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The very weak solution of the Poisson equation with $L^2$ boundary data isdefined by the method of transposition. The finite element solution withregularized 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 domainsalthough the solution remains to be contained in $H^{1/2}(\Omega)$. The reasonis a singularity in the dual problem. In this paper we propose and analyze, asa remedy, both a standard finite element method with mesh grading and a dualvariant of the singular complement method. The error order 1/2 is retained inboth cases also with non-convex domains. Numerical experiments confirm thetheoretical results.

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