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Numerische Mathematik

, Volume 140, Issue 3, pp 677–701 | Cite as

Local error analysis for the Stokes equations with a punctual source term

  • Silvia Bertoluzza
  • Astrid Decoene
  • Loïc Lacouture
  • Sébastien Martin
Article
First Online:
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Revised:

Abstract

The solution of the Stokes problem with a punctual force in source term is not \(H^1 \times \mathbb {L}^2\) and therefore the approximation by a finite element method is suboptimal. In the case of Poisson problem with a Dirac mass in the right-hand side, an optimal convergence for the Lagrange finite elements has been shown on a subdomain which excludes the singularity in \(\mathbb {L}^2\)-norm by Köppl and Wohlmuth. Here we show a quasi-optimal local convergence in \(H^1 \times \mathbb {L}^2\)-norm for a \(\mathbb {P}_{k}/\mathbb {P}_{k-1}\)-finite element method, \(k \geqslant 2\), and for the \(\mathbb {P}_{1}{\mathrm{b}}/\mathbb {P}_{1}\). The error is still analysed on a subdomain which does not contain the singularity. The proof is based on local Arnold and Liu error estimates, a weak version of Aubin–Nitsche duality lemma applied to the Stokes problem and discrete inf-sup conditions. These theoretical results are generalized to a wide class of finite element methods, before being illustrated by numerical simulations.

Mathematics Subject Classification

65M60 65M15 76D07 
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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Silvia Bertoluzza
    • 1
  • Astrid Decoene
    • 2
  • Loïc Lacouture
    • 3
  • Sébastien Martin
    • 4
  1. 1.CNR IMATI Enrico MagenesPaviaItaly
  2. 2.Université Paris Sud, Laboratoire de mathématiques d’Orsay (CNRS-UMR 8628)Orsay cedexFrance
  3. 3.Institut national des Sciences appliquées de Toulouse, GMMToulouse Cedex 4France
  4. 4.Université Paris Descartes, Laboratoire MAP5 (CNRS UMR 8145)Paris cedex 06France

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