Numerical Analysis
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.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.10.012
Chen, Yanlai 1; Hesthaven, Jan S. 1; Maday, Yvon 1, 2; Rodríguez, Jerónimo 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
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Chen, Yanlai; Hesthaven, Jan S.; Maday, Yvon; Rodríguez, Jerónimo. 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. http://www.numdam.org/articles/10.1016/j.crma.2008.10.012/

[1] Barret, A.; Reddien, G. On the reduced basis method, Z. Angew. Math. Mech., Volume 75 (1995) no. 7, pp. 543-549

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

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

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

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