logo AFST

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, Série 6, Tome 21 (2012) no. 4, pp. 651-743.

On considère certaines équations non linéaires abstraites dans un espace de Hilbert séparable H et certaines classes d’équations approchées dans les sous-espaces vectoriels fermés de H. Le résultat principal concerne la stabilité relativement à l’approximation de l’espace H. 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 H. 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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1348
@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, Toulouse},
     volume = {Ser. 6, 21},
     number = {4},
     year = {2012},
     doi = {10.5802/afst.1348},
     zbl = {pre06113107},
     mrnumber = {3052028},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1348/}
}
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, Série 6, Tome 21 (2012) no. 4, pp. 651-743. doi : 10.5802/afst.1348. https://afst.centre-mersenne.org/articles/10.5802/afst.1348/

[1] Adams (R.).— Sobolev spaces, Academic Press (1975). | MR 450957 | Zbl 1098.46001

[2] Deimling (K.).— Nonlinear Functional Analysis, Springer-Verlag (1980). | MR 787404 | Zbl 0559.47040

[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). | Numdam | MR 713114 | Zbl 0546.35058

[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). | MR 435626 | Zbl 0335.35077

[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). | Numdam | MR 481645 | Zbl 0384.35047

[6] Girault (V.), Raviart (P.A.).— Finite element methods for Navier-Stokes equations, Springer-Verlag, Berlin – Heidelberg – New York - Tokyo (1986). | MR 851383 | Zbl 0585.65077

[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). | MR 1104732 | Zbl 0732.76022

[8] Holly (K.), Motyl (E.).— Inversion of the divdiv * – operator and three numerical methods in hydrodynamics, Selected problems of Mathematics, Cracow University of Technology, p. 35-94 (1995). | MR 1438065

[9] Kołodziej (W.).— Wybrane rozdziały analizy matematycznej, Biblioteka Matematyczna, 36, Warszawa (1982). | Zbl 0553.46001

[10] Lions (J.L.).— Quelques méthodes de résolution des problemes aux limites non linèaires, Dunod, Paris (1969). | Zbl 0189.40603

[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). | MR 1812117 | Zbl 1007.76041

[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). | MR 2202974 | Zbl 1089.35045

[14] Nirenberg (L.).— Topics in nonlinear functional analysis, New York (1974). | MR 488102 | Zbl 0286.47037

[15] Rudin (W.).— Functional analysis, McGraw-Hill Book Company, New York (1973). | MR 365062 | Zbl 0867.46001

[16] Saut (J.C.), Temam (R.).— Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4, p. 293-319 (1979). | MR 522714 | Zbl 0462.35016

[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). | MR 570691 | Zbl 0445.76023

[18] Smale (S.).— An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87, p. 861-866 (1965). | MR 185604 | Zbl 0143.35301

[19] Temam (R.).— Navier-Stokes equations. Theory and numerical analysis, North Holland Publishing Company, Amsterdam – New York – Oxford (1979). | MR 603444 | Zbl 0426.35003

[20] Temam (R.).— Navier-Stokes equations and nonlinear functional analysis, SIAM, Philadelphia, Pensylvania (1995). | MR 1318914 | Zbl 0833.35110