Local error estimates of the finite element method for an elliptic problem with a Dirac source term

The solutions of elliptic problems with a Dirac measure right-hand side are not H1 in dimension d ? {2, 3} and therefore the convergence of the finite element solutions is suboptimal in the L2-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 Hs-norm, for s >= 1 on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence in H1-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.