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Several approaches are discussed how to understand the solution of theDirichlet problem for the Poisson equation when the Dirichlet data arenon-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 functionsare considered, and a regularity result is proved. An approach of Berggren isrecovered as the method of transposition with the second variant of testfunctions. A further concept is the regularization of the boundary datacombined with the weak solution of the regularized problem. The effect of theregularization error is studied. The regularization approach is the simplest todiscretize. The discretization error is estimated for a sequence ofquasi-uniform meshes. Since this approach turns out to be equivalent toBerggren's discretization his error estimates are rendered more precisely.Numerical tests show that the error estimates are sharp, in particular that theorder becomes arbitrarily small when the maximal interior angle of the domaintends to $2\pi$.

Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains