Chapter 1 Introduction
Chapter 2 Some preliminaries
2.1 Sobolevspaces
2.2 Finite element methods for elliptic equations
2.2.1 A priori error estimates
2.2.2 A posteriori error estimates
2.2.3 Superconvergence
2.3 Mixed finite element methods
2.3.1 Elliptic equations
2.3.2 Parabolic equations
2.3.3 Hyperbolic equations
2.4 Optimal control problems
2.4.1 Backgrounds and motivations
2.4.2 Some typical examples
2.4.3 Optimality conditions
Chapter 3 Finite element methods for optimal control problems
3.1 Elliptic optimal control problems
3.1.1 Distributed elliptic optimal control problems
3.1.2 Finite element diseretization
3.1.3 m posterinri error estimates
3.2 Parabolic optimal control problems
3.2.1 Fully discrete finite element approximation
3.22 Intermediate error estimates
3.2.3 Superconvergence
3.3 Optimal control problems with oscillating coefficients
3.3.1 Finite element scheme
3.32 Multiscale finite element scheme
3.3.3 Homogenization theory and related estimates
3.3.4 Convergence analysis
3.4 Recovery a posteriori error estimates
3.4.1 Fully discrete finite element scheme
3.4.2 Error estimates of intermediate variables
3.4.3 Superconvergence
3.4A A posteriori error estimates
3.5 Numerical examples
3.5.1 Parabolic optimal control problems
3.5.2 Recovery a posteriori error estimates
Chapter 4 A priori error estimates of mixed finite element methods ~.
4,1 Elliptic optimal control problems
1.1 Mixed finite element scheme
4,1.2 A priori error estimates
4.2 Parabolic optimal control problems
4.2.1 Mixed finite element discretization
4.2.2 Mixed method projection
4.2.3 Intermediate error estimates
4.2.4 A priori error estimates
4.3 Hyperbolic optimal control problems
4.3.1 Mixed finite element methods
4.32 A priori error estimates
4.4 Fourth order optimal control problems
4.4.1 Mixed finite element scheme
4.4.2 L2-error estimates
4.43 L~-error estimates
4.5 Nonlinear optimal control problems
4.5.1 Mixed finite element discretization
4.5.2 Error estimates
4.6 Numerical examples
4.6.1 Elliptic optimal control problems
4.6.2 Fourth order optimal control problems
Chapter 5 A posteriori error estimates of mixed finite element methods-
5.1 Elliptic optimal control problems
5.1.1 Mixed finite element discretization
5.1.2 A posteriori error estimates for control variable
5.1.3 A posteriori error estimates for state variables
5.2 Parabolic optimal control problems
52.1 Mixed finite element approximation
5.2.2 A posteriori error estimates
5.3 Hyperbolic optimal control problems
5.3.1 Intermediate error estimates
5.3.2 A posteriori error estimates for control variable
5.33 A posteriori error estimates for state variables
5A Nonlinear optimal control problems
5.4,1 Mixed finite element discretization
5.4.2 Intermediate error estimates
5.43 A posteriofi error estimates
Chapter 6 Superconvergence of mixed finite element methods
6.1 Elliptic optimal control problems
6.1,1 Recovery operator
6.1.2 Superconvergence property
6.2 Parabolic optimal control problems
62.1 Superconvergence for the intermediate errors
6.2.2 Superconvergence
6.3 Hyperbolic optimal control problems
6.3.1 Superconvergence property
6.32 Superconvergence for the control variable
6.4 Nonlinear optimal control problems
6.4.1 Supereonvergence for the intermediate errors
6.4.2 Global superconvergence
6.4.3 H-t-error estimates
6.5 Numerical examples
6.5,1 Elliptic optimal control problems
6.52 Nonlinear optimal control problems
Chapter 7 Finite volume element methods for optimal control problems
7.1 Elliptic optimal control problems
7.1.1 Finite volume element methods
7.12 L2-error estimates
7.1,3 Hj error estimates
7.1.4 Maximum-norm error estimates
7.2 Parabolic optimal control problems
7.2.1 Crank-Nicolson finite volume scheme
7.2,2 Error estimates of CN-FVEM
7.3 Hyperbolic optimal control problems
7.3.1 Finite volume element methods
7.32 A priori error estimates
7.4 Numerical examples
7.4.1 Elliptic optimal control problems
7.4.2 Parabolic optimal control problems
7.4.3 Hyperbolic optimal control problems
Chapter 8 Variational diseretization methods for optimal control problems ~ ~ ~
8.1 Variational discretization
8.1.1 Variational discretization scbeme
8.122 A priori error estimates
8.1.3 A posteriori error estimates
8~2 Mixed variational discretization
82.1 Mixed finite element approximation and variational discretization
82.2 A priori error estimates for semi-discrete scheme
82.3 A priori error estimates for fully discrete scheme
8.3 Numerical examples
8.3.1 Variational discretizatinn
8.3.2 Mixed variational discretization
Chapter 9 Legendre-Galerkin spectral methods for optimal control problems.
9.1 Elliptic optimal control problems
9.1 1 Legendre-Galerkin spectral approximation
9.1.2 Regularity of the optimal control
9.1.3 A priori error estimates
9.1.4 A posteriori error estimates
9.1.5 The hp spectral element methods
9.2 Parabolic optimal control problems
9.2.1 Legendre-Galerkin spectral methods
9.2.2 A priori error estimates
9.2.3 A posteriori error estimates
9.3 Optimal control problems governed by Stokes equations
9.3.1 Legendre-Galerkin spectral approximation
9.3.2 A priori error estimates
9.3.3 A posteriori error estimates
9.4 Optimal control problems with integral state and control constraints
9.4.1 Legendre-Galerkin spectral scheme
9.4.2 A priori error estimates
9.4.3 A posteriori error estimates
9.5 Numerical examples
9.5.1 Elliptic optimal control problems
9.5.2 Optimal control problems governed by Stokes equations
9.5.3 Optimal control problems with integral state and control constraints
Bibliography
Index
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