Mathematical Problems in Mechanics
On a residual local projection method for the Darcy equation
Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1105-1110.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.06.016
Franca, Leopoldo P. 1, 2; Harder, Christopher 1; Valentin, Frédéric 3

1 University of Colorado Denver, P.O. Box 17364, Campus Box 170, Denver, CO 80217-3364, USA
2 Department of Civil Engineering COPPE/UFRJ, P.O. Box 68506, 21945-970 Rio de Janeiro - RJ, Brazil
3 Laboratório Nacional de Computação Científica (LNCC), Av. Getúlio Vargas, 333, 25651-070 Petrópolis - RJ, Brazil
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Franca, Leopoldo P.; Harder, Christopher; Valentin, Frédéric. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.06.016/

[1] G.R. Barrenechea, F. Valentin, Consistent local projection stabilized finite element methods, Tech. Report 6/2009, LNCC, 2009

[2] Brezzi, F.; Fortin, M. Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, Berlin, New York, 1991

[3] Burman, E. Pressure projection stabilizations for Galerkin approximations of Stokes' and Darcy's problem, Numerical Methods for Partial Differential Equations, Volume 24 (2008), pp. 127-143

[4] Dohrmann, C.; Bochev, P. A stabilized finite element method for the Stokes problem based on polynomial pressure projections, International Journal for Numerical Methods in Fluids, Volume 46 (2004), pp. 183-201

Cited by Sources:

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.