no. 9
Numerical Analysis
Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds
[Approximation par base réduite de l'équation de Burgers visqueuse : bornes d'erreur a posteriori rigoureuses]

Présenté par : Philippe G. Ciarlet

Veroy, Karen1 ; Prud'homme, Christophe1 ; Patera, Anthony T.1
1 Department of Mechanical Engineering, M.I.T., Room 3-264, Cambridge, MA 02139-4307, USA
Comptes Rendus. Mathématique, Tome 337 (2003) no. 9, pp. 619-624.

Nous présentons des bornes d'erreur a posteriori rigoureuses, précises et peu coûteuses pour l'approximation par base réduite de l'équation de Burgers avec la viscosité comme paramètre. Il y a deux composantes essentielles : l'approche de Brezzi, Rappaz et Raviart (Numer. Math. 36 (1980) 1–25) pour l'analyse d'approximations d'équations aux dérivées partielles nonlinéaires elliptiques ; et une procédure hors-ligne/en-ligne pour le calcul efficace des constantes nécessaires de continuité et de stabilité, et de la norme duale du résidu. Les résultats numériques confirment les performances de ces bornes d'erreur.

We present rigorous, sharp, and inexpensive a posteriori error bounds for reduced-basis approximations of the viscosity-parametrized Burgers equation. There are two critical ingredients: the Brezzi, Rappaz and Raviart (Numer. Math. 36 (1980) 1–25) framework for analysis of approximations of nonlinear elliptic partial differential equations; and offline/online computational procedures for efficient calculation of the necessary continuity and stability constants, and of the dual norm of the residual. Numerical results confirm the performance of the error bounds.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.09.023
Veroy, Karen 1 ; Prud'homme, Christophe 1 ; Patera, Anthony T. 1

1 Department of Mechanical Engineering, M.I.T., Room 3-264, Cambridge, MA 02139-4307, USA 
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Veroy, Karen; Prud'homme, Christophe; Patera, Anthony T. Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. Comptes Rendus. Mathématique, Tome 337 (2003) no. 9, pp. 619-624. doi : 10.1016/j.crma.2003.09.023. http://www.numdam.org/articles/10.1016/j.crma.2003.09.023/

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