Comptes Rendus
no. 7
Numerical Analysis
Equivalence between mixed finite element and multi-point finite volume methods

Presented by: Philippe G. Ciarlet

Martin Vohralík1, 2
1 Laboratoire de mathématiques, analyse numérique et EDP, université de Paris-Sud, bâtiment 425, 91405 Orsay, France
2 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic
Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 525-528.

We consider the lowest-order Raviart–Thomas mixed finite element method for elliptic problems on simplicial meshes in two or three space dimensions. This method produces saddle-point type problems for scalar and flux unknowns. We show how to easily eliminate the flux unknowns, which implies an equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. We describe the stencil of the final matrix and give sufficient conditions for its symmetry and positive definiteness. We present a numerical example illustrating the performance of the proposed method.

Nous considérons la méthode des éléments finis mixtes de Raviart–Thomas de plus bas degré pour des problèmes elliptiques sur les maillages composés de triangles en dimension deux d'espace et de tétraèdres en dimension trois d'espace. Cette méthode aboutit à des problèmes de type point-selle pour les inconnues scalaires et les flux. Nous montrons comment facilement éliminer les flux, ce qui implique l'équivalence entre cette méthode et une méthode de type volumes finis à plusieurs points et ceci sans aucune intégration numérique approchée. Nous décrivons le nombre maximal des éléments non nuls sur chaque ligne de la matrice finale et présentons les conditions suffisantes pour qu'elle soit symétrique et définie positive. Nous présentons un essai numérique montrant la performance de la méthode proposée.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.08.004

Martin Vohralík 1, 2

1 Laboratoire de mathématiques, analyse numérique et EDP, université de Paris-Sud, bâtiment 425, 91405 Orsay, France
2 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic 
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Martin Vohralík. Equivalence between mixed finite element and multi-point finite volume methods. Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 525-528. doi : 10.1016/j.crma.2004.08.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.004/

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[6] M. Vohralík, Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes, in preparation

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