Table Of ContentSpringer Monographs ..... in Mathematics
Springer-Verlag Berlin Heidelberg GmbH
Jukka Saranen • Gennadi Vainikko
Periodic Integral
and Pseudodifferential
Equations
with Numerical
Approximation
, Springer
Jukka SaTanen
Department of Mathematical Sciences
University of OuIu
90014 OuIu, Finland
e-mail: [email protected]
Gennadi Vainikko
Institute of Mathematics
Helsinki University of Technology
02150 Espoo, Finland
e-mail: [email protected]
Library of Congress Cataloging·in-Publication Data applied for
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Saranen, Jukka: Periodic integral and pseudodifferential equations
with numerical approximation / Jukka Saranen ; Gennadi Vainikko. -
Berlin ; Heidelberg ; New York; Barcelona; Hong Kong ; London ;
Milan; Paris; Tokyo : Springer, 2002
(Springer monographs in mathematics)
Mathematics Subject Classification (2000):
31A10, 35J05, 35J25, 41A15, 42A15, 45A05, 45E05, 46E35, 47A53, 47G30, 65N38, 65R20
ISSN 1439-7382
ISBN 978-3-642-07538-4 ISBN 978-3-662-04796-5 (eBook)
DOI 10.1007/978-3-662-04796-5
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Originally published by Springer-Verlag Berlin Heidelberg New York in 2002.
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Preface
This is a self-contained course book for graduate/postgradute students and
it can be considered also as a scientific monograph. The purpose of the book
is introduce the reader to the modern treatment of boundary value problems
and integral equations. We do this by using the theory of pseudodifferential
operators. The theory of pseudodifferential operators in ]Rn origins from the
works of Kohn and Nirenberg, Hörmander '65, and others.
In this monograph we cover the theory of periodie pseudodifferential equa
tions, or pseudodifferential equations on the unit circle introduced by Agra
novieh '79. During the last twenty years periodie pseudodifferential operators
occurred to be also a powerful tool in numerieal analysis.
The content of the book includes the following items. We begin with the
Fredholm operators and Krylov subspace methods such as GMRES and con
jugate gradients. Next we derive the classieal boundary integral equations
for two dimensional Laplace and Helmholtz equations. We also treat singular
integral equations on a closed curve and on an open arc. Using a parame
trization of the boundary these problems take a form of periodic integral and
pseudodifferential equations. With this introduction, we present a general the
ory of periodic integral and pseudodifferential equations in the framework of
periodie Sobolev spaces. After that we develop approximate methods to solve
periodie integral equations. The following methods are elaborated: trigono
metrie Galerkin and collocation methods with fully discrete versions and fast
solvers on the basis of those, quadrat ure and spline based methods. The the
ory of periodie pseudodifferential equations is widely used here. The book is
equipped with exercises, apart of whieh extend the main results. Therefore
we recommend to browse the exercises even if you have no intention to solve
any of them.
The treatise of the book is based on the scientific works of the authors,
partly with their colleagues. The content of the book has been arisen from
authors lectures at the University of Oulu (from 1983), the University of
Jyväskylä (1985), the Helsinki University of Technology (from 1993) and the
Seoul National University (1994). Using the materials of the book, a lecturer
can design different courses. First, on the basis of Chapters 1-6 (omitting
Section 1.5 about Krylov subspace methods), a one-semester course can be
presented where the emphasis is put onto the way from boundary value prob-
vi Preface
lems to periodic integral equations and onto well-posedness of those. On the
basis of Chapters 1, 4-7, with abrief discussion of materials of Chapters 2
and 3, a well-motivated course of periodic pseudodifferential equations can
be build. After either of those courses, a one-semester course on numeri
cal methods can be designed with an emphasis on Galerkin and collocation
type methods (Section 1.5, Chapters 8-11) or on quadrat ure and spline type
methods (Chapters 8, 12, 13).
The manuscript of this book has been developed during a long time. A
milestone was a short version by Vainikko '96 which contains only a small part
of the final version. The authors have found much help from their colleagues
and previous students through their constructive criticism. Our special thanks
belong to R. Plato, V. Turunen, P. Oja, A. Pedas, M. Hamina, J. Anttila,
J. Kemppainen, P. Ola, and many others. For expert knowledge on R<1EX 2c
environment and other technical issues, we are grateful to S. Savolainen and
M. Kuukasjärvi. For typing some parts of the monograph we are grateful to
M.Ojala.
Oulu and Espoo Jukka Saranen
June 21, 2001 Gennadi Vainikko
Contents
1. Preliminaries............................................. 1
1.1 Preliminaries from Operator Theory . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Linear Continuous Operators ...................... 2
1.1.2 Linear Compaet Operators ........................ 3
1.1.3 Examples of Compaet Integral Operators. . . . . . . . . . . . 4
1.1.4 Charaeterization of Continuous Projeetors . . . . . . . . . . . 5
1.1.5 Complementable Subspaees . . . . . . . . . . . . . . . . . .. . . . . . 5
1.2 Fredholm Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Results Involving only the Banaeh Spaee X . . . . . . . . . . 7
=
1.2.2 Results Involving X and its Dual X' C(X, C) ...... 8
1.2.3 Dual Systems and Dual Operators. . . . . . . . . . . . . . . . . . 9
1.2.4 Results Involving Dual Systems. . . . . . . . . . . . . . . . . . .. 11
1.3 The Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
1.3.1 Fredholm Operators ofIndex 0.. . . . . . . . . . . . . . . . . . .. 14
1.3.2 The Produet of Fredholm Operators . . . . . . . . . . . . . . .. 15
1.3.3 Fredholm Operators of Nonzero Index. . . . . . . . . . . . . .. 17
1.3.4 Perturbation of Fredholm Operators . . . . . . . . . . . . . . .. 18
1.4 The Regularizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
1.4.1 Connections to Fredholmness ...................... 20
1.4.2 Results of Fredholm Type . . . . . . . . . . . . . . . . . . . . . . . .. 22
1.5 Krylov Subspaee Methods ........ . . . . . . . . . . . . . . . . .. . . . .. 25
1.5.1 GMRES......................................... 25
1.5.2 Another Algorithm of GMRES .... .... .... .. ....... 29
1.5.3 Conjugate Gradients (CGMR and CGME) .......... 30
2. Single Layer and Double Layer Potentials........ ... ... . .. 35
2.1 Classical Boundary Value Problems.. .. . . . . . . . . . . . . . . . . . .. 35
2.2 Fundamental Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
2.3 An Integral Representation of Functions . . . . . . . . . . . . . . . . . .. 44
2.4 Jordan Ares and Curves. . . . . . .. .. .. .. . . . . . . . . . . . . . . . . . .. 48
2.5 Boundary Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54
2.5.1 Single Layer Potential ............................ 54
2.5.2 Double Layer Potential. . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
2.5.3 Normal Derivative of the Single Layer Potential. . . . .. 61
viii Contents
2.5.4 Normal Derivative of the Double Layer Potential. . . .. 66
2.5.5 Calderon Projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
3. Solution of Boundary Value Problems by Integral Equations 71
3.1 Integral Equations for Boundary Value Problems. . . . .. . . . .. 71
3.2 Solution of the Laplace Equation by Integral
Equations of the Second Kind. . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
3.2.1 Solvability of the Boundary Integral Equations . . . . . .. 76
3.2.2 Solution of the Interior Boundary Value Problems. . .. 78
3.2.3 Solution of the Exterior Boundary Value Problems ... 81
3.3 Solution of the Helmholtz Equation by Integral
Equations of the Second Kind. . . . . . . . . . . . . . . . . . . . . . . . . . .. 89
3.3.1 Solvability of the Boundary Integral Equations ...... 89
3.3.2 Solution of the Boundary Value Problems ........... 95
3.3.3 Radiation Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
4. Singular Integral Equations ............................... 105
4.1 Singular Integral Equations in Hölder Spaces ............... 105
4.1.1 Hölder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 105
4.1.2 Cauchy Integral .................................. 106
4.1.3 Cauchy Singular IntegralOperator ................. 110
4.1.4 Operator Bk ..................................... 112
4.1.5 The Commutator of Band Ma •.•••...•.••..••..•• 115
4.1.6 Projection Operators P_ and P+ ................... 116
4.1.7 Winding Number ................................. 117
4.1.8 The Index of the Singular Operator ................. 119
4.1.9 The Index of More General Singular Integral Operator 122
4.2 L2_ Theory of Singular Integral Equations .................. 124
4.2.1 Lax Theorem .................................... 124
4.2.2 Singular Integral Operators on L2(r) ............... 127
5. Boundary Integral Operators in Periodic Sobolev Spaces . 133
5.1 Distributions on the Real Line ........................... 133
5.2 Periodic Distributions ................................... 136
5.3 Periodic Sobolev Spaces ................................. 141
5.4 Finite Part of Hypersingular Integrals ..................... 143
5.5 Spectral Representation of a Convolution Integral Operator .. 148
5.6 Symm's Integral Operator ............................... 150
5.7 Hilbert Integral Operator ................................ 152
5.8 Cauchy Integral Operator on the Unit Cirde ............... 154
5.9 Cauchy Integral Operator on a Jordan Curve .............. 155
5.10 Hypersingular Integral Operator .......................... 156
5.11 Biharmonic Problem .................................... 157
5.12 Operator Interpolation .................................. 161
5.13 Multiplication of Functions in HA ........................ 163
Contents ix
6. Periodie Integral Equations ............................... 167
6.1 Boundedness of Integral Operators Between Sobolev Spaces .. 167
6.1.1 Product of Biperiodic Functions .................... 168
6.1.2 Boundedness of Integral Operators ................. 171
6.2 Fredholmness of Integral Operators Between Sobolev Spaces . 175
6.3 A Class of Periodic Integral Equations .................... 176
6.4 Examples of Periodic Integral Equations ................... 177
6.5 Analysis of the Modified Symm's Equations ................ 180
6.6 A General Class of Periodic Integral Equations ............. 182
6.6.1 Winding Number of a Periodic Function ............. 182
6.6.2 Inversion of Cauchy Singular Operators ............. 185
6.6.3 The Class of Integral Equations .................... 187
6.6.4 Example: the Cauchy Integral Equation ............. 192
6.7 Equations with Analytic Coefficient Functions .............. 193
6.7.1 Spaces of Periodic Analytic Functions .............. 193
6.7.2 Mapping Properties of Periodic Integral Operators .... 195
6.7.3 Analytic Solutions of Integral equations ............. 197
7. Periodic Pseudodifferential Operators .................... 199
7.1 Prolongation of a Function Defined on Z .................. 199
7.2 Two Definitions of PPDO and Their Equivalence ........... 202
7.3 Boundedness of a PPDO ................................ 205
7.4 Asymptotic Expansion of the Symbol ..................... 206
7.5 Amplitudes ............................................ 207
7.6 Asymptotic Expansion of Integral Operators ............... 214
f;
=
7.6.1 Operator (Au)(t) a(t, s) loglsin 7r(t - s)lu(s) ds ... 215
=
7.6.2 Operator (Au)(t) ifol a(t, s) cot7r(t - s)u(s) ds ...... 216
7.7 The Symbol of Dual and Adjoint Operators ................ 217
7.8 The Symbol of the Composition of PPDOs ................ 218
7.9 Pseudolocality ......................................... 221
7.10 Elliptic PPDOs ........................................ 224
7.11 Gärding's Inequality .................................... 227
7.12 Estimation of the Operator Norm ........................ 230
7.13 Classical PPDOs ....................................... 231
7.14 Integral Operator Representation of Classical PPDOs ....... 234
7.15 Functions I\;~(t) ........................................ 237
8. Trigonometrie Interpolation .............................. 239
8.1 Subspace In ........................................... 239
8.2 Orthogonal Projection .................................. 241
8.3 Interpolation Projection ................................. 242
8.3.1 Interpolation of Functions u E H/L .................. 242
8.3.2 Interpolation of Even and Odd Functions ............ 244
8.3.3 Discrete Fourier Thansform ........................ 245
8.3.4 Interpolation of Functions aun, Un Ein ............. 246
x Contents
8.4 Exponential Approximation Order ........................ 248
8.5 Two Dimensional Interpolation ........................... 250
8.5.1 Estimates for u E HI-'1,I-'2 •••.••••••••.••••••.•••••• 250
8.5.2 Estimates for u E HiC)R2) ......................... 252
9. Galerkin Method and Fast Solvers ........................ 255
9.1 Precondition of the Problem ............................. 255
9.2 Galerkin Method for the Preconditioned Problem ........... 257
9.3 Matrix Representation of a PIO .......................... 262
9.4 A Full Discretization .................................... 264
9.4.1 Approximation of the Galerkin Equation ............ 264
9.4.2 Computational Costs ............................. 268
9.4.3 Fast Solvers ..................................... 270
9.5 Using Asymptotic Expansions ............................ 273
9.5.1 Approximation of the Galerkin Equation ............ 273
9.5.2 Two Grid Iteration Method ........................ 276
9.5.3 Fast Solvers ..................................... 277
9.6 StabiIity Estimates ..................................... 279
9.7 Regularization via Discretization ......................... 286
9.8 Standard Galerkin Method .............................. 286
9.8.1 Stability Inequality ............................... 287
9.8.2 Convergence of the Galerkin Method ................ 290
10. Trigonometrie Colloeation ................................ 293
10.1 Collocation Problem .................................... 293
10.2 Full Discretization ...................................... 298
10.3 Modifications .......................................... 302
10.4 Further Discrete Versions ................................ 303
10.4.1 Problem, Methods, Convergence .................... 304
10.4.2 Matrix Form of the Method ....................... 308
10.4.3 Preconditioning and Iteration Solution .............. 311
10.5 Fast Solvers for Lippmann-Schwinger Equation ............ 314
10.5.1 Lippmann-Schwinger Equation ..................... 314
10.5.2 Collocation Solution .............................. 317
10.5.3 Two-Grid Iterations .............................. 319
10.5.4 Matrix Forms of the Methods ...................... 321
10.5.5 Appendix: Fourier Coefficients of K(x) .............. 323
11. Integral Equations on an Open Are . . . . . . . . . . . . . . . . . . . . . . . 325
11.1 Equations on an Interval ................................ 325
11.2 Periodization .......................................... 329
11.2.1 Equations with a Logarithmic Singular Kernel ....... 329
11.2.2 Cauchy-singular Equations ........................ 330
11.2.3 Hypersingular Equations .......................... 332
11.3 Even and Odd Operators ................................ 333
