Optimal a priori error estimates in weighted Sobolev spaces for the Poisson problem with singular sources

Abstract

We study the problem -Δu=f, where f has a point-singularity. In particular, we are interested in f = δx0, a Dirac delta with support in x0, but singularities of the form f ~ |x − x0|−s are also considered. We prove the stability of the Galerkin projection on graded meshes in weighted spaces, with weights given by powers of the distance to x0. We also recover optimal rates of convergence for the finite element method on these graded meshes. Our approach is general and holds both in two and three dimensions. Numerical experiments are shown that verify our results, and lead to interesting observations.