Anisotropic finite elements for elliptic problems with singular data

TL;DR

This paper investigates finite element methods for elliptic problems with singular data, demonstrating optimal convergence rates using graded meshes tailored to the measure's support, supported by numerical validation.

Contribution

It introduces the use of anisotropic graded meshes for measures supported on segments, extending previous isotropic approaches for point singularities.

Findings

Optimal convergence rates achieved with graded meshes

Anisotropic meshes improve accuracy for segment-supported measures

Numerical experiments confirm theoretical results

Abstract

We study the problem $-\Delta u = \gamma$, where $\gamma$ 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 $\gamma$ is a point Dirac delta, and anisotropic graded meshes when $\gamma$ is a measure supported on a segment. Numerical experiments are shown that verify our results, and lead to interesting observations.