Numerische Mathematik

, Volume 129, Issue 3, pp 587–610

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

Article

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.

Mathematics Subject Classification (2000)

Primary 06B10 Secondary 06D05 

References

  1. 1.
    Babuška, I.: Stabilität des Definitionsgebietes mit Rücksicht auf grundlegende Probleme der Theorie des partiellen Differentialgleichungen auch im Zusammenhang mit der Elasticitätstheorie 1, 2. Czechoslov. Math. J. 11(76–105), 165–203 (1961). (in Russian)Google Scholar
  2. 2.
    Babuška, I.: The theory of small changes in the domain of existence in the theory of partial differential equations and its applications. In: Differential Equations and their Applications, pp. 13–26. Academic Press, London (1963)Google Scholar
  3. 3.
    Babuška, I.: The finite element method with penalty. Math. Comput. 27, 221–228 (1973)Google Scholar
  4. 4.
    Bänsch, E., Deckelnick, K.: Optimal error estimates for the Stokes and Navier–Stokes equations with slip-boundary condition. Modélisation mathématique et analyse numérique (RAIRO). M2AN Math. Model. Numer. Anal. 33, 923–938 (1999)Google Scholar
  5. 5.
    Bernardi, C.: Optimal finite element interpolation on curved domains. SIAM J. Numer. Anal. 26, 212–234 (1989)Google Scholar
  6. 6.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. In: Springer Series in Computational Mathematics, vol. 15 (1991)Google Scholar
  7. 7.
    Caglar, A., Liakos, A.: Weak imposition of boundary conditions for the Navier–Stokes equations by a penalty method. Int. J. Numer. Methods Fluids 61, 411–431 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cuvelier, C., Driessen, J.M.: Thermocapillary free boundaries in crystal growth. J. Fluid Mech. 169, 1–26 (1986)CrossRefMATHGoogle Scholar
  9. 9.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, Evolution Problems I, vol. 5. Springer, New York (2000)Google Scholar
  10. 10.
    Dione, I., Tibirna, C., Urquiza, J.M.: Stokes equations with penalized slip boundary conditions. Int. J. Comput. Fluid Dyn. 27, 283–296 (2013)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Gazzola, F., Grunau, H.C., Sweers, G.: Polyharmonic Boundary Value Problems. Springer, New York (2010)CrossRefMATHGoogle Scholar
  12. 12.
    Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations, theory and algorithms. In: Springer Series in Computational Mathematics, vol. 5 (1986)Google Scholar
  13. 13.
    Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988)Google Scholar
  14. 14.
    Kistler, S.F., Scriven, L.E.: Coating flow theory by finite element and asymptotic analysis of the Navier–Stokes system. Int. J. Numer. Methods Fluids 4, 207–229 (1984)Google Scholar
  15. 15.
    Knobloch, P.: Variational crimes in finite element discretization of 3D Stokes equation with nonstandard boundary conditions. East–West J. Numer. Math. 7, 133–158 (1999)MATHMathSciNetGoogle Scholar
  16. 16.
    Lenoir, M.: Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23, 562–580 (1986)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Maury, B.: Numerical analysis of a finite element/volume penalty method. SIAM J. Numer. Anal. 47, 1126–1148 (2009)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Mohammadi, B., Pironneau, O.: Analysis of the \(k-\epsilon \) turbulence model. Res. Appl. Math. Masson, Paris, John Wiley & Sons, Ltd., Chichester (1994)Google Scholar
  19. 19.
    Neto, C., Evans, D.R., Bonaccurso, E., Butt, H.-J., Craig, V.S.J.: Boundary slip in Newtonian liquids: a review of experimental studies. Rep. Prog. Phys. 68, 2859–2897 (2005)CrossRefGoogle Scholar
  20. 20.
    Raviart, P.-A., Thomas, J.-M.: Introduction à l’Analyse Numérique des Équations aux Dérivées Partielles, Dunod (1998)Google Scholar
  21. 21.
    Solonnikov, S.A., Scadilov, V.E.: On a boundary value problem for a stationary system of Navier–Stokes equations. Proc. Steklov Inst. Math. 125, 186–199 (1973)MATHMathSciNetGoogle Scholar
  22. 22.
    Utku, M., Carey, G.F.: Boundary penalty techniques. Comput. Methods Appl. Mech. Eng. 30, 103–118 (1982)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Verfürth, R.: Finite element approximation of steady Navier–Stokes equations with mixed boundary condition. Modélisation Mathématique et Anal. Numérique (RAIRO) 19, 461–475 (1985)MATHGoogle Scholar
  24. 24.
    Verfürth, R.: Finite element approximation of incompressible Navier–Stokes equations with slip boundary condition. Numer. Math. 50, 697–721 (1987)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Verfürth, R.: Finite element approximation of incompressible Navier–Stokes equations with slip boundary condition II. Numer. Math. 59, 615–636 (1991)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Zhong-CI, S.: On the convergence rate of the boundary penalty method. Int. J. Numer. Methods Eng. 20, 2027–2032 (1984)CrossRefMATHGoogle Scholar

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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