Relationship between multiscale enrichment and stabilized finite element methods for the generalized Stokes problem

Gabriel R. Barrenechea; Frédéric Valentin

doi:10.1016/j.crma.2005.09.038

Numerical Analysis

Relationship between multiscale enrichment and stabilized finite element methods for the generalized Stokes problem

Presented by: Olivier Pironneau

Gabriel R. Barrenechea ¹; Frédéric Valentin ²

¹ Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
² Departamento de Matemática Aplicada, Laboratório Nacional de Computação Científica (LNCC), Av. Getúlio Vargas, 333, 25651-070 Petrópolis – RJ, Brazil

Comptes Rendus. Mathématique, Volume 341 (2005) no. 10, pp. 635-640.

Abstract
Résumé

We derive a new stabilized finite element method for the generalized Stokes problem starting from the non-stable continuous $P_{1} / P_{1}$ finite element space enriched with multiscale functions. The stabilization parameter is related with the enrichment functions which are analytically computed from a boundary value problem at the element level leading to a method which is free of constants. Optimal error estimates are obtained in natural norms and numerical tests validate the method.

On propose une nouvelle méthode d'éléments finis stabilisée pour le problème de Stokes généralisé basée sur l'enrichissement de l'espace d'éléments finis continu $P_{1} / P_{1}$ par des fonctions multi-échelles. Le paramètre de stabilisation est donné par la moyenne de la fonction d'enrichissement sur l'élément, qui à son tour est calculée analytiquement par la résolution d'un problème aux limites dans chaque élément. Des estimations d'erreurs optimales sont obtenues et des tests numériques sont présentés.

Received: 2005-06-07
Accepted: 2005-09-27
Published online: 2005-10-27

DOI: 10.1016/j.crma.2005.09.038

Author's affiliations:

Gabriel R. Barrenechea ¹; Frédéric Valentin ²

¹ Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
² Departamento de Matemática Aplicada, Laboratório Nacional de Computação Científica (LNCC), Av. Getúlio Vargas, 333, 25651-070 Petrópolis – RJ, Brazil

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     author = {Gabriel R. Barrenechea and Fr\'ed\'eric Valentin},
     title = {Relationship between multiscale enrichment and stabilized finite element methods for the generalized {Stokes} problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {635--640},
     publisher = {Elsevier},
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     number = {10},
     year = {2005},
     doi = {10.1016/j.crma.2005.09.038},
     language = {en},
}

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Gabriel R. Barrenechea; Frédéric Valentin. Relationship between multiscale enrichment and stabilized finite element methods for the generalized Stokes problem. Comptes Rendus. Mathématique, Volume 341 (2005) no. 10, pp. 635-640. doi : 10.1016/j.crma.2005.09.038. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.038/

References
Cited by

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[2] G.R. Barrenechea; F. Valentin An unusual stabilized finite element method for a generalized Stokes problem, Numer. Math., Volume 92 (2002), pp. 653-677

[3] A. Ern; J.-L. Guermond Theory and Practice of Finite Elements, Springer-Verlag, 2004

[4] L.P. Franca; A. Madureira; F. Valentin Towards multiscale functions: Enriching finite element spaces with local but non bubble-like functions, Comput. Methods Appl. Mech. Engrg., Volume 194 (2005), pp. 3006-3021

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