Stability for a certain class of numerical methods – abstract approach and application to the stationary Navier-Stokes equations

Elżbieta Motyl[1]

  • [1] Department of Mathematics and Computer Sciences, University of Łódź, Poland

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: 4, page 651-743
  • ISSN: 0240-2963

Abstract

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We consider some abstract nonlinear equations in a separable Hilbert space H and some class of approximate equations on closed linear subspaces of H . The main result concerns stability with respect to the approximation of the space H . We prove that, generically, the set of all solutions of the exact equation is the limit in the sense of the Hausdorff metric over H of the sets of approximate solutions, over some filterbase on the family of all closed linear subspaces of H . 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.

How to cite

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Motyl, Elżbieta. "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 21.4 (2012): 651-743. <http://eudml.org/doc/250996>.

@article{Motyl2012,
abstract = {We consider some abstract nonlinear equations in a separable Hilbert space $H$ and some class of approximate equations on closed linear subspaces of $H$. The main result concerns stability with respect to the approximation of the space $H$. We prove that, generically, the set of all solutions of the exact equation is the limit in the sense of the Hausdorff metric over $H$ of the sets of approximate solutions, over some filterbase on the family of all closed linear subspaces of $H$. 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.},
affiliation = {Department of Mathematics and Computer Sciences, University of Łódź, Poland},
author = {Motyl, Elżbieta},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {10},
number = {4},
pages = {651-743},
publisher = {Université Paul Sabatier, Toulouse},
title = {Stability for a certain class of numerical methods – abstract approach and application to the stationary Navier-Stokes equations},
url = {http://eudml.org/doc/250996},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Motyl, Elżbieta
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
DA - 2012/10//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - 4
SP - 651
EP - 743
AB - We consider some abstract nonlinear equations in a separable Hilbert space $H$ and some class of approximate equations on closed linear subspaces of $H$. The main result concerns stability with respect to the approximation of the space $H$. We prove that, generically, the set of all solutions of the exact equation is the limit in the sense of the Hausdorff metric over $H$ of the sets of approximate solutions, over some filterbase on the family of all closed linear subspaces of $H$. 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.
LA - eng
UR - http://eudml.org/doc/250996
ER -

References

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