A Priori Error Analysis for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems

In this article we discuss a priori error estimates for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations and subject to inequality control constraints. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters taking into account the spatial and the temporal regularity of the optimal solution. For the treatment of the control discretization we discuss different approaches extending techniques known from the elliptic case. For detailed proofs and numerical results we refer to [18, 19].