Numerical analysis
A Petrov–Galerkin reduced basis approximation of the Stokes equation in parameterized geometries
[Une méthode d'éléments finis de type Petrov–Galerkin pour l'approximation en base réduite du problème de Stokes]

Présenté par : Olivier Pironneau

Abdulle, Assyr1 ; Budáč, Ondrej1
1 ANMC, Section de mathématiques, École polytechnique fédérale de Lausanne, Switzerland
Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 641-645.

Nous présentons une méthode d'éléments finis de type Petrov–Galerkin pour l'approximation en « bases réduites » du problème de Stokes. La stabilité de notre méthode est établie à l'aide de la théorie inf–sup de Babuška et nous prouvons une borne sur la condition numérique de la matrice du système linéaire « en ligne ». Comparée aux méthodes de type bases réduites existantes, qui sont à la fois stable et dont la condition numérique du système linéaire en ligne peut être controlée, notre méthode a un coût en ligne considerablement plus faible et est applicable à des formulations générales non coercives ne nécessitant pas de structure de type point-selle.

We present a Petrov–Galerkin reduced basis (RB) approximation for the parameterized Stokes equation. Our method, which relies on a reduced solution space and a parameter-dependent test space, is shown to be stable (in the sense of Babuška) and algebraically stable (a bound on the condition number of the online system can be established). Compared to other stable RB methods that can also be shown to be algebraically stable, our approach is among those with the smallest online time cost and it has general applicability to linear non-coercive problems without assuming a saddle-point structure.

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DOI : 10.1016/j.crma.2015.03.019
Abdulle, Assyr 1 ; Budáč, Ondrej 1

1 ANMC, Section de mathématiques, École polytechnique fédérale de Lausanne, Switzerland 
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Abdulle, Assyr; Budáč, Ondrej. A Petrov–Galerkin reduced basis approximation of the Stokes equation in parameterized geometries. Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 641-645. doi : 10.1016/j.crma.2015.03.019. http://archive.numdam.org/articles/10.1016/j.crma.2015.03.019/

[1] Abdulle, A.; Budáč, O. An adaptive finite element heterogeneous multiscale method for the Stokes problem in porous media, Multiscale Model. Simul., Volume 13 (2015), pp. 256-290

[2] A. Abdulle, O. Budáč, A reduced basis finite element heterogeneous multiscale method for Stokes flow in porous media, 2014, in preparation.

[3] Carlberg, K.; Bou-Mosleh, C.; Farhat, C. Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations, Int. J. Numer. Methods Eng., Volume 86 (2011), pp. 155-181

[4] Dahmen, W.; Plesken, C.; Welper, G. Double greedy algorithms: reduced basis methods for transport dominated problems, M2AN Math. Model. Numer. Anal., Volume 48 (2014), pp. 623-663

[5] Gerner, A.-L.; Veroy, K. Certified reduced basis methods for parameterized saddle point problems, SIAM J. Sci. Comput., Volume 34 (2012), p. A2812-A2836

[6] Huynh, D.B.P.; Knezevic, D.J.; Chen, Y.; Hesthaven, J.S.; Patera, A.T. A natural-norm successive constraint method for inf–sup lower bounds, Comput. Methods Appl. Mech. Eng., Volume 199 (2010), pp. 1963-1975

[7] Løvgren, A.E.; Maday, Y.; Rønquist, E.M. A reduced basis element method for the steady Stokes problem, ESAIM Math. Model. Numer. Anal., Volume 40 (2006), pp. 529-552

[8] Rovas, D.V. Reduced-basis output bound methods for parametrized partial differential equations, Massachusetts Institute of Technology, 2003 (PhD thesis)

[9] Rozza, G.; Huynh, D.B.P.; Manzoni, A. Reduced basis approximation and a posteriori error estimation for Stokes flows in parameterized geometries: roles of the inf–sup stability constants, Numer. Math., Volume 125 (2013), pp. 1-38

[10] Rozza, G.; Huynh, D.B.P.; Patera, A.T. Reduced basis approximation and a posteriori error estimation for affinely parameterized elliptic coercive partial differential equations, Arch. Comput. Methods Eng., Volume 15 (2008), pp. 229-275

[11] Rozza, G.; Veroy, K. On the stability of the reduced basis method for Stokes equations in parameterized domains, Comput. Methods Appl. Mech. Eng., Volume 196 (2007), pp. 1244-1260

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