Chapter 1 Introduction
Chapter 2 Some preliminaries
2.1 Sobolev spaces
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 discretization
3.1.3 A posteriori error estimates
3.2 Parabolic optimal control problems
3.2.1 Fully discrete finite element approximation
3.2.2 Intermediate error estimates
3.2.3 Superconvergence
3.3 Optimal control problems with oscillating coefficients
3.3.1 Finite element scheme
3.3.2 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 ofintermediate variables
3.4.3 Superconvergence
3.4.4 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
4.1.1 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
4.3.2 A priori error estimates
4.4 Fourth order optimal control problems
4.4.1 Mixed finite element scheme
4.4.2 L2-error estimates
4.4.3 L ∞-error estimates
4.5 Nonlmear 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
5.2.1 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
5.3.3 A posteriori error estimates for state variables
5.4 Nonlinear optimal control problems
5.4.1 Mixed finite element discretization
5.4.2 Intermediate error estimates
5.4.3 A posteriori 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
6.2.1 Superconvergence for the intermediate errors
6.2.2 Superconvergence
6.3 Hyperbolic optimal control problems
6.3.1 Superconvergence property
6.3.2 Superconvergence for the control variable
6.4 Nonlinear optimal control problems
6.4.1 Superconvergence for the intermediate errors
6.4.2 Global superconvergence
6.4.3 H—1—error estimates
6.5 Numerical examples
6.5.1 Elliptic optimal control problems
6.5.2 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
7.1.2 L2—error estimates
7.1.3 H1 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
7.3.2 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 discretization methods for optimal control problems
8.1 Variational discretization
8.1.1 Variational discretization scheme
8.1.2 A priori error estimates
8.1.3 A posteriori error estimates
8.2 Mixed variational discretization
8.2.1 Mixed finite element approximation and variational discretization
8.2.2 A priori error estimates for semi—discrete scheme
8.2.3 A priori error estimates for fully discrete scheme
8.3 Numerical examples
8.3.1 Variational discretization
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
评论
本书主要介绍了几类*控制问题的算法，包括了椭圆*控制问题、抛物*控制问题、双曲*控制问题、四阶*控制问题等新近热门领域，结合了作者本人在*控制问题方面的研究成果，并根据作者对有限元方法、变分离散方法、混合有限元方法、有限体积法和谱方法的理解和研究生教学要求，全面、客观的评价了这几类*控制问题的数值计算方法，并列举了很多数值算例，阐述了许多新的学术观点，具有较大的学术价值。
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