Éléments finis mixtes minimaux sur les polyèdres

Snorre H. Christiansen

doi:10.1016/j.crma.2010.01.017

Analyse numérique

Éléments finis mixtes minimaux sur les polyèdres

Présenté par : Philippe G. Ciarlet

Snorre H. Christiansen ¹

¹ CMA, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway

Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 217-221.

Résumé (VO)
Abstract

Étant donné un complexe cellulaire constitué de polytopes, plongé dans un espace Euclidien, nous construisons des espaces d'éléments finis de formes différentielles, conformes par rapport à la dérivée extérieure, contenant celles qui sont polynomiales de degré maximal donné, ayant localement la propriété de suite exacte et d'extension, de telle sorte que parmi tous les espaces ayant ces propriétés, ils ont la plus petite dimension. Plus généralement nous construisons, pour tout système d'éléments finis inclus dans un système d'éléments finis compatible, un système d'éléments finis compatible intermédiaire et de dimension minimale.

Given a cellular complex consisting of polytopes, embedded in a Euclidean space, we construct finite element spaces of differential forms, conforming with respect to the exterior derivative, containing those that are polynomial of given maximal degree, having locally the property of exact sequence and extension, so that among all spaces having these properties they have the smallest dimension. More generally we construct, for any finite element system included in a compatible finite element system, an intermediate compatible finite element system of minimal dimension.

Reçu le : 2009-07-01
Accepté le : 2010-01-18
Publié le : 2010-02-01

DOI : 10.1016/j.crma.2010.01.017

Affiliations des auteurs :

Snorre H. Christiansen ¹

¹ CMA, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway

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Snorre H. Christiansen. Éléments finis mixtes minimaux sur les polyèdres. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 217-221. doi : 10.1016/j.crma.2010.01.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.017/

Bibliographie
Cité par

[1] D.N. Arnold; R.S. Falk; R. Winther Finite element exterior calculus, homological techniques, and applications, Acta Numer., Volume 15 (2006), pp. 1-155

[2] S.H. Christiansen A construction of spaces of compatible differential forms on cellular complexes, Math. Models Methods Appl. Sci., Volume 18 (2008) no. 5, pp. 739-757

[3] S.H. Christiansen Foundations of finite element methods for wave equations of Maxwell type, Applied Wave Mathematics, Springer, Berlin, Heidelberg, 2009, pp. 335-393

[4] S.H. Christiansen; R. Winther Smoothed projections in finite element exterior calculus, Math. Comp., Volume 77 (2008) no. 262, pp. 813-829

[5] R. Hiptmair Canonical construction of finite elements, Math. Comp., Volume 68 (1999) no. 228, pp. 1325-1346

[6] J.-C. Nédélec Mixed finite elements in $R^{3}$ , Numer. Math., Volume 35 (1980) no. 3, pp. 315-341

Martin Vohralík; Soleiman Yousef A simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows, Computer Methods in Applied Mechanics and Engineering, Volume 331 (2018), pp. 728-760 | DOI:10.1016/j.cma.2017.11.027 | Zbl:1439.74472
Snorre H. Christiansen On eigenmode approximation for Dirac equations: differential forms and fractional Sobolev spaces, Mathematics of Computation, Volume 87 (2018) no. 310, pp. 547-580 | DOI:10.1090/mcom/3233 | Zbl:1380.65350
Snorre H. Christiansen; Andrew Gillette Constructions of some minimal finite element systems, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 50 (2016) no. 3, pp. 833-850 | DOI:10.1051/m2an/2015089 | Zbl:1343.65135
Snorre H. Christiansen; Francesca Rapetti On high order finite element spaces of differential forms, Mathematics of Computation, Volume 85 (2016) no. 298, pp. 517-548 | DOI:10.1090/mcom/2995 | Zbl:1332.65163
John Thuburn; Colin J. Cotter A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes, Journal of Computational Physics, Volume 290 (2015), pp. 274-297 | DOI:10.1016/j.jcp.2015.02.045 | Zbl:1349.76273
S. H. Christiansen Upwinding in Finite Element Systems of Differential Forms, Foundations of Computational Mathematics, Budapest 2011 (2012), p. 45 | DOI:10.1017/cbo9781139095402.004
Snorre H. Christiansen; Hans Z. Munthe-Kaas; Brynjulf Owren Topics in structure-preserving discretization, Acta Numerica, Volume 20 (2011), pp. 1-119 | DOI:10.1017/s096249291100002x | Zbl:1233.65087

Cité par 7 documents. Sources : Crossref, zbMATH

^☆ This work, conducted as part of the award “Numerical analysis and simulations of geometric wave equations” made under the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) Awards scheme, was supported by funds from the Participating Organizations of EURYI and the EC Sixth Framework Program.

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