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]
Comptes Rendus. Mathématique, Tome 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.

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DOI : 10.1016/j.crma.2011.08.005
Li, Mingxia 1 ; Becker, Roland 2 ; Mao, Shipeng 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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Li, Mingxia; Becker, Roland; Mao, Shipeng. A remark on supercloseness and extrapolation of the quadrilateral Han element for the Stokes equations. Comptes Rendus. Mathématique, Tome 349 (2011) no. 17-18, pp. 1017-1020. doi : 10.1016/j.crma.2011.08.005. http://www.numdam.org/articles/10.1016/j.crma.2011.08.005/

[1] Han, H.D. Nonconforming elements in the mixed finite element method, J. Comput. Math., Volume 2 (1984), pp. 223-233

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

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

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

[6] Ye, X. 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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