[Une méthode de projection local résiduelle pour l'équation de Darcy]
On propose une nouvelle méthode de projection locale symétrique du type résiduel (RELP) pour l'équation de Darcy. La méthode est construite dans un cadre d'enrichissement des espaces d'interpolations par une approche du type Petrov–Galerkin, ce qui nous permet de modifier de façon naturelle la méthode de Galerkin et d'éviter le choix des constantes de stabilisation. L'approche d'enrichissement est basée sur la résolution de problèmes de Darcy locaux, qui dépendent des résidus après un procédé de condensation statique. On démontre que la méthode est stable pour les paires d'éléments finis linéaires continus et discontinus en pression. On établit, ensuite, l'optimalité de l'erreur dans les normes naturelles et on propose une stratégie de reconstruction de champ de vitesse localement conservatif. Les aspects théoriques sont validés numériquement.
A new symmetric local projection method built on residual bases (RELP) makes linear equal-order finite element pairs stable for the Darcy problem. The derivation is performed inside a Petrov–Galerkin enriching space approach (PGEM) which indicates parameter-free terms to be added to the Galerkin method without compromising consistency. Velocity and pressure spaces are augmented using solutions of residual dependent local Darcy problems obtained after a static condensation procedure. We prove the method achieves error optimality and indicates a way to recover a locally mass conservative velocity field. Numerical experiments validate theory.
Accepté le :
Publié le :
Leopoldo P. Franca 1, 2 ; Christopher Harder 1 ; Frédéric Valentin 3
@article{CRMATH_2009__347_17-18_1105_0,
author = {Leopoldo P. Franca and Christopher Harder and Fr\'ed\'eric Valentin},
title = {On a residual local projection method for the {Darcy} equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {1105--1110},
publisher = {Elsevier},
volume = {347},
number = {17-18},
year = {2009},
doi = {10.1016/j.crma.2009.06.016},
language = {en},
}
TY - JOUR AU - Leopoldo P. Franca AU - Christopher Harder AU - Frédéric Valentin TI - On a residual local projection method for the Darcy equation JO - Comptes Rendus. Mathématique PY - 2009 SP - 1105 EP - 1110 VL - 347 IS - 17-18 PB - Elsevier DO - 10.1016/j.crma.2009.06.016 LA - en ID - CRMATH_2009__347_17-18_1105_0 ER -
Leopoldo P. Franca; Christopher Harder; Frédéric Valentin. On a residual local projection method for the Darcy equation. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1105-1110. doi : 10.1016/j.crma.2009.06.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.016/
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Cité par 7 documents. Sources : Crossref, zbMATH
☆ This research was supported by NSF Grant No. 0610039, CNPq No. 306255/2008-1 and 304051/2006-3, FAPERJ No. E-26/100.519/2007 and Projeto Galileu, COPPE/UFRJ.
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