Adapted Numerical Methods for the Poisson Equation with L² Boundary Data in NonConvex Domains

The very weak solution of the Poisson equation with L² boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in the L²(Ω)-norm with order 1/2 in convex domains but has a reduced convergence order in nonconvex domains although the solution remains to be contained in H1/2(Ω). 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 nonconvex domains. Numerical experiments confirm the theoretical results.