Comptes Rendus
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

Karen Veroy1 ; Christophe Prud'homme1 ; Anthony T. Patera1
1 Department of Mechanical Engineering, M.I.T., Room 3-264, Cambridge, MA 02139-4307, USA
Comptes Rendus. Mathématique, Volume 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
Karen Veroy 1 ; Christophe Prud'homme 1 ; Anthony T. Patera 1

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

[1] B.O. Almroth; P. Stern; F.A. Brogan Automatic choice of global shape functions in structural analysis, AIAA J., Volume 16 (1978), pp. 525-528

[2] E. Balmes Parametric families of reduced finite element models: Theory and applications, Mech. Systems and Signal Processing, Volume 10 (1996) no. 4, pp. 381-394

[3] F. Brezzi; J. Rappaz; P.A. Raviart Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math., Volume 36 (1980), pp. 1-25

[4] G. Caloz; J. Rappaz Numerical analysis for nonlinear and bifurcation problems (P.G. Ciarlet; J.-L. Lions, eds.), Techniques of Scientific Computing (Part 2), Handbook of Numerical Anaylsis, V, Elsevier, 1997, pp. 487-637

[5] J.P. Fink; W.C. Rheinboldt On the error behavior of the reduced basis technique for nonlinear finite element approximations, Z. Angew. Math. Mech., Volume 63 (1983), pp. 21-28

[6] L. Machiels; Y. Maday; I.B. Oliveira; A.T. Patera; D.V. Rovas Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2000) no. 2, pp. 153-158

[7] Y. Maday; A.T. Patera; G. Turinici Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 1-6

[8] A.K. Noor; J.M. Peters Reduced basis technique for nonlinear analysis of structures, AIAA J., Volume 18 (1980) no. 4, pp. 455-462

[9] J.S. Peterson The reduced basis method for incompressible viscous flow calculations, SIAM J. Sci. Statist. Comput., Volume 10 (1989) no. 4, pp. 777-786

[10] T.A. Porsching Estimation of the error in the reduced basis method solution of nonlinear equations, Math. Comput., Volume 45 (1985) no. 172, pp. 487-496

[11] C. Prud'homme; D. Rovas; K. Veroy; Y. Maday; A.T. Patera; G. Turinici Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, J. Fluids Engrg., Volume 124 (2002) no. 1, pp. 70-80

[12] K. Veroy; C. Prud'homme; D.V. Rovas; A.T. Patera A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003

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