Local Error Estimates of the Finite Element Method for an Elliptic Problem with a Dirac Source Term

TL;DR

This paper establishes quasi-optimal and optimal convergence rates for finite element solutions of elliptic problems with Dirac sources, using local error estimates and graded meshes in two dimensions.

Contribution

It introduces new local error estimates and convergence results for finite element methods applied to elliptic problems with singular sources, extending previous results to Hs-seminorms.

Findings

Quasi-optimal convergence in Hs-seminorm for s > 0

Optimal convergence in H1-seminorm for lowest order elements

Numerical results confirm theoretical error estimates

Abstract

The solutions of elliptic problems with a Dirac measure in right-hand side are not H1 and therefore the convergence of the finite element solutions is suboptimal. Graded meshes are standard remedy to recover quasi-optimality, namely optimality up to a log-factor, for low order finite elements in L2-norm. Optimal (or quasi-optimal for the lowest order case) convergence has been shown in L2-seminorm, where the L2-seminorm is defined as the L2-norm on a subdomain which excludes the singularity. Here we show a quasi-optimal convergence for the Hs-seminorm, s \textgreater{} 0, and an optimal convergence in H1-seminorm for the lowest order case, on a family of quasi- uniform meshes in dimension 2. This question is motivated by the use of the Dirac measure as a reduced model in physical problems, and a high accuracy at the singularity of the finite element method is not required. Our results are obtained using local Nitsche and Schatz-type error estimates, a weak version of Aubin-Nitsche duality lemma and a discrete inf-sup condition. These theoretical results are confirmed by numerical illustrations.