The Fast Solution of Boundary Integral Equations

by Rjasanow, Sergej; Steinbach, Olaf

• ISBN13: 9780387340418
• ISBN10: 0387340416
• Format: Hardcover
• Copyright: 2007-05-04
• Publisher: Springer-Verlag New York Inc

Summary

The use of surface potentials to describe solutions of partial differential equations goes back to the middle of the 19th century. Numerical approximation procedures, known today as Boundary Element Methods (BEM), have been developed in the physics and engineering community since the 1950s. These methods turn out to be powerful tools for numerical studies of various physical phenomena which can be described mathematically by partial differential equations.

Table of Contents

Preface	p. V
Boundary Integral Equations	p. 1
Laplace Equation	p. 2
Interior Dirichlet Boundary Value Problem	p. 10
Interior Neumann Boundary Value Problem	p. 13
Mixed Boundary Value Problem	p. 17
Robin Boundary Value Problem	p. 19
Exterior Dirichlet Boundary Value Problem	p. 21
Exterior Neumann Boundary Value Problem	p. 22
Poisson Problem	p. 24
Interface Problem	p. 26
Lame Equations	p. 27
Dirichlet Boundary Value Problem	p. 35
Neumann Boundary Value Problem	p. 36
Mixed Boundary Value Problem	p. 37
Stokes System	p. 40
Helmholtz Equation	p. 44
Interior Dirichlet Boundary Value Problem	p. 49
Interior Neumann Boundary Value Problem	p. 50
Exterior Dirichlet Boundary Value Problem	p. 52
Exterior Neumann Boundary Value Problem	p. 54
Bibliographic Remarks	p. 56
Boundary Element Methods	p. 59
Boundary Elements	p. 59
Basis Functions	p. 61
Laplace Equation	p. 65
Interior Dirichlet Boundary Value Problem	p. 65
Interior Neumann Boundary Value Problem	p. 72
Mixed Boundary Value Problem	p. 77
Interface Problem	p. 84
Lame Equations	p. 87
Helmholtz Equation	p. 91
Interior Dirichlet Problem	p. 91
Interior Neumann Problem	p. 93
Exterior Dirichlet Problem	p. 97
Exterior Neumann Problem	p. 98
Bibliographic Remarks	p. 99
Approximation of Boundary Element Matrices	p. 101
Hierarchical Matrices	p. 101
Motivation	p. 101
Hierarchical clustering	p. 108
Block Approximation Methods	p. 112
Analytic Form of Adaptive Cross Approximation	p. 112
Algebraic Form of Adaptive Cross Approximation	p. 119
Bibliographic Remarks	p. 129
Implementation and Numerical Examples	p. 131
Geometry Description	p. 131
Unit Sphere	p. 131
TEAM Problem 10	p. 131
TEAM Problem 24	p. 134
Relay	p. 134
Exhaust manifold	p. 135
Laplace Equation	p. 135
Analytical solutions	p. 135
Discretisation, Approximation and Iterative Solution	p. 137
Generation of Matrices	p. 139
Interior Dirichlet Problem	p. 143
Interior Neumann Problem	p. 149
Interior Mixed Problem	p. 155
Inhomogeneous Interface Problem	p. 160
Linear Elastostatics	p. 162
Generation of Matrices	p. 162
Relay	p. 163
Foam	p. 166
Helmholtz Equation	p. 168
Analytical Solutions	p. 169
Discretisation, Approximation and Iterative Solution	p. 169
Generation of Matrices	p. 170
Interior Dirichlet Problem	p. 171
Interior Neumann Problem	p. 185
Exterior Dirichlet Problem	p. 191
Exterior Neumann Problem	p. 196
Mathematical Foundations	p. 199
Function Spaces	p. 199
Fundamental Solutions	p. 208
Laplace Equation	p. 208
Lame System	p. 209
Stokes System	p. 210
Helmholtz Equation	p. 212
Mapping Properties	p. 213
Numerical Analysis	p. 225
Variational Methods	p. 225
Approximation Properties	p. 231
Numerical Algorithms	p. 239
Numerical Integration	p. 239
Analytic Integration	p. 244
Single Layer Potential for the Laplace operator	p. 246
Double Layer Potential for the Laplace operator	p. 249
Linear Elasticity Single Layer Potential	p. 252
Iterative Solution Methods	p. 256
Conjugate Gradient Method (CG)	p. 256
Generalised Minimal Residual Method (GMRES)	p. 263
References	p. 269
Index	p. 277
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