We consider some abstract nonlinear equations in a separable Hilbert space and some class of approximate equations on closed linear subspaces of . The main result concerns stability with respect to the approximation of the space . We prove that, generically, the set of all solutions of the exact equation is the limit in the sense of the Hausdorff metric over of the sets of approximate solutions, over some filterbase on the family of all closed linear subspaces of . The abstract results are applied to the classical Galerkin method and to the Holly method for the stationary Navier-Stokes equations for incompressible fluid in 2 and 3-dimensional bounded domains.
On considère certaines équations non linéaires abstraites dans un espace de Hilbert séparable et certaines classes d’équations approchées dans les sous-espaces vectoriels fermés de . Le résultat principal concerne la stabilité relativement à l’approximation de l’espace . On prouve que l’ensemble de toutes les solutions de l’équation exacte est la limite dans la métrique de Hausdorff des ensembles des solutions approchées, relativement à certaine base filtrée sur la famille des sous-espaces vectoriels fermés de . Les résultats généraux sont appliqués à la méthode de Galerkin et à la méthode de Holly pour les équations de Navier-Stokes stationnaires dans domaines bornés de dimension 2 et 3.
@article{AFST_2012_6_21_4_651_0,
author = {El\.zbieta Motyl},
title = {Stability for a certain class of numerical methods {\textendash} abstract approach and application to the stationary {Navier-Stokes} equations},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {651--743},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {4},
year = {2012},
doi = {10.5802/afst.1348},
mrnumber = {3052028},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1348/}
}
TY - JOUR AU - Elżbieta Motyl TI - Stability for a certain class of numerical methods – abstract approach and application to the stationary Navier-Stokes equations JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 651 EP - 743 VL - 21 IS - 4 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1348/ DO - 10.5802/afst.1348 LA - en ID - AFST_2012_6_21_4_651_0 ER -
%0 Journal Article %A Elżbieta Motyl %T Stability for a certain class of numerical methods – abstract approach and application to the stationary Navier-Stokes equations %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 651-743 %V 21 %N 4 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1348/ %R 10.5802/afst.1348 %G en %F AFST_2012_6_21_4_651_0
Elżbieta Motyl. Stability for a certain class of numerical methods – abstract approach and application to the stationary Navier-Stokes equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. 4, pp. 651-743. doi : 10.5802/afst.1348. https://afst.centre-mersenne.org/articles/10.5802/afst.1348/
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