A Posteriori Estimates for Partial Differential Equations

by Repin, Sergey

• ISBN13: 9783110191530
• ISBN10: 3110191539
• Format: Hardcover
• Copyright: 2008-09-30
• Publisher: De Gruyter

Summary

This book deals with the reliable verification of the accuracy of approximate solutions which is one of the central problems in modern applied analysis. After giving an overview of the methods developed for models based on partial differential equations, the author derives computable a posteriori error estimates by using methods of the theory of partial differential equations and functional analysis. These estimates are applicable to approximate solutions computed by various methods.

Table of Contents

Introduction	p. 1
A priori and a posteriori methods of error estimation	p. 1
Book structure	p. 2
The error control problem	p. 5
Mathematical background and notation	p. 8
Overview	p. 22
Error indicator by Runge	p. 22
Prager-Synge estimate	p. 23
Mikhlin estimate	p. 25
Ostrowski estimates for contractive mappings	p. 26
Error estimates based on monotonicity	p. 30
A posteriori error indicators for finite element approximations	p. 31
Poisson's equation	p. 45
The variational method	p. 45
The method of integral identities	p. 50
Properties of a posteriori estimates	p. 52
Two-sided bounds in combined norms	p. 57
Modifications of estimates	p. 59
How can one use functional a posteriori estimates in practical computations?	p. 65
Linear elliptic problems	p. 75
Two-sided estimates for stationary diffusion problem	p. 75
The stationary reaction-diffusion problem	p. 81
Diffusion problems with convective term	p. 87
Notes for the chapter	p. 103
Elasticity	p. 104
The linear elasticity problem	p. 104
Estimates for displacements	p. 107
Estimates for stresses	p. 109
Isotropic linear elasticity	p. 110
Notes for the chapter	p. 116
Incompressible viscous fluids	p. 117
The Stokes problem	p. 117
A posteriori estimates for the stationary Stokes problem	p. 123
Generalized Stokes problem	p. 140
The Oseen problem	p. 151
Stationary Navier-Stokes problem for d = 2	p. 153
Notes for the chapter	p. 156
Generalizations	p. 158
Linear elliptic problem	p. 158
Elliptic problems with lower terms	p. 171
Problems with solutions defined in subspaces	p. 173
Derivation of a posteriori estimates from saddle point relations	p. 176
Nonlinear problems	p. 178
Variational inequalities	p. 178
General elliptic problem. Variational method	p. 186
General elliptic problem. Nonvariational method	p. 191
A posteriori estimates for special classes of nonlinear elliptic problems	p. 196
Notes for the chapter	p. 214
Other problems	p. 218
Differential equations of higher order	p. 218
Equations with the operator curl	p. 224
Evolutionary problems	p. 229
A posteriori estimates for optimal control problems	p. 242
Estimates for nonconforming approximations	p. 254
Uncertain data	p. 265
Error estimates in terms of functionals and nonenergy norms	p. 279
Bibliography	p. 291
Index	p. 314
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