%0 Journal Article
%T Superconvergence for General Convex Optimal Control Problems Governed by Semilinear Parabolic Equations
%A Yongquan Dai
%A Yanping Chen
%J ISRN Applied Mathematics
%D 2014
%R 10.1155/2014/579047
%X We will investigate the superconvergence for the semidiscrete finite element approximation of distributed convex optimal control problems governed by semilinear parabolic equations. The state and costate are approximated by the piecewise linear functions and the control is approximated by piecewise constant functions. We present the superconvergence analysis for both the control variable and the state variables. 1. Introduction Finite element approximation of optimal control problems plays a very important role in numerical methods for these problems. There have been extensive studies on this aspect, for example, [1¨C14]. A systematic introduction of finite element method for PDEs and optimal control problems can be found in, for example, [15¨C18]. The superconvergence of nonlinear parabolic problem was studied in [19]. In [20], superconvergence was obtained for parabolic optimal control problems with convex control constraints, where the state partial differential equations are linear. Optimal control problems governed by nonlinear parabolic state equations, a priori error estimates of finite element approximation, were studied in, for example, [21, 22]. In this paper, we will study the superconvergence of both the control variable and the state variables for this problem. The model optimal control problem that we shall study in detail is the following convex optimal control problem: Here, the bounded open set is a convex polygon or has the smooth boundary . Let be a linear continuous operator from to and . Assume that , with being a symmetric matrix and, for any vector , there is a constant satisfying Here, denotes the admissible set of the control variable, which is defined by In this paper, we adopt the standard notation for Sobolev spaces on with a norm given by a semi-norm given by We set . For , we denote We denote by the Banach space of all integrable functions from into , with norm and the standard modification for , where . Similarly, one can define the spaces and . The details can be found in [23]. In addition, and denote general positive constants independent of . The plan of the paper is as follows. In Section 2, we shall give a brief review on the finite element method and then construct the approximation scheme for the optimal control problem. In Section 3, we shall give some preliminaries and some intermediate error estimates. In Section 4, superconvergence results for both control and state variables were derived. In Section 5, we give a numerical example to demonstrate our theoretical results. In the last section we make a conclusion and
%U http://www.hindawi.com/journals/isrn.applied.mathematics/2014/579047/