Skip to main content
SpringerLink
Log in

Uniform preconditioners for the time dependent Stokes problem

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

An Erratum to this article was published on 02 February 2005

Summary.

Implicit time stepping procedures for the time dependent Stokes problem lead to stationary singular perturbation problems at each time step. These singular perturbation problems are systems of saddle point type, which formally approach a mixed formulation of the Poisson equation as the time step tends to zero. Preconditioners for discrete analogous of these systems are discussed. The preconditioners uses standard positive definite elliptic preconditioners as building blocks and lead to condition numbers which are bounded uniformly with respect to the time step and the spatial discretization. The construction of the discrete preconditioners is related to the mapping properties of the corresponding continuous system.

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. Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element method for the Stokes equations. CALCOLO 21, 337–344 (1984)

    MATH  MathSciNet  Google Scholar 

  2. Arnold, D.N., Falk, R.S., Winther, R.: Preconditioning discrete approximations of the Reissner–Mindlin plate model. M 2AN 31, 517–557 (1997)

    Google Scholar 

  3. Bercovier, M., Pironneau, O.: Error estimates for finite element method solution of the Stokes problem in primitive variables. Numer. Math. 33, 211–224 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bergh, J., Löfström, J.: Interpolation spaces. Springer Verlag, 1976

  5. Braess, D., Blömer, C.: A multigrid method for a parameter dependent problem in solid mechanics. Numer. Math 57, 747–761 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bramble, J.H., Pasciak, J.E.: Iterative techniques for time dependent Stokes problem. Comput. Math. Appl. 33, 13–30 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brenner, S.C.: Multigrid methods for parameter dependent problems. RAIRO M 2AN 30(3), 265–297 (1996)

    Google Scholar 

  8. Brezzi, F.: On the existence, uniqueness and approximation of saddle–point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8, 129–151 (1974)

    MATH  MathSciNet  Google Scholar 

  9. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer Verlag, 1991

  10. Cahouet, J., Chabard, J.P.: Some fast 3D finite element solvers for the generalized Stokes problem. Int. J. Numer. Meth. Fluids 8, 869–895 (1988)

    Article  MathSciNet  Google Scholar 

  11. Crouzeix, M., Raviart, P.A.: Conforming and non–conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7, 33–76 (1973)

    MathSciNet  Google Scholar 

  12. Dean, E.J., Glowinski, R.: On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow. In Proceedings: Incompressible computational fluid dynamics; Trends and advances, M. D. Gunzburger and R. A. Nicolaides, (eds.), Cambridge University Press, 1993

  13. Elman, H.C.: Preconditioners for saddle point problems arising in computational fluid dynamics. Appl. Numer. Math. 43, 75–89 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Elman, H.C., Silvester, D.J., Wathen, A.J.: Block preconditioners for the discrete incompressible Navier–Stokes equations. Int. J. Numer. Meth. Fluids 40, 333–344 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations. Springer Verlag 1986

  16. Hackbusch, W.: Iterative solution of large sparse systems of equations. Springer Verlag 1994

  17. Haug, E., Winther, R.: A domain embedding preconditioner for the Lagrange multiplier system. Math. Comp. 69, 65–82 (1999)

    MathSciNet  Google Scholar 

  18. Klawonn, A.: An optimal preconditioner for a class of saddle point problems with a penalty term. SIAM J. Sci. Comput. 19, 540–552 (1998)

    MATH  MathSciNet  Google Scholar 

  19. Loghin, D., Wathen, A.J.: Schur complement preconditioners for the Navier-Stokes equations. Int. J. Numer. Meth. Fluids 40, 403–412 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mayday, Y., Meiron, D., Patera, A., Ronquist, E.: Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations. SIAM J. Sci. Comput. 14, 310–337 (1993)

    MathSciNet  Google Scholar 

  21. Mardal, K.A., Tai, X.-C., Winther, R.: A robust finite element method for Darcy–Stokes flow. SIAM J. Numer. Anal. 40, 1605–1631 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nečas, J.: Equations aux dérivée partielles. Presses de l’Université de Montréal 1965

  23. Olshanskii, M.A., Reusken, A.: Navier–Stokes equations in rotation form: A robust multigrid solver for the velocity problem. SIAM J. Sci. Comp. 23, 1683–1706 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Paige, C.C., Sauders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pavarino, L.F.: Indefinite overlapping Schwarz methods for time-dependent Stokes problems. Comput. Meth. Appl. Mech. Eng. 187, 35–51 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pironneau, O.: The finite element method for fluids. John Wiley & Sons, 1989

  27. Rusten, T., Vassilevski, P.S., Winther, R.: Interior penalty preconditioners for mixed finite element approximations of elliptic problems. Math. Comp. 65, 447–466 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rusten, T., Winther, R.: A preconditioned iterative method for saddle point problems. SIAM J. Matrix Anal. 13, 887–904 (1992)

    MATH  MathSciNet  Google Scholar 

  29. Silvester, D., Wathen, A.: Fast iterative solution of stabilized Stokes systems. Part II: Using block diagonal preconditioners. SIAM J. Numer. Anal. 31, 1352–1367 (1994)

    MATH  MathSciNet  Google Scholar 

  30. Stenberg, R.: Error analysis of some finite element methods for the Stokes problem. Math. Comp. 54, 495–508 (1990)

    MATH  MathSciNet  Google Scholar 

  31. Turek, S.: Efficient Solvers for Incompressible Flow Problem. Springer Verlag 1999

  32. Verfürth, R.: Error estimates for a mixed finite element approximation of the Stokes equation. R.A.I.R.O. Anal. Numer. 18, 175–182 (1984)

    MATH  Google Scholar 

  33. Verfürth, R.: A multilevel algorithm for mixed problems. SIAM J. Numer. Anal 21, 264–271 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wittum, G.: Multigrid methods for Stokes and Navier-Stokes equations. Numer. Math. 54, 543–564 (1989)

    Article  MathSciNet  Google Scholar 

  35. Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Scientific Computing and Department of Informatics, Simula Research Laboratory, University of Oslo, 134, 1325, Lysaker, Norway

    Kent-Andre Mardal

  2. Centre of Mathematics for Applications and Department of Informatics, University of Oslo, 1053 Blindern, 0316, Oslo, Norway

    Ragnar Winther

Authors
  1. Kent-Andre Mardal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ragnar Winther
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kent-Andre Mardal.

Additional information

Mathematics Subject Classification: 65M55, 65

Revised version received September 18, 2003

An erratum to this article is availabel at http://dx.doi.org/10.1007/s00211-005-0663-9.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mardal, KA., Winther, R. Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98, 305–327 (2004). https://doi.org/10.1007/s00211-004-0529-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-004-0529-6

Keywords

Search

Navigation