Comptes Rendus
no. 2
Numerical Analysis
Accelerating the method of finite element patches using approximately harmonic functions
[Acceleration de la méthode des patches d'éléments finis en utilisant des fonctions approximativement harmoniques]

Présenté par : Olivier Pironneau

Jiwen He1 ; Alexei Lozinski1 ; Jacques Rappaz2
1 Dept. of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204-3008, USA
2 Section of Mathematics, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland
Comptes Rendus. Mathématique, Volume 345 (2007) no. 2, pp. 107-112.

Nous présentons une nouvelle variante de la méthode de décomposition de domaines avec un recouvrement complet et des maillages non nécessairement emboîtés pour la résolution numérique des problèmes elliptiques avec des données multi-échelle. La nouveauté de la méthode consiste dans la restriction de l'espace des fonctions éléments finis grossières à être des approximations des fonctions harmoniques dans le sous-domaine sur lequel une triangulation fine est appliquée. Des expériences numériques confirment une augmentation du taux de convergence par rapport à une méthode proposée précédemment.

We present a new variant of a domain decomposition method with complete overlap and not necessarily nested grids to solve numerically elliptic problems with multi-scale data. The novelty of the method consists in restricting finite element functions on the coarse grid to be approximately harmonic inside the subdomain where a finer triangulation is applied. Numerical experiments confirm an increase in the convergence rate over a previously proposed method.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.06.006
Jiwen He 1 ; Alexei Lozinski 1 ; Jacques Rappaz 2

1 Dept. of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204-3008, USA
2 Section of Mathematics, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland 
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     author = {Jiwen He and Alexei Lozinski and Jacques Rappaz},
     title = {Accelerating the method of finite element patches using approximately harmonic functions},
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     pages = {107--112},
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     number = {2},
     year = {2007},
     doi = {10.1016/j.crma.2007.06.006},
     language = {en},
}
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Jiwen He; Alexei Lozinski; Jacques Rappaz. Accelerating the method of finite element patches using approximately harmonic functions. Comptes Rendus. Mathématique, Volume 345 (2007) no. 2, pp. 107-112. doi : 10.1016/j.crma.2007.06.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.06.006/

[1] F. Brezzi; J.-L. Lions; O. Pironneau Analysis of a chimera method, C. R. Acad. Sci. Paris, Ser. I, Volume 332 (2003), pp. 655-660

[2] R. Glowinski; J. He; A. Lozinski; J. Rappaz; J. Wagner Finite element approximation of multi-scale elliptic problems using patches of elements, Numer. Math., Volume 101 (2005) no. 4, pp. 663-687

[3] R. Glowinski; J. He; J. Rappaz; J. Wagner Approximation of multi-scale elliptic problems using patches of finite elements, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 679-684

[4] J. He, A. Lozinski, J. Rappaz, Some results on the method of finite element patches, Technical report, Ecole Polytechnique Fédérale de Lausanne, Switzerland, 2007, in preparation

[5] F. Hecht; O. Pironneau; K. Ohtsuka Freefem++ http://www.freefem.org (ver. 4.12)

[6] V. Rezzonico, Multiscale algorithm with patches of finite element and applications, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland, 2007

[7] J.R. Shewchuk Triangle http://www.cs.cmu.edu/quake/triangle.html

[8] J. Wagner, Finite element methods with patches and applications, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland, 2006

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