Adapted numerical methods for the numerical solution of the Poisson equation with $L^2$ boundary data in non-convex domains

The very weak solution of the Poisson equation with $L^2$ boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in the $L^2(\Omega)$-norm with order $1/2$ in convex domains but has a reduced convergence order in non-convex domains although the solution remains to be contained in $H^{1/2}(\Omega)$. The reason is a singularity in the dual problem. In this paper we propose and analyze, as a remedy, both a standard finite element method with mesh grading and a dual variant of the singular complement method. The error order 1/2 is retained in both cases also with non-convex domains. Numerical experiments confirm the theoretical results.