  This article deals with error estimates for the finite element approximation
of variational normal derivatives and, as a consequence, error estimates for
the finite element approximation of Dirichlet boundary control problems with
energy regularization. The regularity of the solution is carefully carved out
exploiting weighted Sobolev and H\"older spaces. This allows to derive a sharp
relation between the convergence rates for the approximation and the structure
of the geometry, more precisely, the largest opening angle at the vertices of
polygonal domains. Numerical experiments confirm that the derived convergence
rates are sharp.
