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://www.numdam.org/articles/10.1016/j.crma.2015.03.019/

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