Mixed Finite Element Methods And Superconvergence For Some Optimal Control Problems

There have existed many works about the optimal control problems governed by partial differential equations.Most of these researches aim at the standard finite element method while there doesn't seem to exist much works on theoretical analysis of mixed finite element methods.Mixed finite element methods have many advantages.For example,in computational fluid control problem both the scalar variable and its flux variable can be approximated in the same accuracy by mixed methods.It has many significance to extend the standard finite element method to mixed finite element methods for optimal control problems.In this paper,we will investigate the error estimates and superconvergence properties for some optimal control problems by mixed methods.The paper consists of two parts.In the first part,we study the elliptic optimal control problem.In the analysis,we transform a minimization problem to a coupled system of state equation,co-state equation and a variational inequality .Then,we discretize the obtained optimality condition by mixed finite element method and approximate the states variable(scalar and vector) and control variable with different finite element spaces.First,for quadratic and general convex functional,we investigate the maximum error estimates for optimal control problem with obstacle constraint set.More importantly,we introduce a special projection operator aiming at the low regularity of control variable.Then,only for the quadratic functional,we give a priori error estimate for one class of optimal control problem with special constraint set.At last,some numerical experiments are proposed.In the second part,we discuss the parabolic optimal control problem.There have existed some researches fromⅤ.Thomée for parabolic equation which don't focus on the optimal control problem.First,we study a priori error estimates for this problem by mixed methods.What follows is the superconvergence analysis under rectangulation.In these works,the difficulty is how to use the convexity and continous differentiability of the objective functional.Moreover,we need to construct some intermediate variables with which we can deompose the finite clement error into several parts.At last,we prove that the projection of state,costate variables and control variable are superclose,which is O(h^3/2),to their approximations.