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

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Abstract

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.

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Authors and Affiliations

  1. Groupe Interdisciplinaire de Recherche en Éléments Finis (GIREF), Département de mathématiques et de statistique, Université Laval, 1045 avenue de la Médecine, Quebec, QC, G1V 0A6, Canada

    Ibrahima Dione & José M. Urquiza

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  1. Ibrahima Dione
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  2. José M. Urquiza
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Correspondence to José M. Urquiza.

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Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Dione, I., Urquiza, J.M. Penalty: finite element approximation of Stokes equations with slip boundary conditions. Numer. Math. 129, 587–610 (2015). https://doi.org/10.1007/s00211-014-0646-9

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