Comptes Rendus
Partial Differential Equations/Numerical Analysis
On the preconditioned conjugate gradient solution of a Stokes problem with Robin-type boundary conditions
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 903-908.

We discuss the solution by a preconditioned conjugate gradient algorithm of a Stokes problem with Robin-type boundary conditions. This Stokes problem is encountered when applying an appropriate operator-splitting scheme to the time-discretization of a system modeling the interaction of an incompressible viscous fluid with a deformable thin elastic structure. The main contribution of this Note is the identification of a preconditioner operating in the pressure space that reduces substantially the number of iterations when compared to a conjugate gradient algorithm equipped with the canonical scalar product of L2. The results of numerical experiments show the validity of our approach.

Dans cette Note on étudie la résolution d'un problème de Stokes avec conditions aux limites du type Robin par un algorithme de gradient conjugué préconditionné. On rencontre ce type de problèmes lorsque l'on applique certains schémas de discrétisation en temps par décomposition d'opérateurs à la résolution numérique du système d'équations aux derivées partielles modélisant l'interaction d'un fluide visqueux incompressible avec une structure élastique mince. La contribution principale de cette Note est l'identification d'un opérateur de préconditionnement agissant dans l'espace des pressions (en l'occurrence L2(Ω)) ; cet opérateur réduit considérablement le nombre d'itérations nécesssaires à la convergence, lorsque l'on compare à l'algorithme de gradient conjugué opérant dans L2(Ω) muni de son produit scalaire canonique. Les résultats d'essais numériques confirment la validité de notre approche.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.05.014
Roland Glowinski 1, 2; Giovanna Guidoboni 2

1 Laboratoire Jacques-Louis Lions, université P. et M. Curie, 4, place Jussieu, 75005 Paris, France
2 Department of Mathematics, University of Houston, PGH 651, Houston, TX 77204-3476, USA
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Roland Glowinski; Giovanna Guidoboni. On the preconditioned conjugate gradient solution of a Stokes problem with Robin-type boundary conditions. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 903-908. doi : 10.1016/j.crma.2009.05.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.014/

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[7] R. Glowinski; G. Guidoboni; T.-W. Pan Wall-driven incompressible viscous flow in a two-dimensional semi-circular cavity, J. Comput. Phys., Volume 216 (2006) no. 1, pp. 76-91

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