Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains
For numerical analysts solving PDEs on non-convex domains with rough boundary data, this provides rigorous error estimates and clarifies relationships between different weak formulations.
The paper analyzes discretization of the Poisson equation with non-smooth Dirichlet data, focusing on non-convex domains. It proves regularity results for the transposition method, shows equivalence to Berggren's approach, and provides sharp error estimates that degrade as the maximal interior angle approaches 2π.
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 $2π$.