Anisotropic finite elements for elliptic problems with singular data

Abstract

We study the problem $-Δu = γ$, where $γ$ is a singular measure, with support on a curve or a point. We prove that optimal rates of convergence for the finite element method can be obtained using properly graded meshes. In particular, we consider isotropic graded meshes when $γ$ is a point Dirac delta, and anisotropic graded meshes when $γ$ is a measure supported on a segment. Numerical experiments are shown that verify our results, and lead to interesting observations.