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

TL;DR

This paper investigates discretization methods for solving the Poisson equation with non-smooth boundary data on non-convex domains, analyzing regularization techniques and providing sharp error estimates supported by numerical tests.

Contribution

It introduces a regularization approach for the Dirichlet problem with non-smooth data, connecting it to Berggren's method and deriving precise error estimates on non-convex domains.

Findings

Regularization simplifies discretization of the Poisson problem with non-smooth data.

Error estimates are sharp and depend on the domain's interior angle.

Numerical tests confirm the theoretical error bounds, especially for non-convex domains.

Abstract

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 $2\pi$.