The Convergence and Superconvergence of a MFEM for Elliptic Optimal Control Problems

Abstract

In this paper, we investigate a mixed finite element method (MFEM) for the elliptic optimal control problems (OCPs) with a distributive control. The state variable and adjoint state variable are approximated by the conforming rectangular $Q_{11}+Q_{01}\times Q_{10}$ elements pair. The discrete B-B condition is satisfied automatically, which is usually considered to be the key point of the MFEM. The control is then obtained by the orthogonal projection through the adjoint state. Optimal orders of convergence are derived for the above mentioned variables. Furthermore, supercloseness and superconvergence results are also established under certain reasonable regularity assumptions. Some numerical results are provided to verify the theoretical analysis. At last, the proposed method is extended to some other low order conforming and nonconforming elements.