A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations

Yanlai Chen; Jan S. Hesthaven; Yvon Maday; Jerónimo Rodríguez

doi:10.1016/j.crma.2008.10.012

Numerical Analysis

A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations
[Une évaluation monotone de brones fiables pour la constante inf-sup dans la méthode de l'approximation des bases réduites]

Présenté par : Olivier Pironneau

Yanlai Chen ¹ ; Jan S. Hesthaven ¹ ; Yvon Maday ^1,² ; Jerónimo Rodríguez ³

¹ Division of Applied Mathematics, Brown University, 182, George St, Providence, RI 02912, USA
² UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
³ Laboratoire POEMS, 32, boulevard Victor, 75739 Paris cedex 15, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 23-24, pp. 1295-1300.

Résumé
Abstract

Un ingrédient fondamental de l'analyse a posteriori pour l'approximation par méthode par bases réduites de problèmes coercifs ou non coercifs est la définition de bornes fiables pour la constante de coercivité ou la constante inf-sup. Dans cette Note, nous généralisons et améliorons la méthode d'optimisation linéaire de contraintes successives présentées dans Huynh (2007), en proposant une version monotone de cet algorithme qui conduit à la fois à des évaluations plus stables et un nombre réduit de calculs hors ligne.

For accurate a posteriori error analysis of the reduced basis method for coercive and non-coercive problems, a critical ingredient lies in the evaluation of a lower bound for the coercivity or inf-sup constant. In this short Note, we generalize and improve the successive constraint method first presented by Huynh (2007) by providing a monotonic version of this algorithm that leads to both more stable evaluations and fewer offline computations.

Reçu le : 2008-06-17
Accepté le : 2008-10-21
Publié le : 2008-11-20

DOI : 10.1016/j.crma.2008.10.012

Affiliations des auteurs :

Yanlai Chen ¹ ; Jan S. Hesthaven ¹ ; Yvon Maday ^{1, 2} ; Jerónimo Rodríguez ³

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     title = {A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations},
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Yanlai Chen; Jan S. Hesthaven; Yvon Maday; Jerónimo Rodríguez. A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations. Comptes Rendus. Mathématique, Volume 346 (2008) no. 23-24, pp. 1295-1300. doi : 10.1016/j.crma.2008.10.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.10.012/

Bibliographie
Cité par

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[2] D.B.P. Huynh; G. Rozza; S. Sen; A.T. Patera A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007), pp. 473-478

[3] Y. Maday; A.T. Patera; D.V. Rovas A blackbox reduced-basis output bound method for noncoercive linear problems (D. Cioranescu; J.L. Lions, eds.), Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, vol. XIV, Elsevier Science B.V., 2002, pp. 533-569

[4] N.C. Nguyen; K. Veroy; A.T. Patera Certified real-time solution of parametrized partial differential equations (S. Yip, ed.), Handbook of Materials Modeling, Springer, 2005, pp. 1523-1558

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

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