A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

In this paper, we propose a new mixed finite element method, called stabilized mixed finite element method, for the approximation of optimal control problems constrained by a first-order elliptic system. This method is obtained by adding suitable elementwise least-squares residual terms for the primal state variable y and its flux σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}. We prove the coercive and continuous properties for the new mixed bilinear formulation at both continuous and discrete levels. Therefore, the finite element function spaces do not require to satisfy the Ladyzhenkaya–Babuska–Brezzi consistency condition. Furthermore, the state and flux state variables can be approximated by the standard Lagrange finite element. We derive optimality conditions for such optimal control problems under the concept of Discretization-then-Optimization, and then a priori error estimates in a weighted norm are discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the stabilized method.