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Local error estimates of the finite element method for an elliptic problem with a Dirac source term
Author(s) -
Bertoluzza Silvia,
Decoene Astrid,
Lacouture Loïc,
Martin Sébastien
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22186
Subject(s) - mathematics , finite element method , dirac measure , mathematical analysis , singularity , measure (data warehouse) , dirac (video compression format) , norm (philosophy) , dirac equation , lemma (botany) , dirac algebra , dirac spinor , mathematical physics , quantum mechanics , physics , neutrino , thermodynamics , ecology , poaceae , database , biology , computer science , law , political science
The solutions of elliptic problems with a Dirac measure right‐hand side are notH 1in dimension d ∈ { 2 , 3 } and therefore the convergence of the finite element solutions is suboptimal in theL 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 theH s‐norm, for s ≥ 1 on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence inH 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.