no. 12
Numerical Analysis
Stability of the finite element Stokes projection in W1,∞
[Stabilité dans W1,∞ de la projection de Stokes par éléments finis]

Présenté par : Philippe G. Ciarlet

Girault, Vivette1 ; Nochetto, Ricardo H.2 ; Scott, Ridgway3
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, Tome 338 (2004) no. 12, pp. 957-962.

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,∞ (Ω)×L ∞ (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,∞ (Ω)×L ∞ (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,∞ (Ω)×L ∞ (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.

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Accepté le :
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DOI : 10.1016/j.crma.2004.04.005
Girault, Vivette 1 ; Nochetto, Ricardo H. 2 ; Scott, Ridgway 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 
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Girault, Vivette; Nochetto, Ricardo H.; Scott, Ridgway. Stability of the finite element Stokes projection in W1,∞. Comptes Rendus. Mathématique, Tome 338 (2004) no. 12, pp. 957-962. doi : 10.1016/j.crma.2004.04.005. http://www.numdam.org/articles/10.1016/j.crma.2004.04.005/

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