Local Error Estimates of the Finite Element Method for an Elliptic Problem with a Dirac Source Term

Silvia Bertoluzza (1) , Astrid Decoene (2) , Loïc Lacouture (2) , Sébastien Martin (3)

1 Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R.
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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.

Mots clés: finite element method; local error estimates.; Dirichlet problem; Green function; Dirac measure

Domaines: Analyse numérique [math.NA]

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-01150745v2⟩