An energy space finite element approach for elliptic Dirichlet boundary control problems

In this paper we present a finite element analysis for a Dirichlet boundary control problem where the Dirichlet control is considered in a convex closed subspace of the energy space $$H^{1/2}(\Gamma )$$H1/2(Γ). As an equivalent norm in $$H^{1/2}(\Gamma )$$H1/2(Γ) we use the energy norm induced by the so-called Steklov–Poincaré operator which realizes the Dirichlet to Neumann map, and which can be implemented by using standard finite element methods. The presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems, and their finite element approximations. Some numerical results are given, which confirm on one hand the theoretical estimates, but on the other hand indicate the differences when modelling the control in $$L_2(\Gamma )$$L2(Γ).