Numerische Mathematik

, Volume 129, Issue 3, pp 587–610 | Cite as

Penalty: finite element approximation of Stokes equations with slip boundary conditions

  • Ibrahima Dione
  • José M. Urquiza


We consider the finite element approximation of the stationary Stokes equations with slip boundary conditions on a domain with a smooth curved boundary. The slip boundary condition is imposed weakly with the penalty method on polygonal domains approaching the smooth domain. For Taylor-Hood elements, we derive error estimates which depend on the penalty parameter \(\varepsilon \), the disctretization parameter \(h\) and the approximation error of the normal to the boundary. In particular, if in the penalty term we use the normal to the polygonal boundary, the best convergence order is \(2/3\) and it is obtained with \(\varepsilon =c \, h^{2/3}\). This convergence result shows that Babuška’s paradox associated to Stokes equations with slip boundary conditions is circumvented. A numerical example illustrates the theoretical results, notably that regularized normal approximations give better approximations and convergence orders.

Mathematics Subject Classification (2000)

Primary 06B10 Secondary 06D05 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Groupe Interdisciplinaire de Recherche en Éléments Finis (GIREF), Département de mathématiques et de statistiqueUniversité LavalQuebecCanada

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