Local Error Estimates of the Finite Element Method for an Elliptic Problem with a Dirac Source Term
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.
Keywords
Cite
@article{arxiv.1505.03032,
title = {Local Error Estimates of the Finite Element Method for an Elliptic Problem with a Dirac Source Term},
author = {Silvia Bertoluzza and Astrid Decoene and Loïc Lacouture and Sébastien Martin},
journal= {arXiv preprint arXiv:1505.03032},
year = {2015}
}