Comptes Rendus
no. 3
Partial differential equations/Numerical analysis
A slack approach to reduced-basis approximation and error estimation for variational inequalities
[Approximation bases réduites et estimateur d'erreur pour les inéquations variationnelles via une approche par variable d'écart]

Présenté par : Philippe G. Ciarlet

Zhenying Zhang1 ; Eduard Bader1 ; Karen Veroy1
1 Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Germany
Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 283-289.

Nous proposons une nouvelle approche pour le calcul d'approximations bases réduites pour des inégalités variationnelles du premier type. Les trois principales composantes de cette approche sont : (i) une approximation utilisant des variables d'écart pour la solution ; (ii) une approximation primale pour le multiplicateur de Lagrange ; (iii) une borne supérieure a posteriori de l'erreur sur la solution approchée. La stricte faisabilité de l'approximation primale par variable d'écart nous permet deux améliorations majeures par rapport aux méthodes existantes. La première est de pouvoir borner, a posteriori, de façon précise, l'erreur commise. La deuxième est l'utilisation d'une décomposition hors ligne/en ligne grâce à laquelle le coût de calcul de cette borne reste complètement indépendant de la (grande) dimension originale du problème. Les résultats numériques présentent une comparaison des performances entre cette nouvelle approche et les méthodes existantes.

We propose a novel approach for computing certified reduced-basis approximations to solutions to variational inequalities of the first kind. The proposed approach has three components: (i) a slack-based approximation for the solution; (ii) a primal approximation for the Lagrange multiplier; and (iii) a posteriori bounds for the error in the combined primal-slack variable approximation. The strict feasibility of the primal-slack approximations leads to two significant improvements upon existing methods. First, it provides a posteriori error bounds that are significantly sharper than existing bounds. Second, it enables a full offline–online computational decomposition, in which the online cost to compute the error bound is completely independent of the dimension of the original (high-dimensional) problem. Our numerical results allow us to compare the performance of the proposed and existing approaches.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.024
Zhenying Zhang 1 ; Eduard Bader 1 ; Karen Veroy 1

1 Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Germany 
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Zhenying Zhang; Eduard Bader; Karen Veroy. A slack approach to reduced-basis approximation and error estimation for variational inequalities. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 283-289. doi : 10.1016/j.crma.2015.10.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.024/

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[3] R. Glowinski Numerical Methods for Nonlinear Variational Problems, Springer, 1984

[4] B. Haasdonk; J. Salomon; B. Wohlmuth A reduced basis method for parametrized variational inequalities, SIAM J. Numer. Anal., Volume 50 (2012) no. 5, pp. 2656-2676

[5] N.A. Kikuchi; J.T. Oden Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Springer, 1988

[6] J.L. Lions; G. Stampacchia Variational inequalities, Commun. Pure Appl. Math., Volume 20 (1967) no. 3, pp. 493-519

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

[8] G. Rozza; K. Veroy On the stability of the reduced basis method for Stokes equations in parametrized domains, Comput. Methods Appl. Mech. Eng., Volume 196 (2007) no. 7, pp. 1244-1260

[9] 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 (AIAA paper 2003-3847)

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