Superconvergence Of Rectangular Mixed Finite Element Methods For Constrainted Quadratic Optimal Control Problems

Recently,optimal control problems are utilized widely in many aspects of modern life.They have various application backgrounds in the operation of physical,social and economic processes.Finite element approximation of optimal control problem plays a very important role in numerical method of these problems.The aim task of this paper is to investigate the superconvergence property of the numerical solution of a quadratic convex optimal control by using rectangular mixed finite element methods.The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions.We prove the superconvergence of O(h^s+1)(for some 0＜s≤1)for the control variable with the lowest order Raviart-Thomas mixed finite elements based on rectangulation.We derive the local and global L² superconvergence analysis by applying operator interpolation technique.And the article is composed of three parts.In the first part,we present some examples of optimal control for elliptic systems.In the second part,we first construct a discretized scheme for the optimal control problem(1.1-1.3),next we consider the local L² superconvergence,then some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Last,we carry out the global L² superconvergence of rectangular lowest order Raviart-Thomas mixed finite element methods.In the third part,two numerical examples are present to demonstrate our theoretical results.Finally,we summarize the main ideas of this article and make some comments on the prospect of optimal control problem.