Fully Discrete $H$1-Galerkin Mixed Finite Element Methods for Parabolic Optimal Control Problems

Abstract

In this paper, we investigate a priori and a posteriori error estimates of fully discrete $H$1-Galerkin mixed finite element methods for parabolic optimal control problems. The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. First, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. Second, by use of energy approach, we derive a posteriori error estimates for optimal control problems, assuming that only the underlying mesh is static. A numerical example is presented to verify the theoretical results on a priori error estimates.