Skip to main content
SpringerLink
Log in

A second order accuracy for a full discretized time-dependent Navier–Stokes equations by a two-grid scheme

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We study a second-order two-grid scheme fully discrete in time and space for solving the Navier–Stokes equations. The two-grid strategy consists in discretizing, in the first step, the fully non-linear problem, in space on a coarse grid with mesh-size H and time step Δt and, in the second step, in discretizing the linearized problem around the velocity u H computed in the first step, in space on a fine grid with mesh-size h and the same time step. The two-grid method has been applied for an analysis of a first order fully-discrete in time and space algorithm and we extend the method to the second order algorithm. This strategy is motivated by the fact that under suitable assumptions, the contribution of u H to the error in the non-linear term, is measured in the L 2 norm in space and time, and thus has a higher-order than if it were measured in the H 1 norm in space. We present the following results: if h 2 = H 3 = (Δt)2, then the global error of the two-grid algorithm is of the order of h 2, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abboud, H., Girault, V., Sayah, T.: Second-order two-grid finite element scheme for the fully discrete transient Navier–Stokes equations. In: Proceeding Conférence Euro Méditerranéenne sur les Biomathématiques, Caire, Egypte (2007)

  2. Abboud H., Sayah T.: A full discretization of the time-dependent Navier–Stokes equations by a two-grid scheme. M2AN Math. Model. Numer. Anal 42(1), 141–174 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Abboud, H., Girault V., Sayah, T.: Two-grid finite element scheme for the fully discrete time-dependent Navier–Stokes problem. In: Abboud, H., et al. (eds.) C. R. Acad. Sci. Paris, Ser. I 341 (2005)

  4. Adams R.-A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  5. Ait Ou Amni A., Marion M.: Nonlinear Galerkin methods and mixed finite element: two-grid algorithms for the Navier–Stokes equations. Numer. Math. 62, 189–213 (1994)

    Google Scholar 

  6. Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  7. Foias C., Manley O., Temam R.: Modelization of the interaction of small and large eddies in two dimensional turbulent flows. RAIRO Modél. Anal. Numér. 22, 93–114 (1988)

    MATH  MathSciNet  Google Scholar 

  8. Girault V., Lions J.-L.: Two-grid finite-element schemes for the steady Navier–Stokes problem in polyhedra. Portugal Math. 58, 25–57 (2001)

    MATH  MathSciNet  Google Scholar 

  9. Girault V., Lions J.-L.: Two-grid finite-element schemes for the transient Navier–Stokes equations. M2AN 35, 945–980 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Girault, V., Raviart, P.-A.: Finite Element Methods for the Navier–Stokes Equations. Theory and Algorithms. In: Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

  11. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)

  12. Hecht, F., Pironneau, O.: FreeFem++, see: http://www.freefem.org

  13. Ladyzenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. (In Rusian, 1961), First English translation, Gordon & Breach, New York (1963)

  14. Layton W.: A two-level discretization method for the Navier–Stokes equations. Comput. Math. Appl. 26(2), 33–38 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Layton W., Lenferink W.: Two-level Picard-defect corrections for the Navier–Stokes equations at high Reynolds number. Appl. Math. Comput. 69(2/3), 263–274 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lions J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  17. Lions J.-L., Magenes E.: Problèmes aux limites non homogènes et applications I. Dunod, Paris (1968)

    Google Scholar 

  18. Marion M., Temam R.: Navier–Stokes equations: theory and approximation. In: Ciarlet, P.G., Lions, J.-L. (eds) Handbook of Numerical Analysis, vol. VI, pp. 503–688. Elsevier, Amsterdam (1998)

    Google Scholar 

  19. Marion M., Temam R.: Nonlinear Galerkin methods: the finite element case. Numer. Math. 57, 1–22 (1990)

    Article  MathSciNet  Google Scholar 

  20. Marion M., Temam R.: Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26, 1139–1157 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  21. Nečas J.: Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967)

    Google Scholar 

  22. Temam R.: Une méthode d’approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98, 115–152 (1968)

    MathSciNet  Google Scholar 

  23. Wheeler M.F.: A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM. J. Numer. Anal. 10(4), 723–759 (1973)

    Article  MathSciNet  Google Scholar 

  24. Xu, J.: Some Two-Grid Finite Element Methods, Tech. Report, P.S.U (1992)

  25. Xu J.: A novel two-grid method of semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  26. Xu J.: Two-grid finite element discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 33, 1759–1777 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculté des Sciences et de Génie Informatique, Université Saint-Esprit de Kaslik, B.P. 446, Jounieh, Lebanon

    Hyam Abboud

  2. Faculté des Sciences II, Université Libanaise, Fanar, Lebanon

    Hyam Abboud

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris 6), Boîte Courrier 187, 4 place Jussieu, 75252, Paris Cedex 05, France

    Vivette Girault

  4. Faculté des Sciences, Université Saint-Joseph, B.P. 11-514, Riad El Solh, Beyrouth, 1107 2050, Lebanon

    Toni Sayah

Authors
  1. Hyam Abboud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vivette Girault
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Toni Sayah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyam Abboud.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abboud, H., Girault, V. & Sayah, T. A second order accuracy for a full discretized time-dependent Navier–Stokes equations by a two-grid scheme. Numer. Math. 114, 189–231 (2009). https://doi.org/10.1007/s00211-009-0251-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-009-0251-5

Mathematics Subject Classification (2000)

Search

Navigation