Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T00:13:58.894Z Has data issue: false hasContentIssue false
  • English
  • Français

A penalty algorithm for the spectral element discretization of the Stokes problem*

Published online by Cambridge University Press:  02 August 2010

Christine Bernardi ,
Adel Blouza ,
Nejmeddine Chorfi  and
Nizar Kharrat
Christine Bernardi
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr
Adel Blouza
Affiliation:
Laboratoire de Mathématiques Raphaël Salem (U.M.R. 6085 C.N.R.S.), Université de Rouen, avenue de l'Université, B.P. 12, 76801 Saint-Étienne-du-Rouvray, France. Adel.Blouza@univ-rouen.fr
Nejmeddine Chorfi
Affiliation:
Department of Mathematics, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia. nejmeddine.chorfi@fst.rnu.tn
Nizar Kharrat
Affiliation:
Faculté des Sciences de Bizerte, ENIT-LAMSIN, B.P. 37, 1002 Tunis Le Belvédère, Tunisia. nizar.kharrat@enit.rnu.tn
Get access

Abstract

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

Keywords

Stokes problemspectral elementspenalty algorithm
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 2 , March 2011 , pp. 201 - 216
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ben Belgacem, F., Bernardi, C., Chorfi, N. and Maday, Y., Inf-sup conditions for the mortar spectral element discretization of the Stokes problem. Numer. Math. 85 (2000) 257–281. CrossRef
M. Bercovier, Régularisation duale des problèmes variationnels mixtes : application aux éléments finis mixtes et extension à quelques problèmes non linéaires. Thèse de Doctorat d'État, Université de Rouen, France (1976).
Bercovier, M., Perturbation of mixed variational problems. Application to mixed finite element methods. RAIRO Anal. Numér. 12 (1978) 211–236. CrossRef
Bernardi, C., Indicateurs d'erreur en h – N version des éléments spectraux. RAIRO Modél. Math. Anal. Numér. 30 (1996) 1–38. CrossRef
Bernardi, C. and Maday, Y., Polynomial approximation of some singular functions. Appl. Anal. 42 (1991) 1–32. CrossRef
C. Bernardi and Y. Maday, Spectral Methods, in Handbook of Numerical Analysis V, P.G. Ciarlet and J.-L. Lions Eds., North-Holland (1997) 209–485.
Bernardi, C. and Maday, Y., Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math. Mod. Meth. Appl. Sci. 9 (1999) 395–414. CrossRef
C. Bernardi, B. Métivet and R. Verfürth, Analyse numérique d'indicateurs d'erreur, in Maillage et adaptation, P.-L. George Ed., Hermès (2001) 251–278.
Bernardi, C., Girault, V. and Hecht, F., A posteriori analysis of a penalty method and application to the Stokes problem. Math. Mod. Meth. Appl. Sci. 13 (2003) 1599–1628. CrossRef
C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques & Applications 45. Springer-Verlag (2004).
Carey, G.F. and Krishnan, R., Penalty approximation of Stokes flow. Comput. Meth. Appl. Mech. Eng. 35 (1982) 169–206. CrossRef
Carey, G.F. and Krishnan, R., Penalty finite element method for the Navier–Stokes equations. Comput. Meth. Appl. Mech. Eng. 42 (1984) 183–224. CrossRef
Carey, G.F. and Krishnan, R., Convergence of iterative methods in penalty finite element approximation of the Navier–Stokes equations. Comput. Meth. Appl. Mech. Eng. 60 (1987) 1–29. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms . Springer-Verlag (1986).
Maday, Y., Meiron, D., Patera, A.T. and Rønquist, E.M., Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations. SIAM J. Sci. Comput. 14 (1993) 310–337. CrossRef
D.S. Malkus and E.T. Olsen, Incompressible finite elements which fail the discrete LBB condition, in Penalty-Finite Element Methods in Mechanics, Phoenix, Am. Soc. Mech. Eng., New York (1982) 33–50.