Local error estimates of the finite element method for an elliptic problem with a Dirac source term. Issue 1 (19th September 2017)

• Record Type: Journal Article
• Authors: Bertoluzza, Silvia; Decoene, Astrid; Lacouture, Loïc; Martin, Sébastien
• Is Part Of: Numerical methods for partial differential equations. Volume 34:Issue 1(2018)
• Page Start: 97
• Page End: 120
• Publication Date: 2017-09-19
• Subjects: Dirichlet problem -- Dirac measure -- green function -- finite element method -- local error estimates; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22186
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : The solutions of elliptic problems with a Dirac measure right‐hand side are not H 1 in dimension d ∈ { 2, 3 } and therefore the convergence of the finite element solutions is suboptimal in the L 2 ‐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 H s ‐norm, for s ≥ 1 on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence in H 1 ‐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.