Superconvergence Properties For Finite Element Method Of Optimal Control Problems Governed By Elliptic And Parabolic Equations

Optimal control problems have been widely met in all kinds of practical problems, such as, temperature control problems, air pollution control problems, Stokes flow control problems, electrochemical machining design problems, etc. So it's significant to study the efficient numerical method for these problems. Finite element approximation of optimal control problems plays a very important role in numerical methods for these problems. There have been extensively studies on this aspect. Most of these works are focus on the priori or posteriori error estimates while there doesn't seem to exist much works on the superconvergence analysis, especially for the superconvergence analysis of the nonlinear optimal control problems. Superconvergence analysis is important and useful in numerical methods for solving partial differential equations.In this paper, we shall investigate superconvergence properties for some optimal control problems. And numerical examples are presented to illustrate the theoretical results. The main results are as follows:In the first part of this paper, we study the semi-linear elliptic optimal control problems. We first transform the minimization problem into the couple system of state equation, co-state equation and variational inequality. Due to the different regularity of the control variable and the state variable, we use different finite element spaces to approximation them. For control variable, we use piece-wise constant functions to approximate, while for state and co-state variable, we use piecewise linear functions. Then, by introducing intermediate variables to divide the error into several parts and making some assumption, we get our superconvergence properties for the quadratic optimal control problem, and some applications of the superconvergence properties are presented. At last, we extend the results of quadratic optimal control problem into general convex optimal control problem. Numerical examples are presented to demonstrate our theoretical results.In the second part of this paper, we discuss the parabolic optimal control problems. First, we study the superconvergence of the semi-discrete finite element method for the optimal control problems governed by linear parabolic equations. What follows is the superconvergence properties for the semi-linear parabolic optimal control problem. In these works, we need to introduce intermediate variables with which we can divided the error into several parts and make some assumption of the objective function. Then, we get our superconvergence properties between the finite element solutions and the projection of the exact solutions.