Comptes Rendus
no. 17-18
Numerical Analysis
A remark on supercloseness and extrapolation of the quadrilateral Han element for the Stokes equations
[Une remarque concernant la super-approximation et lʼextrapolation de lʼélement fini de Han pour les équations de Stokes]

Présenté par : Olivier Pironneau

Mingxia Li1 ; Roland Becker2 ; Shipeng Mao3
1 School of Information Engineering, China University of Geosciences (Beijing), Beijing, 100083, PR China
2 Laboratoire de mathématiques appliquées and INRIA Bordeaux Sud-Ouest, université de Pau, 64013 Pau cedex, France
3 LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences (CAS), Beijing, 100190, PR China
Comptes Rendus. Mathématique, Volume 349 (2011) no. 17-18, pp. 1017-1020.

Nous présentons une analyse de la super-convergence de lʼespace dʼélément finis de Han pour les équations de Stokes. Il est démontré que la différence entre la solution discrète et lʼinterpolé naturel de la solution nʼest pas de lʼordre supérieur ( « supercloseness »). Basé sur notre analyse, nous proposons une modification de lʼopérateur dʼinterpolation qui possède cette propriété. Cela permet la construction dʼun schéma dʼextrapolation simple qui est de lʼordre trois.

We analyze the supercloseness properties of the nonconforming quadrilateral Han finite element for the Stokes equations. It is shown that the difference between the discrete solution and the natural interpolation of the continuous solution does not have the supercloseness property. Based on this analysis, we propose a modified interpolation operator which allows for such a result. It is then used to obtain a simple third-order extrapolation scheme.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.08.005

Mingxia Li 1 ; Roland Becker 2 ; Shipeng Mao 3

1 School of Information Engineering, China University of Geosciences (Beijing), Beijing, 100083, PR China
2 Laboratoire de mathématiques appliquées and INRIA Bordeaux Sud-Ouest, université de Pau, 64013 Pau cedex, France
3 LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences (CAS), Beijing, 100190, PR China 
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     title = {A remark on supercloseness and extrapolation of the quadrilateral {Han} element for the {Stokes} equations},
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Mingxia Li; Roland Becker; Shipeng Mao. A remark on supercloseness and extrapolation of the quadrilateral Han element for the Stokes equations. Comptes Rendus. Mathématique, Volume 349 (2011) no. 17-18, pp. 1017-1020. doi : 10.1016/j.crma.2011.08.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.08.005/

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[2] Q. Lin; L. Tobiska; A. Zhou Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation, IMA J. Numer. Anal., Volume 25 (2005), pp. 160-181

[3] S. Mao; S. Chen; D. Shi Convergence and superconvergence of a nonconforming finite element on anisotropic meshes, Int. J. Numer. Anal. Model., Volume 4 (2007), pp. 16-38

[4] R. Rannacher; S. Turek Simple nonconforming quadrilateral stokes element, Numer. Methods Partial Differential Equations, Volume 8 (1992), pp. 97-111

[5] S. Turek Efficient Solvers for Incompressible Flow Problems, Lecture Notes in Computational Science and Engineering, vol. 6, Springer-Verlag, Berlin, 1999

[6] X. Ye Superconvergence of nonconforming finite element method for the Stokes equations, Numer. Methods Partial Differential Equations, Volume 18 (2002), pp. 143-154

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