Error estimates for Neumann boundary control problems with energy regularization

A Neumann boundary control problem for a second order elliptic state equation is considered which is regularized by an energy term which is equivalent to the H−1/2(Γ)-norm of the control. Both the unconstrained and the control constrained cases are investigated. The regularity of the state, control, and co-state variables is studied with particular focus on the singularities due to the corners of the two-dimensional domain. The state and co-state are approximated by piecewise linear finite elements. For the approximation of the control variable we take carefully designed spaces of piecewise linear or piecewise constant functions, such that an inf-sup condition is satisfied. Bounds for the discretization error are proved for all three variables in dependence on the largest interior angle of the domain. Numerical tests suggest that these bounds are optimal in the unconstrained case but too pessimistic in the control constrained case with non-convex domains.