%0 Journal Article
%T A Posteriori Error Estimates for a Semidiscrete Parabolic Integrodifferential Control on Multimeshes
%A Wanfang Shen
%J Discrete Dynamics in Nature and Society
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/481295
%X We extend the existing techniques to study semidiscrete adaptive finite element approximation schemes for a constrained optimal control problem governed by parabolic integrodifferential equations. The control problem involves time accumulation and the control constrain is given in an integral obstacle sense. We first prove the uniqueness and existence of the solution of this optimal control problem. We then derive the upper a posteriori error estimators for both the state and the control approximation, which are useful indicators in adaptive multimesh finite element approximation schemes. 1. Introduction For the last twenty years great progress has been made on standard finite element approximation of optimal control problems governed by elliptic or parabolic equations; see, for example, [1¨C9], although it is impossible to give even a very brief review here. More specifically, research on finite element approximation of parabolic optimal control problems can be found in, for example, [10, 11]. Systematic introduction of the finite element method for PDEs and optimal control problems can be found in, for example, [5, 12, 13]. In many real modeling applications it is important to consider time accumulation. Parabolic integrodifferential equations and their control of this nature appear in heat conduction in materials with memory, population dynamics, and viscoelasticity; see, for example, Friedman and Shinbrot [14], Heard [15], and Renardy et al. [16]. This calls for more studies on the finite element approximation on integrodifferential equations. For instance, finite element methods for parabolic integrodifferential equations with a smooth kernel have been discussed in, for example, Cannon et al. [17], Sloan and Thom¨Ĥe [18], and Yanik and Fairweather [19]. Recently adaptive finite element method has been widely studied and applied in practical control computations. In order to obtain a numerical solution of acceptable accuracy for the optimal control problem, the finite element meshes have to be refined according to a mesh refinement scheme. Adaptive finite element approximation uses a posteriori indicators to guide the mesh refinement procedure. Only the area where the error indicator is larger will be refined so that a higher density of nodes is distributed over the area where the solution is difficult to approximate. In this sense efficiency and reliability of adaptive finite element approximation rely very much on the error indicator used. Adaptive finite element schemes have been widely studied for some optimal control problems governed by elliptic
%U http://www.hindawi.com/journals/ddns/2012/481295/