Optimal Local Error Estimates for Finite Element Methods with Measure-Valued Sources

TL;DR

This paper establishes optimal local error estimates for finite element methods solving elliptic problems with measure-valued sources, showing that convergence loss is confined locally near singularities.

Contribution

The paper introduces a weak solution framework and interior estimate techniques to derive optimal local error bounds for FEM with measure-valued sources on Lipschitz domains.

Findings

Global convergence rates are limited by source singularity.

Optimal local error estimates are achieved away from the source.

Numerical experiments confirm theoretical predictions.

Abstract

We study finite element approximations of second-order elliptic problems with measure-valued right-hand sides supported on lower-dimensional sets. The exact solution generally lacks $H^1$-regularity due to the source singularity, which limits global convergence rates of numerical methods. Using a very weak solution framework, we establish well-posedness and global error estimates for standard Lagrange finite element methods on Lipschitz polyhedral/polygonal domains. By using interior estimates techniques, we prove optimal local $L^2$- and $H^1$-error estimates in subdomains that are strictly separated from the support of the measure. Extensive numerical experiments are provided to verify the theoretical results. These results show that for Lagrange FEMs solving elliptic problems with singular right-hand sides, the loss of global convergence is purely local, and that optimal convergence rates still hold away from the singular source.