A Priori Error Analysis for the Finite Element Approximation of Elliptic Dirichlet Boundary Control Problems

This article presents recent results of an a priori error analysis for the finite element approximation of Dirichlet boundary control problems governed by elliptic partial differential equations. For a standard model problem error estimates are proven for the primal variable, the control, as well as the associated adjoint variable. These estimates are of optimal order with respect to the solution's regularity to be expected on polygonal domains. The proofs rely on the Euler-Lagrange formulation of the optimal control problem and employ standard duality techniques and optimal-order L-P error estimates for the finite element Ritz projection. These estimates improve corresponding results in the literature and are supported by computational experiments. The details are contained in [9].