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We study the problem $-\Delta u = \gamma$, where $\gamma$ is a singularmeasure, with support on a curve or a point. We prove that optimal rates ofconvergence for the finite element method can be obtained using properly gradedmeshes. In particular, we consider isotropic graded meshes when $\gamma$ is apoint Dirac delta, and anisotropic graded meshes when $\gamma$ is a measuresupported on a segment. Numerical experiments are shown that verify ourresults, and lead to interesting observations.

Anisotropic finite elements for elliptic problems with singular data