  Several approaches are discussed how to understand the solution of the
Dirichlet problem for the Poisson equation when the Dirichlet data are
non-smooth such as if they are in $L^2$ only. For the method of transposition
(sometimes called very weak formulation) three spaces for the test functions
are considered, and a regularity result is proved. An approach of Berggren is
recovered as the method of transposition with the second variant of test
functions. A further concept is the regularization of the boundary data
combined with the weak solution of the regularized problem. The effect of the
regularization error is studied. The regularization approach is the simplest to
discretize. The discretization error is estimated for a sequence of
quasi-uniform meshes. Since this approach turns out to be equivalent to
Berggren's discretization his error estimates are rendered more precisely.
Numerical tests show that the error estimates are sharp, in particular that the
order becomes arbitrarily small when the maximal interior angle of the domain
tends to $2\pi$.
