Reliable Methods for Computer Simulation (eBook)

Front Cover 1
Reliable Methods for Computer Simulation: Error Control and A Posteriori Estimates 4
Copyright Page 5
Contents 6
Chapter 1. Introduction 12
1.1 Sources of errors affecting the reliability of numerical solutions 12
1.2 The main approaches to error estimation 14
Chapter 2. Mathematical background 18
2.1 Basic notation 18
2.2 Sobolev spaces 22
2.3 Generalized solutions and their approximation 28
Chapter 3. A posteriori estimates for iteration methods 34
3.1 The Banach fixed point theorem 34
3.2 Iteration methods for bounded linear operators 39
3.3 Iterational methods for integral equations 43
3.4 Iteration methods for ordinary differential equations 46
3.5 Notes for the Chapter 47
Chapter 4. A posteriori estimates for finite element approximations 50
4.1 Methods based upon estimates of the negative norm of the equation residual 51
4.2 Evaluation of negative norms of residuals 56
4.3 Error estimation methods using adjoint problems 65
4.4 Methods based upon post-processing of finite element approximations 69
4.5 Error estimation of goal-oriented quantities by gradient averaging techniques 75
4.6 Notes for the Chapter 85
Chapter 5. Foundations of duality theory 90
5.1 Convex sets and functionals 90
5.2 Dual functionals 97
5.3 Differentiation of convex functionals 107
5.4 Lower semicontinuous functionals and existence of minimizers 112
5.5 Uniformly convex functionals 124
5.6 Integral functionals 129
Chapter 6. Two-sided a posteriori estimates for linear elliptic problems 136
6.1 Basic relations 136
6.2 Two-sided estimates of deviations 142
6.3 Properties of two-sided estimates 146
6.4 Linear elliptic equations of the second order 152
6.5 The linear elasticity problem 161
6.6 Linear elliptic equations of the fourth order 165
6.7 Relationship with other methods 168
6.8 Error estimates taking account of the indeterminacy in the data 174
6.9 Error estimation in terms of linear functionals 185
6.10 Practical implementation 187
6.11 Comments 215
6.12 Notes for the Chapter 218
Chapter 7. A posteriori estimates for nonlinear variational problems 220
7.1 Nonlinear variational problems 220
7.2 General form of the error majorant for the functional J(v) = G'(.v) + F(v ) 223
7.3 Variational problems with nonhomogeneous boundary conditions 227
7.4 Error majorant for the case F ( v ) = < l , v>
7.5 Properties of compound functionals 234
7.6 Two-sided error estimates 242
7.7 A class of nonlinear variational problems 245
7.8 Comments 247
Chapter 8. A posteriori estimates for variational inequalities 256
8.1 Variational inequalities 256
8.2 Problems with obstacles 257
8.3 The elasto-plastic torsion problem 265
8.4 A model problem with a friction type boundary condition 266
8.5 Variational problems in the theory of viscous fluids 270
8.6 Examples 283
8.7 Notes for the Chapter 290
Bibliography 292
Notation 312
Index 314