A finite element method for elliptic Dirichlet boundary control problems

Abstract: We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in $H^{1/2}(\Gamma)$. To avoid computing the latter norm numerically, we realize it using the $H^{1}(\Omega)$ norm of the harmonic extension of the control. We propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier. In the case of convex polygonal domains, optimal error estimates in the $H^1$ and $L^2$ norm are proven. We also consider and analyze the case of control constrained problems.