  This paper is concerned with approximations and related discretization error
estimates for the normal derivatives of solutions of linear elliptic partial
differential equations. In order to illustrate the ideas, we consider the
Poisson equation with homogeneous Dirichlet boundary conditions and use
standard linear finite elements for its discretization. The underlying domain
is assumed to be polygonal but not necessarily convex. Approximations of the
normal derivatives are introduced in a standard way as well as in a variational
sense. On general quasi-uniform meshes, one can show that these approximate
normal derivatives possess a convergence rate close to one in $L^2$ as long as
the singularities due to the corners are mild enough. Using boundary
concentrated meshes, we show that the order of convergence can even be doubled
in terms of the mesh parameter while increasing the complexity of the discrete
problems only by a small factor. As an application, we use these results for
the numerical analysis of Dirichlet boundary control problems, where the
control variable corresponds to the normal derivative of some adjoint variable.
