Adaptive finite element method for elliptic optimal control problems: convergence and optimality
In this paper we consider the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints. We use variational discretization concept to discretize the control variable and piecewise linear and continuous finite elements to approximate the state variable. Based on the well-established convergence theory of AFEM for elliptic boundary value problems, we rigorously prove the convergence and quasi-optimality of AFEM for optimal control problems with respect to the state and adjoint state variables, by using the so-called perturbation argument. Numerical experiments confirm our theoretical analysis.
Finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold
This paper is devoted to the study of finite element approximations to parabolic optimal control problems with controls acting on a lower dimensional manifold. The manifold can be a point, a curve, or a surface which may be independent of time or evolve in the time horizon, and is assumed to be strictly contained in the space domain. At first, we obtain the first order optimality conditions for the control problems and the corresponding regularity results. Then, for the control problems we consider the fully discrete finite element approximations based on the piecewise constant discontinuous Galerkin scheme for time discretization and piecewise linear finite elements for space discretization, and variational discretization to the control variable. A priori error estimates are finally obtained for the fully discretized control problems and supported by numerical examples.
Multiscale Approach for Optimal Design in Conductivity of Composite Materials
This paper discusses the multiscale approach for optimal design in conductivity of composite materials. The homogenization method and the multiscale asymptotic method are presented. The associated numerical algorithms and the convergence analysis are provided. Finally, numerical examples are carried out to confirm the validity of the algorithm.
A multilevel correction method for optimal controls of elliptic equation
We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving an optimization problem on the finest finite element space is transformed into a series of solutions of linear boundary value problems by the multigrid method on multilevel meshes and a series of solutions of optimization problems on the coarsest finite element space. Our proposed scheme, instead of solving a large scale optimization problem in the finest finite element space, solves only a series of linear boundary value problems and the optimization problems in a very low dimensional finite element space, and thus can improve the overall efficiency of the solution of optimal control problems governed by PDEs.
Approximations of Elliptic Optimal Control Problems with Controls Acting on a Lower Dimensional Manifold
In this paper, we study finite element approximations to some elliptic optimal control problems with controls acting on a lower dimensional manifold which can be a point, a curve, or a surface. We use piecewise linear finite elements to approximate state variables, while utilizing the variational discretization to approximate control variables. We derive several a priori error estimates for optimal controls from different cases depending on the dimensions of the computational domain and the manifold where controls act. We ends up with extensive numerical experiments which confirm our theoretical findings.
- 有限元超收敛及后验误差估计, Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods, Science Press, 2008-08
- PDE最优控制问题的自适应有限元方法, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, 2008-11
- 椭圆方程有限元整体超收敛及其应用, Global Superconvergence of Finite Element Methods for Elliptic Equations and Its Applications, 科学出版社, 2012-06