Numerical Approximation Methods for Elliptic Boundary Problems

by Steinbach, Olaf

• ISBN13: 9780387313122
• ISBN10: 0387313125
• Format: Hardcover
• Copyright: 2007-12-01
• Publisher: Springer Nature

Summary

This book presents a unified theory of the Finite Element Method and the Boundary Element Method for a numerical solution of second order elliptic boundary value problems. This includes the solvability, stability, and error analysis as well as efficient methods to solve the resulting linear systems. Applications are the potential equation, the system of linear elastostatics and the Stokes system. While there are textbooks on the finite element method, this is one of the first books on Theory of Boundary Element Methods.

Table of Contents

Boundary Value Problems	p. 1
Potential Equation	p. 1
Linear Elasticity	p. 5
Plane Elasticity	p. 9
Incompressible Elasticity	p. 12
Stokes System	p. 12
Helmholtz Equation	p. 15
Exercises	p. 17
Function Spaces	p. 19
The Spaces C[superscript k]([Omega]), C[superscript k,k]([Omega]) and L[subscript p]([Omega])	p. 19
Generalized Derivatives and Sobolev Spaces	p. 22
Properties of Sobolev Spaces	p. 25
Distributions and Sobolev Spaces	p. 29
Sobolev Spaces on Manifolds	p. 35
Exercises	p. 39
Variational Methods	p. 41
Operator Equations	p. 41
Elliptic Operators	p. 46
Operators and Stability Conditions	p. 48
Operator Equations with Constraints	p. 50
Mixed Formulations	p. 52
Coercive Operators	p. 57
Variational Formulations of Boundary Value Problems	p. 59
Potential Equation	p. 59
Dirichlet Boundary Value Problem	p. 61
Lagrange Multiplier Methods	p. 64
Neumann Boundary Value Problem	p. 67
Mixed Boundary Value Problem	p. 70
Robin Boundary Value Problems	p. 71
Linear Elasticity	p. 72
Dirichlet Boundary Value Problem	p. 76
Neumann Boundary Value Problem	p. 77
Mixed Boundary Value Problems	p. 79
Stokes Problem	p. 79
Helmholtz Equation	p. 85
Exercises	p. 86
Fundamental Solutions	p. 89
Laplace Operator	p. 90
Linear Elasticity	p. 96
Stokes Problem	p. 101
Helmholtz Equation	p. 105
Exercises	p. 109
Boundary Integral Operators	p. 111
Newton Potential	p. 111
Single Layer Potential	p. 118
Adjoint Double Layer Potential	p. 120
Double Layer Potential	p. 124
Hypersingular Boundary Integral Operator	p. 128
Properties of Boundary Integral Operators	p. 136
Ellipticity of the Single Layer Potential	p. 139
Ellipticity of the Hypersingular Boundary Integral Operator	p. 144
Steklov-Poincare Operator	p. 148
Contraction Estimates of the Double Layer Potential	p. 149
Mapping Properties	p. 152
Linear Elasticity	p. 155
Stokes System	p. 165
Helmholtz Equation	p. 167
Exercises	p. 169
Boundary Integral Equations	p. 171
Dirichlet Boundary Value Problem	p. 172
Neumann Boundary Value Problem	p. 175
Mixed Boundary Conditions	p. 179
Robin Boundary Conditions	p. 181
Exterior Boundary Value Problems	p. 181
Helmholtz Equation	p. 183
Exercises	p. 186
Approximation Methods	p. 187
Galerkin-Bubnov Methods	p. 187
Approximation of the Linear Form	p. 190
Approximation of the Operator	p. 191
Galerkin-Petrov Methods	p. 193
Mixed Formulations	p. 195
Coercive Operators	p. 199
Exercises	p. 202
Finite Elements	p. 203
Reference Elements	p. 203
Form Functions	p. 211
Trial Spaces	p. 216
Quasi Interpolation Operators	p. 225
Exercises	p. 227
Boundary Elements	p. 229
Reference Elements	p. 229
Trial Spaces	p. 233
Finite Element Methods	p. 243
Dirichlet Boundary Value Problem	p. 243
Neumann Boundary Value Problem	p. 253
Finite Element Methods with Lagrange Multipliers	p. 255
Exercises	p. 261
Boundary Element Methods	p. 263
Dirichlet Boundary Value Problem	p. 263
Neumann Boundary Value Problem	p. 274
Mixed Boundary Conditions	p. 281
Robin Boundary Conditions	p. 287
Exercises	p. 289
Iterative Solution Methods	p. 291
The Method of Conjugate Gradients	p. 291
A General Preconditioning Strategy	p. 299
An Application in Boundary Element Methods	p. 302
A Multilevel Preconditioner in Finite Element Methods	p. 306
Solution Methods for Saddle Point Problems	p. 319
Fast Boundary Element Methods	p. 327
Hierarchical Cluster Methods	p. 328
Approximation of the Stiffness Matrix	p. 332
Taylor Series Representations	p. 336
Series Representations of the Fundamental Solution	p. 340
Adaptive Cross Approximation	p. 344
Wavelets	p. 351
Exercises	p. 366
Domain Decomposition Methods	p. 367
References	p. 375
Index	p. 383
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