Table Of ContentMathematical Analysis and Numerical Methods
for Science and Technology
Springer
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan
Paris
Singapore
Tokyo
Robert Dautray Jacques-Louis Lions
Mathematical Analysis
and Numerical
Methods for Science
and Technology
Volume 4
Integral Equations and Numerical Methods
With the Collaboration of
Michel Artola, Philippe B6nilan, Michel Bernadou, Michel Cessenat,
Jean-Claude N6d6lec, Jacques Planchard, Bruno Scheurer
Translated from the French by John C. Amson
t
Springer
Robert Dautray
12 rue du Capitaine Scott
75015 Paris, France
Jacques-Louis Lions
College de France
3 rue d'Ulm
75231 Paris Cedex 5, France
Title of the French original edition:
Analyse mathematique et calcul numerique pour les sciences
et les techniques, Masson, S. A.
a
© Commissariat 1' Energie Atomique, Paris 1984, 1985
With 67 Figures
Mathematics Subject Classification (1980): 3l-XX, 35-XX, 41-XX, 42-XX, 44-XX,
45-XX, 46-XX, 47-XX, 65-XX, 73-XX, 76-XX, 78-XX, 80-XX, 81-XX
Library of Congress Cataloging. in-Publication Data
Dautray. Robert. Mathematical analysis and numerical methods for science and technology.
Translation of: Analyse mathematique et caleul numerique pour les sciences et les techniques.
Includes indexes. Includes bibliographical references
Contents: - v. 2 Functional and varitional methods/with the collaboration of Michel Artola ... let al.l
- v. 4 Integral equations an numerical methods/with the collaboration of M. Artola ... let al.l
translated from the French by John C. Amson.
I. Mathematical analysis. 2. Numerical analysis. I. Lions, Jacques Louis. II. Title.
QA300.D34313 1990 88-15089
ISBN-13: 978-3-540-66100-9 e-ISBN-13: 978-3-642-61531-3
001: 10.1007/978-3-642-61531-3
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Introduction to Volume 4
This fourth volume (which contains chapters X to XIII) continues the study
of linear stationary boundary value problems and related questions begun in
volumes 1 and 2, and the spectral theory of elliptic operators developed and ex
emplified in volume 3.
The operators arising in models are differential when they correspond to local
phenomena (see chapter V, § 1). Non-local phenomena (for example, action of a
force at a distance in space, in electromagnetism; or memory in time in viscoelastic
phenomena; or again, abrupt change in a gas particle's velocity in a collision, and
the consequential finite variation of velocity in a neutron transport velocity space)
cannot be modelled using only these differential operators. In particular, integral
operators playa large role in such models. These models then become integral
equations (or integro-differential equations, as for example in the case of trans
port equations; see chapter I, § 5). The corresponding equations are studied in
chapter XI.
There are some stationary problems which are elliptic in one part of the domain
where the phenomenon is being studied, and hyperbolic in another, for example
the case of flow in a nozzle where the flow is subsonic in one part of the nozzle
(elliptic problem) and supersonic in another part (hyperbolic problem). Such
problems are studied in chapter X.
Chapter XII and XIII are totally devoted to numerical methods. Thefinite element
method is studied in chapter XII in a concrete case, involving the elasticity of a
solid body, which is fundamental for engineering. Eigenvalue calculations are
treated equally in both the case where the operator in question is self-adjoint and
the case where it is non self-adjoint, as for instance in the multigroup diffusion of
neutrons.
Chapter XIII briefly treats various numerical methods for integral equations.
In very many applications (for example, in electromagnetism), the integral oper
ators are singular. Their properties are studied in the appendix "Singular Inte
grals" .
We give below the authors of the different contributions, chapter by chapter.
Chapter X: B. Scheurer
Chapter XI: J. C. N6d61ec (principal), M. Artola, M. Cessenat
Chapter XII: M. Bernadou (principal), J. Planchard
Chapter XIII: J. C. Ned61ec
Appendix "Singular Integrals": P. Benilan.
VI Introduction to Volume 4
Equally we thank P. G. Ciarlet, G. Fournet, R. Glowinski, B. Mercier, P. Raviart,
R. Sentis, L. Tartar, H. Vi viand for reading certain sections of the text and for
their advice.
To M. Cessenat we address our very particular thanks for the permanent and
eminent contributions which he has continued to make in this volume 4 as in the
previous volumes 1, 2 and 3, as well as for his detailed and constructive clarifica
tions.
We recall the important and indispensable role played by J. M. Moreau of which
we listed the various aspects in the preface to volume 1 and for which we will not
be able to thank him enough.
We have placed at the end of this volume 4 its table of notations.
R. Dautray, J.-L. Lions
Practical Guide for the Reader
1. Designation of subdivisions of the text:
number of a chapter: in Roman numerals
number of major division of a chapter: the sign § followed by a numeral
number of section: a numeral following the above
number of a sub-section: a numeral following the above.
Example: II, § 3.5.2, denotes chapter II, § 3, section 5, subsection 2.
2. Within each division (§), the equations, definitions, theorems, propositions,
corollaries, lemmas, remarks and examples are each numbered consecutively
beginning with the number 1.
3. The table of notations used is placed at the end of each volume.
Table of Contents
Chapter X. Mixed Problems and the Tricomi Equation
Introduction . . . . . . . . . . . . . . . . 1
§ 1. Description and Formulation of the Problem 2
1. Stationary Plane Flow of a Compressible Fluid . 2
2. Solution in the Hodograph Plane; The Frankl' Equation . 9
§ 2. Methods for Solving Problems of Mixed Type ..... 15
1. An Example of a Well-Posed Boundary Value Problem
for the Frankl' Equation . . . . 15
2. Particular Solutions ..... . 21
3. Existence and Uniqueness Results 23
Bibliographic Commentary 31
Chapter XI. Integral Equations
Introduction . . . . . . . . 33
Part A. Solution Methods Using Analytic Functions and Sectionally
Analytic Functions 38
Introduction . . . . . . . . 38
§ 1. The Wiener-Hopf Method 38
Introduction. Wiener-Hopf Equations 38
1. The Wiener-Hopf Method . . . . 39
2. Decomposition of an Analytic Function Defined in a Strip
in the Complex Plane . . . . . . . . . . . . . . . . 42
3. Factorisation of an Analytic Function Defined in a Strip
in the Complex Plane . . . . . . . . . . . . . 43
4. Application to the Wiener-Hopf Integral Equation
of the Second Kind. . . . . . . 45
5. Application to the Milne Problem 47
6. Application to the Dock Problem. 53
VIII Table of Contents
§ 2. Sectionally Analytic Functions 57
Introduction . . . . . . . . 57
1. S. Analytic Functions. . . 58
2. Cauchy Integrals and Plemelj Formulas 59
3. The Poincare-Bertrand Formula and the Hilbert
Inversion Formula 65
§ 3. The Hilbert Problem 74
Introduction . . . . 74
1. The Hilbert Problem in the Case where L is a Contour 74
2. The Hilbert Problem in the Case where L is an Arc 77
3. The Hilbert Problem in the Case of a Straight Line 81
4. Some Problems Reducible to a Hilbert Problem 83
§ 4. Application to Some Problems in Physics 91
Introduction . . . . . . . . . . . . . 91
1. Simple Layer and Double Layer Problems 91
2. Determination of the Charge Density on the Surface
of a Cylindrical Body at Potential V . . . . 93
3. The Problem of the Thin Aerofoil Profile . . 97
4. Plane Elasticity and the Biharmonic Equation 103
Part B. Integral Equations Associated with Elliptic Boundary
Value Problems in Domains in IR 3 114
§ 1. Study of Certain Weighted Sobolev Spaces . . . . . . 114
Introduction . . . . . . . . . . . . . . . . . . . 114
§ 2. Integral Equations Associated with the Boundary Value Problems
of Electrostatics . . . . . . . . . . . . . . 119
1. Integral Representations ......... 119
2. Dirichlet Problems Relative to the Operator L1 122
3. Neumann Problems Relative to the Operator L1 130
§ 3. Integral Equations Associated with the Helmholtz Equation 141
§ 4. Integral Equations Associated with Problems of Linear Elasticity 148
§ 5. Integral Equations Associated with the Stokes System . . . .. 152
Chapter XII. Numerical Methods for Stationary Problems
Introduction . . . . . . . . . . . . . . . . . . . . 160
1. The Basic Ideas of Finite Difference Methods and Finite
Element Methods . . . . . . . . . . . . . . . . 160
2. Comparison of the Two Methods. Field of Applications
of the Finite Element Method . . . . . . . . . . . 168
Table of Contents IX
3. The Different Topics Treated in this Chapter XII 170
4. The Lax-Milgram Theorem and Sobolev Spaces 171
§ 1. Principal Aspects of the Finite Element Method Applied
to the Problem of Linear Elasticity . . . . . . . . . 173
1. Variational Formulation of the Continuous Problem 173
2. Construction of Approximation Function Spaces . 179
3. The First Approximation Problem (Ph!) . . . . . 192
4. Numerical Quadrature Schemes and the Definition
of the Second Approximation Problem (Ph2) 194
5. Error Estimates . . . . . . . . . . . . 197
6. Numerical Implementation . . . . . . . 229
§ 2. Treatment of Domains with Curved Boundaries 240
1. "Exact" Triangulation of the Domain Q . . 241
2. Construction of an "Approximate" Triangulation of the Domain Q 242
3. Examples of the Construction of the Mappings FK 249
4. Definition of Curved Finite Elements of Class <6'0 . . 251
5. Estimation of the Interpolation Error . . . . . . . 256
6. Application to the Solution of the Problem of Plane
Linear Elasticity . . . . . . . . . . . . 257
§ 3. A Non Conforming Method of Finite Elements 271
1. The Wilson Finite Element . . . . 272
2. Estimation of the Interpolation Error . . . 272
3. The Space X of Finite Elements . . . . . 275
h
4. The Discrete Problem. Abstract Error Estimate . 276
5. The Bilinear Lemma . . . . . 285
6. Estimation of the Error II u-u Ilh . . . . . . 286
h
§ 4. Applications to the Problems of Plates and Shells 290
1. Approximation of the Problems of Plates 291
2. Approximation of the Problems of Shells 301
§ 5. Approximation of Eigenvalues and Eigenvectors 316
1. Some Results from the Spectral Theory of Differential Operators 317
2. The Approximate Problem ........ 320
3. Estimation of the Errors IA r Ahj I, 1 ",;j:-;:;; 3 Mh 323
4. Estimation of the Errors II Uj-Uhj 11,1 :-;:;;j:-;:;; 3 Mh 328
5. Numerical Solutions . . . . . . . . . . . 331
§ 6. An Example of the Approximate Calculation for a Problem
of the Eigenvalues of a Non Self-Adjoint Operator 332
1. The Neutron Diffusion Equations Recalled 332
2. The Critical Problem with Two Energy Groups . 334
3. Determination of the Positive Solution . . . . 339
4. Extension to the Case Where the Number of Neutron (Kinetic)
Energy Groups is Greater than Two ........... 345
x Table of Contents
5. The Eigenvalue Problem Connected with the Evolution Problem
of Neutron Diffusion 349
6. Some Comments 356
Review of Chapter XII 357
Chapter XIII. Approximation of Integral Equations by Finite Elements.
Error Analysis
Introduction . . . . . . . . . . . 359
§ 1. The Case of a Polyhedral Surface 359
1. The Simple Layer Potential Case for the Dirichlet Problem . 359
2. Study of the Potential of a Double Layer
for the Neumann Problem in 1R 3 . . . . . . . . . 362
3. Study of the Exterior Neumann Problem Represented
by a Simple Layer . . . . . . . 364
§ 2. The Case of a Regular Closed Surface . . . . . . . . 367
1. The Approximation of Surfaces . . . . . . . . . 367
2. Notions on the Error Generated by the Approximation
of the Surface . . . . . . . . . . . . . . . . . . 368
Appendix. "Singular Integrals"
Introduction . . . . . . . . 371
§ 1. Operator, Convolution Operator, Integral Operator 371
§ 2. The Hilbert Transformation 382
§ 3. Generalities on Singular Integral Operators. 393
§ 4. Operators with Symbols, Operators on L2 403
§ 5. The Calderon-Zygmund Theorem 409
§ 6. Marcinkiewicz Spaces . . . . . 414
1. Definitions . . . . . . . . 414
2. Application to the Homogeneous Convolution Kernel 417
3. The Hilbert Transformation in the Space L 1 (1R) 418
4. Operators of Weak Type. The Marcinkiewicz Theorem 421
5. The Maximal Hardy-Littlewood Operator.
Proof of Lemma 1 in § 2 424
Bibliography. . . 427
Table of Notations 441
Index . . . . . . 455
Contents of Volumes 1-3, 5, 6 489
