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

top
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.

How to cite

top

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

top
  1. Adams (R.).— Sobolev spaces, Academic Press (1975). Zbl1098.46001MR450957
  2. Deimling (K.).— Nonlinear Functional Analysis, Springer-Verlag (1980). Zbl0559.47040MR787404
  3. Foiaş (C.), Saut (J.C.).— Remarques sur les équations de Navier-Stokes stationaires, Ann. Scuola Norm. Sup. Pisa, Sèrie IV, 10, p. 19-177 (1983). Zbl0546.35058MR713114
  4. Foiaş (C.), Temam (R.).— Structure of the set of stationary solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. Vol. XXX, p. 149-164 (1977). Zbl0335.35077MR435626
  5. Foiaş (C.), Temam R..— Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Ann. Scuola Norm. Sup. Pisa, Sèrie IV, 5, p. 29-63 (1978). Zbl0384.35047MR481645
  6. Girault (V.), Raviart (P.A.).— Finite element methods for Navier-Stokes equations, Springer-Verlag, Berlin – Heidelberg – New York - Tokyo (1986). Zbl0585.65077MR851383
  7. Holly (K.).— Some application of the implicit function theorem to the stationary Navier-Stokes equations, Ann. Polon. Math. LIV.2, p. 93-100 (1991). Zbl0732.76022MR1104732
  8. Holly (K.), Motyl (E.).— Inversion of the – operator and three numerical methods in hydrodynamics, Selected problems of Mathematics, Cracow University of Technology, p. 35-94 (1995). MR1438065
  9. Kołodziej (W.).— Wybrane rozdziały analizy matematycznej, Biblioteka Matematyczna, 36, Warszawa (1982). Zbl0553.46001
  10. Lions (J.L.).— Quelques méthodes de résolution des problemes aux limites non linèaires, Dunod, Paris (1969). Zbl0189.40603
  11. Marcinkowska (H.).— Dystrybucje, przestrzenie Sobolewa, równania różniczkowe, Biblioteka Matematyczna, 75, Warszawa (1993). 
  12. Motyl (E.).— The stationary Navier-Stokes equations – application of the implicit function theorem to the problem of stability, Univ. Iagell. Acta Math. XXXVIII, p. 227-277 (2000). Zbl1007.76041MR1812117
  13. Motyl (E.).— A new method of calculation of the pressure in the stationary Navier-Stokes equations, J. Comp. and Appl. Math. 189, p. 207-219 (2006). Zbl1089.35045MR2202974
  14. Nirenberg (L.).— Topics in nonlinear functional analysis, New York (1974). Zbl0286.47037MR488102
  15. Rudin (W.).— Functional analysis, McGraw-Hill Book Company, New York (1973). Zbl0867.46001MR365062
  16. Saut (J.C.), Temam (R.).— Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4, p. 293-319 (1979). Zbl0462.35016MR522714
  17. Saut (J.C.), Temam (R.).— Generic properties of Navier-Stokes equations: Genericity with respect to the boundary values, Indiana Univ. Math. J. 29, p. 427-446 (1980). Zbl0445.76023MR570691
  18. Smale (S.).— An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87, p. 861-866 (1965). Zbl0143.35301MR185604
  19. Temam (R.).— Navier-Stokes equations. Theory and numerical analysis, North Holland Publishing Company, Amsterdam – New York – Oxford (1979). Zbl0426.35003MR603444
  20. Temam (R.).— Navier-Stokes equations and nonlinear functional analysis, SIAM, Philadelphia, Pensylvania (1995). Zbl0833.35110MR1318914

NotesEmbed ?

top

You must be logged in to post comments.