is the variational formulation, which is the starting point of the Galerkin method, and the infinite element method and the spectral method belong to this catalogue. We investigate some typical equations only. No attempt has been made to provide a comprehensive theory on general equations. For example in many cases we consider equations with constant coefficients instead of that with variable coefficients. The reason is that we intend to show the most important facts by means of some simple models. Secondly it is obvious that the complete theory for some general equations may deserve an entire book, which is not our objective. The theory of the Navier-Stokes equations is the subject of a number of outstanding works. We will briefly present some basic results for the exterior problems of the Navier-Stokes equations. We will consider a bounded domain D.c in two dimensional space M.2 or in three dimensional space K 3 . Let dQ, be its boundary. For simplicity we assume that dfl is a connected curve or surface, which is Lipschitz continuous and piecewise smooth. Let 17 be the exterior part of the closed domain fic, the complement, then Q is an unbounded domain. The dimension of the domain f2 is denoted by dim(fi).

1

Numerical Methods for Exterior

Problems

•x

x

Fig. 1 If the space dimension is one, the domain Q.c is reduced to an interval (a, b), and the domain Q is divided into two independent parts: (—00, a) and (b, +00). We consider the latter one, and it has no harm in assuming b = 0, so we will consider the interval (0, 00). In the sequel it will be assumed that functions and some related spaces will always be real-valued, unless we explicitly state that some functions are complex-valued in some individual sections. We will always denoted by C a generic constant that may not be the same at different occasions.

1.1

Harmonic equation—potential theory

First of all we investigate a typical exterior boundary value problem: the exterior problem of the Laplace equation: Ai

where for two dimensional case, A Q2

(1.1)

0, • + •£=?, and a; = {xl,x2) O2

G

and

O2

for three dimensional case, A = -g^ + -g^ + j ^ , and x = (x\, X2, £3) G . We consider the Dirichlet boundary condition: u = g,

x £ d£l,

(1.2)

Exterior Problems of Partial Differential

Equations

3

where g is a continuous function denned on dQ,. We also consider the Neumann boundary condition:

P = g,

xeffl,

(1.3)

ov where g is also a continuous function defined on dfl, and v is the unit outward normal vector. Let us consider classical solutions of the problems. A function u is a solution to the Dirichlet problem, if it is continuous on Q, twice continuously differentiable on f2, and it satisfies the equation (1.1) and the boundary condition (1.2). A function u is a solution to the Neumann problem, if it and its first order derivatives are continuous on fi, twice continuously differentiable on Q, and it satisfies the equation (1.1) and the boundary condition (1.3). However we are interested in well-posed problems, then some boundary conditions at the infinity are required. J u is a bounded function on fl, dim(fi) — 2, . . \ lim^i^oo u = 0, dim(fi) = 3. We study uniqueness first. The strong maximum principle is applied here. Theorem 1. Let G be a bounded domain, u satisfies the equation (1.1) in G, and u is continuous on the closed domain G, then if u assumes its maximum (minimum) value at an interior point x £ G, then u is a constant on G. Moreover, if the first order derivatives ofu are continuous on G, u is not a constant, and u assumes its maximum (minimum) value at a boundary point x £ dG, then | ^ > 0(< 0) at this point. Theorem 2. Ifu\,U2 are two solutions satisfying (1.1),(1.2) and (1-4), or if dim(Cl) = 3 andui,U2 are two solutions satisfying (1.1),(1.3) and (1-4), then u\ = u 0, then V\QQ > 0. v is also a harmonic function. For an arbitrary x £ Q, let R\ > Ro be large enough, such that x £ B(0, Ri) and v\r=n1 > 0, then by the maximum principle v > 0 on 0,f]B(O,Ri).

4

Numerical Methods for Exterior

Problems

Therefore u(x) > —e\n(r/Ro). Being the same u(x) < eln(r/Ro). But e is arbitrary, so u{x) = 0, and x is arbitrary, so u = 0 on fl. 2. Three dimensional Dirichlet problem The proof is analogous. We take an auxiliary function v = u + £, then following the same lines we can prove v > 0, then get u = 0 on CI. 3. Three dimensional Neumann problem Being the same we take an auxiliary function v = u + e. Then we consider a ball B(0, R\) D Clc. Let i?i be large enough, such that v\r=R1 > 0. If v is a constant on Q. f] B(0, R\), then v > 0 on this domain. If v is not a constant and if v assumes its minimum value at a point x e dCl, then by the strong maximum principle f^ < 0, but ^ = 0 on the boundary dfl, so it is impossible. Therefore v assumes its minimum value at r = Ri and also v > 0 on nf]B(0,Ri). Then in either case u(x) > -e on Qf]B(0,Ri). Being the same u(x) < e. But e is arbitrary, so u = 0 for |a;| < R\. R\ can be arbitrary large, so u = 0 on fl. • Two dimensional Neumann problem is more complicated. We have the following result: Theorem 3. / / dim(Cl) = 2 and (1.1),(1.3) and (1-4) is solvable, then L„ g(x) dsx = 0, and ifui, u^ are two solutions to this problem, then u\—U2 is a constant on Ct. Proof.

Let u be a solution to the problem. Let B(0, Ro) C fic as the R2

previous. We define v(x) = u(r^x), and denote by fii and dfli the image of fi and dfl under the mapping x —* j^px, then v is also a harmonic function on Cii and satisfies the Neumann boundary condition (1.3) on dfli. We notice that O is a boundary point of fii, and we will show that it can be removed, that is, if we set Q.? = fli IJ{^} a n ( ^ define a suitable value of v(0), then v is a harmonic function on 1^2• We consider a small disk B{0, R2) C f^, and solve a Dirichlet boundary value problem on it with the boundary value t i | r = ^ 2 . Let the solution be iv, which can be expressed in terms of the Poisson integral: (see Section 2.1, for example) V ( m - ^ 2 - P 2 [2v W W W j ~ 2TT J0 R* + p*-2R2Pcoa{6-0')

M

'

where (p, 9) are polar coordinates. Using the auxiliary functions v — w ± eln(r/i?2) and the same argument as the previous theorem, we can verify that U = I » O D B(Q,R2) \ {O}. Therefore suffice it to set v(0) = w(0).

Exterior Problems of Partial Differential

5

Equations

Integrating the equation (1.1) on CI2 we get JQa | ^ dsx = 0. Returning to the domain f2, it can be verified that Jdn g(x) dsx = 0. Let u be a solution to the corresponding homogeneous problem. We claim that u is a constant. Because v is the solution to (1.1) on CI2 with homogeneous Neumann boundary condition, then if v were not a constant then it should assume its maximum value at one point of the boundary dQ,\ and | ^ > 0 at this point, which is impossible. Therefore v is a constant and so is u. • We notice that the boundary conditions at the infinity (1.4) are different for different dimensions. This fact is essential. For example let fi = {x; \x\ > 1} and we consider the Dirichlet boundary condition u \dQ = 1- We have one solution u = 1 in two dimensional case. Since bounded solution is unique, there is no solution satisfying lim^i^oo u = 0. On the other hand bounded solutions are not unique for this example in three dimensional case, because we have two solutions u = 1 and u = 1/r, where r = \x\. We apply potential theory to study existence. Consider two dimensional case first. For simplicity we assume that dfl is a smooth curve and we will relax this restriction for the variational formulation later on. It is known that the fundamental solution is u = J- In -, which is the solution to the equation -Au

= 6,

(1.5)

where S is the Dirac ^-function. By this fundamental solution, the solution to the Neumann boundary value problem is expressed in terms of a single layer potential:

U{x) =

hjdnu{y)ln\x^\ds^

(L6)

where sy is the differential on the curve, and the solution to the Dirichlet boundary value problem is expressed in terms of a double layer potential: u x

( ) = 7T / oo. Proof.

The single layer potential can be rewritten as

u x

w ( ) — 7^ (y) In ~, r dsv -\ In -. / K v ' 2* Jen KJ \x-y\ 2TT \x-y0\Jdn

uj(y) dsv, yy > yi

where yo S flc is a fixed point. Since lim|;c|_>00 In -f^zij = 0 uniformly with respect to y £ dQ and limi^i^oo In -n^z—r = —oo, the conclusion follows immediately. • The kernel of the double layer potential is not weakly singular, so it does not define a continuous function on M.2. However one property of the double layer potential is the following: Lemma 2. For the double layer potential (1.7) if a is a continuous function on dQ,, then the following limits exist: lim u{x) = -a(x0) 2 _ > Xo x Gfl,xo £ dCl

+ — / a(y)—\n2?r JdQ dvy

x

lim u(x) = --o-(x0) 2 x->x0 x € f2c, x0 S d£l Proof.

+— 2?r

J™

-dsy, \x0 - y\

a{y)-—ln-dsy. dv y \x° ~ y\

We consider the case of cr = 1 first. For x ^ y it holds that d

1

duy

\x — y\

_ (x - y) • uy _ COS((:E - y),

\x — y\2

vy)

\x — y\

For a fixed point i £ I 2 , we construct polar coordinates (p, 0) with the origin x. Then for y G d£l dB =

_coB{{x-y)uy) \x-y\

Exterior Problems of Partial Differential

7

Equations

Therefore -2TT,

In-

dl/ / .dn y

dsv

-IT,

\x-y\

x G Qc, x G dQ.,

0,

(1.8)

iea

For general cases let XQ G dQ, be a fixed point. We set v x

()

d = 7T2n

Jan

( 0, we consider a local neighborhood U oi XQ, then take an arc s C dQ, XQ G S, and s is small enough so that

^(a(y)-,(*0)) A h _ ! _ * , ,

< e

for all x G U. Then we take x close enough to xo, such t h a t — / (o~(y) — o-(x0)) < —— In 1*Jan\a \dvy \x-y\

: — - — In -. r > ds% < e. y duy \x0 - y\ j

Then

M

TJJ^-'^W*-*l n

i

_ J L In dv„

ds, FO

is a continuous

+ — / w(2/)—In 27r xo Jon + — 2TT Jan

uj(y)-—\ndvx

•day,

function

(1.9)

-y\

-dsy. \x0 - y\

(1.10)

8

Numerical Methods for Exterior

Problems

Proof. For each point x £ dQ we consider the straight line perpendicular to the curve, then we take derivatives along this line. By this way we extend the normal derivatives -Jj^- to a neighborhood of dCl. Let v x

( ) = 7T I 2TT

W

(V) \ ^—

Jon

\dvx

ln

1 \x-y\

f + -5— l n 1 dvy

f?

ds

y

\x - y\ J

For xo S dCl the kernel _d_ 1 d 1 _ cos((y - x), vx) + cos((x - y), vy) dvx \x-y\ dvy \x-y\ \^-y\ is bounded near XQ. Then following the same argument as the previous lemma we can show that v is continuous at XQ, then applying the results of the previous lemma we get the jump results. • Having expressed the solutions of the Dirichlet and Neumann boundary value problems in terms of double layer potentials and single layer potentials, we apply the previous lemmas and get the integral equations for the densities a and u>: 1. Dirichlet exterior problem: -a(x)+ *

a(y)K(x,y)dsy=g(x),

(1.11)

Jan

where K(x, y) = £.£- In ^ . 2. Dirichlet interior problem: -l B be a compact operator. Then the following alternative holds. Either, 1. There is a unique solution u £ B satisfying the equation u + Ku = f, for all f e B. Or 2. The corresponding homogeneous equation u + Ku = 0 admits k > 0 linearly independent solutions, its dual equation v + K*v = 0, v G B*, also admits k linearly independent solutions, i.e., the dimensions of null spaces are equal, and the equation u + Ku = f is solvable if and only if vf = 0,VveB*,v + K*v = 0. Theorem 5. The integral equations (1.12) and (1.13) admit unique solutions. Proof. Let w be a solution to the homogeneous equation associated with (1.13). Then by the equation - - / u>(x)dsx+ / / cj(y)K(y,x)dsydsx ^ Jan Jan Jdn

= 0.

By exchanging the order of integrals and noting (1.8) we have OJ(X)

dsx = 0.

JdS lan

With this density u> we consider a single layer potential u defined by (1.6). We notice that u> is a solution to the homogeneous equation. Then by Lemma 3 we get ^ j r = 0, . By the uniqueness of Neumann exterior problems u is a constant on CI, then by the uniqueness of Dirichlet interior problems u is equal to the same constant on f2c, so -^r = 0. We apply Lemma 3 again and get w = 0. Then by the Fredholm theorem the equations admit unique solutions. • Theorem 6. The homogeneous integral equation associated with(l.ll)or (1.14) admit one nonzero solution, and any two solutions are linearly dependent. Proof. The homogeneous equation associated with (1.11) has a nonzero solution a = 1. By the Fredholm theorem the homogeneous equation associated with (1.14) also admits nonzero solutions. Let ui be one of them, then we claim that / u>(x) dsx ± 0. Jan

10

Numerical Methods for Exterior

Problems

Otherwise with this density w we consider s a single layer potential u defined by (1.6), then by Lemma 3 we get -^ = 0 . By the uniqueness of Neumann interior problems u is a constant on f2c, then by Lemma 1 and the uniqueness of Dirichlet exterior problems u is equal to the same constant on fi, so -g^f = 0. We apply Lemma 3 again and get u = 0, which is not a nonzero solution. Let u>i,cJ2 be two solutions to the homogeneous equation associated with (1.14). We consider a linear combination c\U\ + C2W2, cf + c\ ^ 0, such that Jgn(ciu>i(x) + C2U>2(x)) dsx — 0. Then by the above argument c\W\ + C2LO2 = 0) so they are linearly dependent. • Theorem 7. The problem (1.1),(1.2),(1.4)

admits a unique solution.

Proof. The uniqueness has been given. We prove existence here. We take a constant c such that Jdn(g(x) + c)ui(x) dsx = 0, where w is a nonzero solution to the homogeneous equation associated with (1.14). Then we solve the equation (1.11) with a right hand side g + c. Owing to the Predholm alternative theorem it is solvable. The desired solution to (1.1),(1.2),(1.4) is u — c, where u is the double layer potential (1.7). • Theorem 8. The sufficient and necessary condition for (1.1),(1.3),(1.4) admitting a solution is Jdn g(x) dsx = 0, and the solutions are unique up to adding an arbitrary constant. Proof. We solve the equation (1.13), then construct a single layer potential u. If Jdng(x)dsx — 0, then one finds that Jgiloj(x)dsx = 0 by integrating both sides of (1.13). Then by Lemma 1 u is bounded, so it satisfies the boundary condition (1.4). • Next we investigate three dimensional problems. Using single layer potential and double layer potential we also get integral equations (1.11)(1.14), where K(x, y ) = X _ | _ _ i _ . Parallel to two dimensional case, we have the following theorems. We will omit most of the proof and only indicate the difference in the argument. Theorem 9. The integral equations (1.12) and (1.13) admit unique solutions. Theorem 10. The homogeneous integral equation associated with (1.11) or (1.14) admit one nonzero solution, and any two solutions are linearly dependent. Proof.

Being the same they admit nonzero solutions.

Exterior Problems of Partial Differential

Equations

11

Let wj,W2 be two solutions to the homogeneous equation associated with (1.14). Let ui,U2 be two single layer potentials with densities wi,W2They are constants in flc. We consider a linear combination c\W\ + C2W2, Cj + c | 7^ 0, such that u — c\Ui + C2U2 = 0 in fic. Since u = 0 on dil, by the uniqueness of the Dirichlet problems, u = 0 on Cl. Therefore ciu>\ + C20J2 = 0. They are linearly dependent. • Theorem 11. The problem (1.1),(1.2),(1.4)

admits a unique solution.

Proof. The uniqueness has been given. We prove existence here. We take a constant c such that fan(g(x) +c)w(x) dsx — 0, where w is a nonzero solution to the homogeneous equation associated with (1-14). Then we solve the equation (1.11) with a right hand side g + c. We may assume that the solution ui constructed in the proof of Theorem 10 satisfies u = l in Q,c. Then the desired solution is u — cui, where u is the double layer potential (1.7). • Theorem 12. The problem (1.1),(1.3),(1.4)

admits a unique solution.

Proof. This is because the corresponding integral equation admits a unique solution. • The results in this section are collected in the following tables: Table 1. Boundary value problems condition for solutions to expressions existence homogeneous problems 2-D Dirichlet none 0 double layer — c 2-D Neumann Jgds = 0 constant single layer +C 3-D Dirichlet none 0 double layer — cu\ single layer 3-D Neumann none 0 Table 2. Integral equations

2-D Dirichlet 2-D Neumann 3-D Dirichlet 3-D Neumann

condition for existence / gu: ds = 0 none J gu> ds = 0

none

solutions to homogeneous equations constant 0 constant 0

12

1.2

Numerical Methods for Exterior

Problems

Poisson equations

We investigate the Poisson equation, -Au

= f,

x£tt,

(1.15)

in this section. For simplicity we assume that / £ Co(fi), that is, / is continuous with a bounded support. The uniqueness of the problem (1.15),(1.2),(1.4) or the problem (1.15),(1.3),(1.4) comes from the results in the previous section directly, so we study existence here. For two dimensional case the Newton potential is 1

«(*) = T" )w / f(v)ln rk -^ v\ dv>

( L16 )

where / is extended to the entire space. If / is extended by zero, the integral is in fact taken on fi. It can be verified that u satisfies the equation (1.15). Analogous to Lemma 1, we have Lemma 4. For the Newton potential (1.16), the limit lim^i-nx, u(x) = 0 if and only if JR2 f(y) dy = 0, otherwise the function u tends to infinity. We extend the function / to Q,c so that JK2 f(y) dy = 0. Let UQ be the corresponding Newton potential, then u — UQ satisfies a boundary value problem of the Laplace equation, so the results in the previous section can be applied here. The Dirichlet problem admits a unique solution. As for the Neumann problem we notice that / f(x)dx Jn

=- / -—^dsx. Jan dv

Thus the sufficient and necessary condition of existing a solution is / f(x) dx+ g(x) dsx = 0, Ja Jan and the solutions are unique up to adding an arbitrary constant. For three dimensional case the Newton potential is

<X) = T- I f(y)r^—\dy-

(L17)

4?r 7 R 3 \x - y\ We can define / = 0 on ftc then verify that both the Dirichlet problem and the Neumann problem are uniquely solvable.

Exterior Problems of Partial Differential

1.3

13

Equations

Poisson equations—variational formulation

To study the variational formulation and weak solutions the (complex) LaxMilgram theorem is applied: Theorem 13. Let H be a Hilbert space, M,a > 0, and a(-, •) be a sesquilinear form on 7i with 1. \a(u,v)\< M\\u\\\\v\\, \/u,v£H, 2. |a(u,u)| > a||u|| 2 , Vu e U. Let f be a bounded linear functional on "H. Then the equation a(u,v) = f(v),

VveTi

admits a unique solution u G 7i, and the mapping f —> u is bounded. We list some notations which are commonly used: Wm'p{Q) and Hm(Q), m £ M.,1 < p < oo, are Sobolev spaces, and || • || m , p , || • || m are norms in these spaces. The semi-norms are denoted by | • | m p, \ • \m respectively. The L 2 inner product is denoted by (•, •). W™'P(Q) and H™^) are the closures of Cg°(Q.) in Wm'P(Q) and Hm(Q) respectively. Hm(Cl) is a Hilbert space which is chosen as the basic space in many variational formulations of partial differential equations. However we have seen two particular solutions in Section 1.1, namely, u = 1 for two dimensional exterior problems and u = 1/r for three dimensional exterior problems. They are certainly not in the space L2(tt). So some different spaces have to be introduced. The following lemmas are useful in introducing some new spaces. Lemma 5. We assume that dim (£1) = 3 and u € Co°(fi), then f T^^

dxgf£* fc=i 3

dx -Kk J £-? fc=1

,- ,, F-2/1

J,

\*-y?

dx.

14

Numerical Methods for Exterior

Problems

Using the Cauchy inequality to get

' f u2{x) J-, {xk - Vkf

u2(x)

/ 1 Ma dx ^ 2 J \x-y\z

\j

^ u t Y?k=i ^glffi

\x~y\'

\x-y\*£^

=

\Wu(x)\2dx

dx

*' w n i c h g i y e s the result.

a

Lemma 6. We assume that dim (Q) = 2, 5 ( 0 , 1 ) C Qc and u £ Co°(fi), then

u2{x)

r

f Nu(x)

dx Ro in Q later on. We consider the variational formulation of (1.15),(1.2) first. Let g G Hll2{d£l). By the inverse trace theorem of Sobolev spaces there is a function wo G HX{D,), such that uo|an =