Stability of the finite element Stokes projection in <i>W</i><sup>1,∞</sup>

Vivette Girault; Ricardo H. Nochetto; Ridgway Scott

doi:10.1016/j.crma.2004.04.005

Numerical Analysis

Stability of the finite element Stokes projection in W^1,∞
[Stabilité dans W^1,∞ de la projection de Stokes par éléments finis]

Présenté par : Philippe G. Ciarlet

Vivette Girault ¹ ; Ricardo H. Nochetto ² ; Ridgway Scott ³

¹ Laboratoire Jacques-Louis Lions, université P. et M. Curie, 75252 Paris cedex 05, France
² Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742–4015, USA
³ Department of Mathematics and the Computation Institute, University of Chicago, Chicago, IL 60637–1581, USA

Comptes Rendus. Mathématique, Volume 338 (2004) no. 12, pp. 957-962.

Résumé
Abstract

Nous démontrons que la norme du maximum du gradient de la vitesse et celle de la pression, calculés par des méthodes d'éléments finis usuelles pour discrétiser le problème de Stokes, sont bornées indépendamment du pas de la discrétisation. La démonstration est basée sur des estimations à poids dans L² pour des fonctions de Green associées au problème de Stokes et sur une condition inf–sup à poids. Le domaine est un polygone ou un polyèdre à frontière lipschitzienne dont les angles intérieurs satisfont des conditions suffisantes convenables pour assurer que la solution exacte est aussi bornée dans $W^{1, \infty} (Ω) \times L^{\infty} (Ω)$ . La famille de triangulations est uniformément régulière. Nous employons une propriété de super-approximation que nous démontrons pour des espaces d'éléments finis couramment utilisés.

We prove stability of the finite element Stokes projection in the product space $W^{1, \infty} (Ω) \times L^{\infty} (Ω)$ . The proof relies on weighted L² estimates for regularized Green's functions associated with the Stokes problem and on a weighted inf–sup condition. The domain is a polygon or a polyhedron with a Lipschitz-continuous boundary, satisfying suitable sufficient conditions on the inner angles of its boundary, so that the exact solution is bounded in $W^{1, \infty} (Ω) \times L^{\infty} (Ω)$ . The family of triangulations is shape-regular and quasi-uniform. The finite element spaces satisfy a super-approximation property, which is shown to be valid for commonly used stable finite element spaces.

Reçu le : 2004-04-02
Accepté le : 2004-04-06
Publié le : 2004-05-12

DOI : 10.1016/j.crma.2004.04.005

Affiliations des auteurs :

Vivette Girault ¹ ; Ricardo H. Nochetto ² ; Ridgway Scott ³

@article{CRMATH_2004__338_12_957_0,
     author = {Vivette Girault and Ricardo H. Nochetto and Ridgway Scott},
     title = {Stability of the finite element {Stokes} projection in {\protect\emph{W}\protect\textsuperscript{1,\ensuremath{\infty}}}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {957--962},
     publisher = {Elsevier},
     volume = {338},
     number = {12},
     year = {2004},
     doi = {10.1016/j.crma.2004.04.005},
     language = {en},
}

TY  - JOUR
AU  - Vivette Girault
AU  - Ricardo H. Nochetto
AU  - Ridgway Scott
TI  - Stability of the finite element Stokes projection in W1,∞
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 957
EP  - 962
VL  - 338
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2004.04.005
LA  - en
ID  - CRMATH_2004__338_12_957_0
ER  -

%0 Journal Article
%A Vivette Girault
%A Ricardo H. Nochetto
%A Ridgway Scott
%T Stability of the finite element Stokes projection in W1,∞
%J Comptes Rendus. Mathématique
%D 2004
%P 957-962
%V 338
%N 12
%I Elsevier
%R 10.1016/j.crma.2004.04.005
%G en
%F CRMATH_2004__338_12_957_0

Vivette Girault; Ricardo H. Nochetto; Ridgway Scott. Stability of the finite element Stokes projection in W^1,∞. Comptes Rendus. Mathématique, Volume 338 (2004) no. 12, pp. 957-962. doi : 10.1016/j.crma.2004.04.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.005/

Bibliographie
Cité par

[1] S. Brenner; L.R. Scott The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, NY, 1994

[2] H. Chen, Pointwise error estimates for finite element solutions of the Stokes problem, Preprint, 2002

[3] P.G. Ciarlet; P.-A. Raviart Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., Volume 2 (1973), pp. 17-31

[4] M. Dauge Stationary Stokes and Navier–Stokes systems on two or three-dimensional domains with corners, SIAM J. Math. Anal., Volume 20 (1989), pp. 74-97

[5] R. Durán; A. Muschietti An explicit right inverse of the divergence operator which is continuous in weighted norms, Studia Math., Volume 148 (2001), pp. 207-219

[6] R. Durán; R.H. Nochetto; J. Wang Sharp maximum norm error estimates for finite element approximations of the Stokes problem in 2-D, Math. Comp., Volume 51 (1988), pp. 1177-1192

[7] V. Girault; R.H. Nochetto; L.R. Scott Maximum-norm stability of the finite-element Stokes projection, J. Math. Pures Appl. (2004) (in press)

[8] V. Girault; L.R. Scott A quasi-local interpolation operator preserving the discrete divergence, Calcolo, Volume 40 (2003), pp. 1-19

[9] P. Grisvard Elliptic Problems in Nonsmooth Domains, Pitman Monographs and Studies in Math., vol. 24, Pitman, Boston, MA, 1985

[10] R. Rannacher; L.R. Scott Some optimal error estimates for linear finite element approximations, Math. Comp., Volume 38 (1982), pp. 437-445

[11] A. Schatz Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids, Math. Comp., Volume 67 (1998), pp. 877-899

[12] E.M. Stein Note on singular integrals, Proc. Amer. Math. Soc., Volume 8 (1957), pp. 250-254

Cité par Sources :

Commentaires - Politique