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.

DOI : 10.5802/afst.1348
Motyl, Elżbieta 1

1 Department of Mathematics and Computer Sciences, University of Łódź, Poland
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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, Série 6, Tome 21 (2012) no. 4, pp. 651-743. doi : 10.5802/afst.1348. http://www.numdam.org/articles/10.5802/afst.1348/

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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