CALCOLO

, Volume 18, Issue 3, pp 255–291 | Cite as

Methodes d'elements finis mixtes pour les equations de stokes et de Navier-Stokes dans un polygone non convexe

  • C. Bernardi
  • G. Raugel
Article

Abstract

We study a mixed finite element method for the steady-state Navier-Stokes equations in a polygon which is not necessarily convex. To take into account the signularities of the solution near the corners, we introduce weighted Sobolev spaces and prove the convergence of the method. The use of a non-uniformly regular family of triangulations allows us to get best error estimates.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographie

  1. [1]
    J. Bergh, J. Löfström,Interpolation Spaces. An introduction. (1976), Springer-Verlag.Google Scholar
  2. [2]
    C. Bernardi,Thèse de 3ème cycle, Paris 6 (1979).Google Scholar
  3. [3]
    C. Bernardi, G. Raugel,Méthodes mixtes pour les équations de Navier-Stokes dans un ouvert polygonal plan. Rapport interne du laboratoire d'Analyse Numérique, Université Paris 6 (1980).Google Scholar
  4. [4]
    L. Cattabriga,Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova31, (1961), 1–33.MathSciNetGoogle Scholar
  5. [5]
    P. G. Ciarlet,The finite element method for elliptic problems (1978), North Holland.Google Scholar
  6. [6]
    P. Clement,Approximation by finite element functions using local regularization. RAIRO9, no 2 (1975), 77–84.MathSciNetGoogle Scholar
  7. [7]
    C. Dauge, Thèse de 3ème Cycle, Nantes (1980).Google Scholar
  8. [8]
    V. Girault, P. A. Raviart,An analysis of a mixed finite element method for the Navier-Stokes equations. Numer. Math.33, (1979), 235–271.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    V. Girault, P. A. Raviart,Finite element approximation of the Navier-Stokes equations. Lecture Notes no 749, Springer-Verlag (1979).Google Scholar
  10. [10]
    P. Grisvard,Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyedral domain. Numerical Solution of Partial Differential Equations III, Synspade 1975, (1976), Academic Press, 207–274.Google Scholar
  11. [11]
    P. Grisvard,Singularité des solutions du problème de Stokes dans un polygone. Publications de l'université de Nice (1978).Google Scholar
  12. [12]
    R. B. Kellogg, J. E. Osborn,A regularity result for the Stokes problem in a convex polygon. J. Functional Analysis21 no 4 (1976), 397–431.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    V. A. Kondratiev,Boundary value problems for elliptic equations in domains with conical or angular points. Trudi Mosk. Mat. Obš.16, (1966), 209–292.Google Scholar
  14. [14]
    V. A. Kondratiev,Asymptotic behavior of a solution of the Navier-Stokes equation near the angular part of the boundary. Prikl. Mat. Meh.31, (1967), 118–123, J. Appl. Math. Mech.31 (1967), 125–129.Google Scholar
  15. [15]
    A. Kufner,Einige Eigenschaften der Sobolevschen Raüme mit Belegungsfunktion. Czech. Math. J.15, 90 (1965), 597–620.MathSciNetGoogle Scholar
  16. [16]
    O. A. Ladyzhenskaya,The mathematical theory of viscous incompressible flow. (1962), Gordon Breach, New-York.Google Scholar
  17. [17]
    R. Lozi,Résultats numériques de régularité du problème de Stokes et du laplacien itéré dans un polygone. RAIRO12, no 3 (1978), 267–282.MATHMathSciNetGoogle Scholar
  18. [18]
    J. E. Osborn,Regularity of solutions of the Stokes problem in a polygonal domain. Numerical Solution of Partial Differential Equations III, Synspade 1975. (1976), Academic Press, 393–411.Google Scholar
  19. [19]
    G. Raugel, Thèse de 3ème cycle, Rennes (1978).Google Scholar
  20. [20]
    A. H. Schatz, L. B. Wahlbin,Maximum norm estimates in the finite element method on plane polygonal domains, Part I. Math. Comput.32, (1978), 73–109.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    A. H. Schatz, L. B. Wahlbin,Maximum norm estimates in the finite element method on plane polygonal domains, Part II, Refinements. Math. Comput.33, (1979), 465–492.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    R. Scholz,A mixed method for 4 th order problems using linear finite elements. RAIRO12, no 1 (1978), 85–90.MATHMathSciNetGoogle Scholar
  23. [23]
    J. B. Seif,On the Green's function for the biharmonic equation on a infinite wedge. Trans. Amer. Math. Soc.182, (1973), 241–260.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Instituto di Elaborazione della Informazione del CNR 1981

Authors and Affiliations

  • C. Bernardi
    • 1
  • G. Raugel
    • 2
  1. 1.Laboratoire d'Analyse Numérique (La 189)CNRS et Université de Paris VIParis Cedex 05
  2. 2.UER de Mathématiques et Informatique, Laboratoire d'Analyse NumériqueCNRS et Université de Rennes IRennes Cedex

Personalised recommendations