Some simple error estimates of finite volume approximate solution for parabolic equations

Abdallah Bradji

doi:10.1016/j.crma.2008.03.023

Numerical Analysis

Some simple error estimates of finite volume approximate solution for parabolic equations

Presented by: Philippe G. Ciarlet

Abdallah Bradji ¹

¹ Faculty of Physics and Mathematics, Charles University Prague, Sokolovská 83, 18675 Praha 8, Czech Republic

Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 571-574.

Abstract
Résumé

An implicit finite volume scheme for parabolic equations, in which the approximate initial condition is an “orthogonal projection” of the exact initial function, is considered.

In this Note, we prove that the error estimate is of order $h + k$ (where h and k are, respectively, the mesh size of the space discretization and the mesh size of the time discretization) on the discrete norms of $L^{\infty} (0, T; H_{0}^{1} (Ω))$ and $W^{1, \infty} (0, T; L^{2} (Ω))$ . From these results, error estimates can be derived for the approximations of the fluxes across the interfaces between neighbouring control volumes and of the first derivative of the unknown solution with respect to the time.

On considère un schéma implicite de volumes finis pour les problèmes paraboliques. La condition initiale a été discrétisée en utilisant une « projection orthogonale ».

Dans cette Note, on démontre que l'estimation d'erreur est d'ordre $h + k$ (h et k étant, respectivement, le pas de discrétisation en espace et le pas de discrètisation en temps) en normes discrètes dans $L^{\infty} (0, T; H_{0}^{1} (Ω))$ et dans $W^{1, \infty} (0, T; L^{2} (Ω))$ . Ces estimations nous permettent d'obtenir des estimations des approximations des flux au travers des interfaces entre les volumes voisins de contrôle et de la dérivée par rapport au temps de la solution inconnue.

Received: 2007-11-01
Accepted: 2008-03-12
Published online: 2008-04-24

DOI: 10.1016/j.crma.2008.03.023

Author's affiliations:

Abdallah Bradji ¹

¹ Faculty of Physics and Mathematics, Charles University Prague, Sokolovská 83, 18675 Praha 8, Czech Republic

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     title = {Some simple error estimates of finite volume approximate solution for parabolic equations},
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Abdallah Bradji. Some simple error estimates of finite volume approximate solution for parabolic equations. Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 571-574. doi : 10.1016/j.crma.2008.03.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.023/

References
Cited by

[1] R. Eymard; T. Gallouët; R. Herbin A cell–centered finite volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension, IMA Journal of Numerical Analysis, Volume 26 (2006), pp. 326-353

[2] R. Eymard; T. Gallouët; R. Herbin Finite volume methods (P.G. Ciarlet; J.L. Lions, eds.), Handbook of Numerical Analysis, vol. VII, 2000, pp. 723-1020

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