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Preprints, Working Papers, ... Year : 2015

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 > 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.
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Dates and versions

hal-01150745 , version 2 (16-07-2015)
hal-01150745 , version 3 (25-11-2015)
hal-01150745 , version 4 (02-02-2016)
hal-01150745 , version 5 (06-02-2016)

Identifiers

  • HAL Id : hal-01150745 , version 5

Cite

Silvia Bertoluzza, Astrid Decoene, Loïc Lacouture, Sébastien Martin. Local Error Estimates of the Finite Element Method for an Elliptic Problem with a Dirac Source Term. 2015. ⟨hal-01150745v5⟩
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