A penalty algorithm for the spectral element discretization of the Stokes problem
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 2, pp. 201-216.

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.

DOI : 10.1051/m2an/2010038
Classification : 76D07, 76M22
Mots clés : Stokes problem, spectral elements, penalty algorithm
@article{M2AN_2011__45_2_201_0,
     author = {Bernardi, Christine and Blouza, Adel and Chorfi, Nejmeddine and Kharrat, Nizar},
     title = {A penalty algorithm for the spectral element discretization of the {Stokes} problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {201--216},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {2},
     year = {2011},
     doi = {10.1051/m2an/2010038},
     mrnumber = {2804636},
     zbl = {1267.76023},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010038/}
}
TY  - JOUR
AU  - Bernardi, Christine
AU  - Blouza, Adel
AU  - Chorfi, Nejmeddine
AU  - Kharrat, Nizar
TI  - A penalty algorithm for the spectral element discretization of the Stokes problem
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2011
SP  - 201
EP  - 216
VL  - 45
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010038/
DO  - 10.1051/m2an/2010038
LA  - en
ID  - M2AN_2011__45_2_201_0
ER  - 
%0 Journal Article
%A Bernardi, Christine
%A Blouza, Adel
%A Chorfi, Nejmeddine
%A Kharrat, Nizar
%T A penalty algorithm for the spectral element discretization of the Stokes problem
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2011
%P 201-216
%V 45
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2010038/
%R 10.1051/m2an/2010038
%G en
%F M2AN_2011__45_2_201_0
Bernardi, Christine; Blouza, Adel; Chorfi, Nejmeddine; Kharrat, Nizar. A penalty algorithm for the spectral element discretization of the Stokes problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 2, pp. 201-216. doi : 10.1051/m2an/2010038. http://www.numdam.org/articles/10.1051/m2an/2010038/

[1] F. Ben Belgacem, C. Bernardi, N. Chorfi and Y. Maday, Inf-sup conditions for the mortar spectral element discretization of the Stokes problem. Numer. Math. 85 (2000) 257-281. | MR | Zbl

[2] 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).

[3] M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods. RAIRO Anal. Numér. 12 (1978) 211-236. | Numdam | MR | Zbl

[4] C. Bernardi, Indicateurs d'erreur en h - N version des éléments spectraux. RAIRO Modél. Math. Anal. Numér. 30 (1996) 1-38. | Numdam | MR | Zbl

[5] C. Bernardi and Y. Maday, Polynomial approximation of some singular functions. Appl. Anal. 42 (1991) 1-32. | MR | Zbl

[6] 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. | MR | Zbl

[7] C. Bernardi and Y. Maday, Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math. Mod. Meth. Appl. Sci. 9 (1999) 395-414. | MR | Zbl

[8] 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.

[9] C. Bernardi, V. Girault and F. Hecht, A posteriori analysis of a penalty method and application to the Stokes problem. Math. Mod. Meth. Appl. Sci. 13 (2003) 1599-1628. | MR | Zbl

[10] C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques & Applications 45. Springer-Verlag (2004). | MR | Zbl

[11] G.F. Carey and R. Krishnan, Penalty approximation of Stokes flow. Comput. Meth. Appl. Mech. Eng. 35 (1982) 169-206. | MR | Zbl

[12] G.F. Carey and R. Krishnan, Penalty finite element method for the Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 42 (1984) 183-224. | MR | Zbl

[13] G.F. Carey and R. Krishnan, Convergence of iterative methods in penalty finite element approximation of the Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 60 (1987) 1-29. | MR | Zbl

[14] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms . Springer-Verlag (1986). | MR | Zbl

[15] Y. Maday, D. Meiron, A.T. Patera and E.M. Rønquist, Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations. SIAM J. Sci. Comput. 14 (1993) 310-337. | MR | Zbl

[16] 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. | MR | Zbl

Cité par Sources :