Error Analysis for a Finite Element Approximation of Elliptic Dirichlet Boundary Control Problems

We consider the Galerkin finite element approximation of an elliptic Dirichlet boundary control model problem governed by the Laplacian operator. The analytical setting of this problem uses L² controls and a “very weak” formulation of the state equation. However, the corresponding finite element approximation uses standard continuous trial and test functions. For this approximation, we derive a priori error estimates of optimal order, which are confirmed by numerical experiments. The proofs employ duality arguments and known results from the Lᵖ error analysis for the finite element Dirichlet projection.