目录
- Contents
Chapter 1 Introduction 1
Chapter 2 Some preliminaries 5
2.1 Sobolev spaces 5
2.2 Finite element methods for elliptic equations 8
2.2.1 A priori error estimates 9
2.2.2 A posteriori error estimates 15
2.2.3 Superconvergence 17
2.3 Mixed finite element methods 19
2.3.1 Elliptic equations 19
2.3.2 Parabolic equations 25
2.3.3 Hyperbolic equations 26
2.4 Optimal control problems 31
2.4.1 Backgrounds and motivations 31
2.4.2 Some typical examples 32
2.4.3 Optimality conditions 34
Chapter 3 Finite element methods for optimal control problems 36
3.1 Elliptic optimal control problems 36
3.1.1 Distributed elliptic optimal control problems 36
3.1.2 Finite element discretization 37
3.1.3 A posteriori error estimates 38
3.2 Parabolic optimal control problems 44
3.2.1 Fully discrete finite element approximation 45
3.2.2 Intermediate error estimates 46
3.2.3 Superconvergence 50
3.3 Optimal control problems with oscillating coefficients 54
3.3.1 Finite element scheme 55
3.3.2 Multiscale finite element scheme 56
3.3.3 Homogenization theory and related estimates 57
3.3.4 Convergence analysis59
3.4 Recovery a posteriori error estimates 63
3.4.1 Fully discrete finite element scheme 65
3.4.2 Error estimates of intermediate variables 65
3.4.3 Superconvergence 68
3.4.4 A posteriori error estimates 72
3.5 Numerical examples 74
3.5.1 Parabolic optimal control problems 74
3.5.2 Recovery a posteriori error estimates 77
Chapter 4 A priori error estimates of mixed finite element methods 81
4.1 Elliptic optimal control problems 81
4.1.1 Mixed finite element scheme 82
4.1.2 A priori error estimates 84
4.2 Parabolic optimal control problems 92
4.2.1 Mixed finite element discretization 92
4.2.2 Mixed method projection95
4.2.3 Intermediate error estimates 98
4.2.4 A priori error estimates 101
4.3 Hyperbolic optimal control problems 106
4.3.1 Mixed finite element methods 107
4.3.2 A priori error estimates 109
4.4 Fourth order optimal control problems 116
4.4.1 Mixed finite element scheme116
4.4.2 L2-error estimates 119
4.4.3 L∞-error estimates 124
4.5 Nonlinear optimal control problems 128
4.5.1 Mixed finite element discretization 129
4.5.2 Error estimates 131
4.6 Numerical examples 132
4.6.1 Elliptic optimal control problems 132
4.6.2 Fourth order optimal control problems 134
Chapter 5 A posteriori error estimates of mixed finite element methods .136
5.1 Elliptic optimal control problems 136
5.1.1 Mixed finite element discretization 136
5.1.2 A posteriori error estimates for control variable138
5.1.3 A posteriori error estimates for state variables 141
5.2 Parabolic optimal control problems 145
5.2.1 Mixed finite element approximation 146
5.2.2 A posteriori error estimates 148
5.3 Hyperbolic optimal control problems 161
5.3.1 Intermediate error estimates 161
5.3.2 A posteriori error estimates for control variable164
5.3.3 A posteriori error estimates for state variables 166
5.4 Nonlinear optimal control problems 175
5.4.1 Mixed finite element discretization 175
5.4.2 Intermediate error estimates 176
5.4.3 A posteriori error estimates 181
Chapter 6 Superconvergence of mixed finite element methods 183
6.1 Elliptic optimal control problems 183
6.1.1 Recovery operator 183
6.1.2 Superconvergence property 184
6.2 Parabolic optimal control problems 185
6.2.1 Superconvergence for the intermediate errors 189
6.2.2 Superconvergence 193
6.3 Hyperbolic optimal control problems 197
6.3.1 Superconvergence property 198
6.3.2 Superconvergence for the control variable 200
6.4 Nonlinear optimal control problems 201
6.4.1 Superconvergence for the intermediate errors 201
6.4.2 Global superconvergence207
6.4.3 H.1-error estimates 209
6.5 Numerical examples 211
6.5.1 Elliptic optimal control problems 211
6.5.2 Nonlinear optimal control problems 213
Chapter 7 Finite volume element methods for optimal control problems 216
7.1 Elliptic optimal control problems 216
7.1.1 Finite volume element methods 218
7.1.2 L2-error estimates 222
7.1.3 H1 error estimates 225
7.1.4 Maximum-norm error estimates 226
7.2 Parabolic optimal control problems 227
7.2.1 Crank-Nicolson finite volume scheme 228
7.2.2 Error estimates of CN-FVEM 235
7.3 Hyperbolic optimal control problems 239
7.3.1 Finite volume element methods 240
7.3.2 A priori error estimates 241
7.4 Numerical examples 249
7.4.1 Elliptic optimal control problems 249
7.4.2 Parabolic optimal control problems 251
7.4.3 Hyperbolic optimal control problems 253
Chapter 8 Variational discretization methods for optimal control problems 256
8.1 Variational discretization 256
8.1.1 Variational discretization scheme 257
8.1.2 A priori error estimates 258
8.1.3 A posteriori error estimates 262
8.2 Mixed variational discretization 267
8.2.1 Mixed finite element approximation and variational discretization269
8.2.2 A priori error estimates for semi-discrete scheme 271
8.2.3 A priori error estimates for fully discrete scheme 277
8.3 Numerical examples 284
8.3.1 Variational discretization 284
8.3.2 Mixed variational discretization 288
Chapter 9 Legendre-Galerkin spectral methods for optimal control problems 290
9.1 Elliptic optimal control problems 290
9.1.1 Legendre-Galerkin spectral approximation 291
9.1.2 Regularity of the optimal control 293
9.1.3 A priori error estimates 295
9.1.4 A posteriori error estimates 297
9.1.5 The hp spectral element methods 300
9.2 Parabolic optimal control problems 306
9.2.1 Legendre-Galerkin spectral methods 306
9.2.2 A priori error estimates 309
9.2.3 A posteriori error estimates 316
9.3 Optimal control problems governed by Stokes equations 326
9.3.1 Legendre-Galerkin spectral approximation 327
9.3.2 A priori error estimates 335
9.3.3 A posteriori error estimates 341
9.4 Optimal control problems with integral state and control constraints 346
9.4.1 Legendre-Galerkin spectral scheme 346
9.4.2 A priori error estimates 349
9.4.3 A posteriori error estimates 355
9.5 Numerical examples 364
9.5.1 Elliptic optimal control problems 364
9.5.2 Optimal control problems governed by Stokes equations 371
9.5.3 Optimal control problems with integral state and control constraints 374
Bibliography 378
Index 384
京公网安备 11010102004214号