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 right‐hand side are not in dimension and therefore the convergence of the finite element solutions is suboptimal in the ‐norm. In this article, we address the numerical analysis of the finite element method for the Laplace equation with Dirac source term: we consider, in dimension 3, the Dirac measure along a curve and, in dimension 2, the punctual Dirac measure. The study of this problem is motivated by the use of the Dirac measure as a reduced model in physical problems, for which high accuracy of the finite element method at the singularity is not required. We show a quasioptimal convergence in the ‐norm, for on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence in ‐norm is shown on a family of quasiuniform meshes. 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.