A posteriori error control for distributed elliptic optimal control problems with control constraints discretized by [formula omitted]-finite elements

Abstract

A distributed elliptic control problem with control constraints is considered, which is formulated as a three field problem and consists of two variational equations for the state and the co-state variables as well as of a variational inequality for the control variable. Two discretization approaches with hp-finite elements are discussed: In the first discretization approach all variables (state, co-state and control) are discretized. A semi smooth Newton method is introduced for solving the resulting algebraic system. In the second discretization approach only the state and the co-state variables are discretized, whereas the control is determined by projection. A simple fixed point scheme is presented for the iterative solution of this approach. The main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates, which enables hp-adaptive mesh refinements. In particular, the estimates can be applied to the iteration solutions, so that they can be used as a stopping criterion of the iterative solution schemes. In several numerical experiments the order of convergence of the (adaptive) discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied.