  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 with order $1/2$ in convex domains but has
a reduced convergence order in non-convex domains. As a remedy, a dual variant
of the singular complement method is proposed. The error order of the convex
case is retained. Numerical experiments confirm the theoretical results.
