[Un complexe dual d'éléments finis sur le raffinement barycentrique]
Un maillage simplicial sur une surface orientée bidimensionnelle donne lieu à un complexe d'espaces d'éléments finis centré sur l'espace de Raviart–Thomas de champs de vecteurs à divergence conforme et naturellement isomorphe au complexe des cochaînes simpliciales. Sur le raffinement barycentrique d'un tel maillage, nous construisons des espaces d'éléments finis formant un complexe , centré sur des champs de vecteurs à rotationnel conforme, naturellement isomorphe au complexe des chaînes simpliciales sur le maillage de départ et tel que soit en dualité avec . En termes de formes différentielles, on obtient un analogue de la dualité de Hodge pour les éléments finis.
A simplicial mesh on an oriented two-dimensional surface gives rise to a complex of finite element spaces centered on divergence conforming Raviart–Thomas vector fields and naturally isomorphic to the simplicial cochain complex. On the barycentric refinement of such a mesh, we construct finite element spaces forming a complex , centered around curl conforming vector fields, naturally isomorphic to the simplicial chain complex on the original mesh and such that is in duality with . In terms of differential forms this provides a finite element analogue of Hodge duality.
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@article{CRMATH_2005__340_6_461_0,
author = {Buffa, Annalisa and Christiansen, Snorre H.},
title = {A dual finite element complex on the barycentric refinement},
journal = {Comptes Rendus. Math\'ematique},
pages = {461--464},
publisher = {Elsevier},
volume = {340},
number = {6},
year = {2005},
doi = {10.1016/j.crma.2004.12.022},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.12.022/}
}
TY - JOUR AU - Buffa, Annalisa AU - Christiansen, Snorre H. TI - A dual finite element complex on the barycentric refinement JO - Comptes Rendus. Mathématique PY - 2005 SP - 461 EP - 464 VL - 340 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.12.022/ DO - 10.1016/j.crma.2004.12.022 LA - en ID - CRMATH_2005__340_6_461_0 ER -
%0 Journal Article %A Buffa, Annalisa %A Christiansen, Snorre H. %T A dual finite element complex on the barycentric refinement %J Comptes Rendus. Mathématique %D 2005 %P 461-464 %V 340 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.12.022/ %R 10.1016/j.crma.2004.12.022 %G en %F CRMATH_2005__340_6_461_0
Buffa, Annalisa; Christiansen, Snorre H. A dual finite element complex on the barycentric refinement. Comptes Rendus. Mathématique, Tome 340 (2005) no. 6, pp. 461-464. doi : 10.1016/j.crma.2004.12.022. http://www.numdam.org/articles/10.1016/j.crma.2004.12.022/
[1] A boundary-element solution of the Leontovitch problem, IEEE Trans. Antennas and Propagation, Volume 47 (1999) no. 10, pp. 1597-1605
[2] Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991
[3] On traces for functional spaces related to Maxwell's equations, Part I: An integration by parts formula in Lipschitz polyhedra, Math. Methods Appl. Sci., Volume 21 (2001) no. 1, pp. 9-30
[4] On traces for functional spaces related to Maxwell's equations, Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Methods Appl. Sci., Volume 21 (2001) no. 1, pp. 31-48
[5] A preconditioner for the electric field integral equation based on Calderon formulas, SIAM J. Numer. Anal., Volume 40 (2002) no. 3, pp. 1100-1135
[6] Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer-Verlag, 2001
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