  The paper deals with finite element approximations of elliptic Dirichlet
boundary control problems posed on two-dimensional polygonal domains. Error
estimates are derived for the approximation of the control and the state
variables. Special features of unconstrained and control constrained problems
as well as general quasi-uniform meshes and superconvergence meshes are
carefully elaborated. Compared to existing results, the convergence rates for
the control variable are not only improved but also fully explain the observed
orders of convergence in the literature. Moreover, for the first time, results
in non-convex domains are provided.
