Computing and Visualization in Science

, Volume 11, Issue 3, pp 169–180 | Cite as

Smoothed aggregation multigrid for a Stokes problem

Regular article

Abstract

We discuss advantages of using algebraic multigrid based on smoothed aggregation for solving indefinite linear problems. The ingredients of smoothed aggregation are used to construct a black-box monolithic multigrid method with indefinite coarse problems. Several techniques enforcing inf–sup stability conditions on coarse levels are presented. Numerical experiments are designed to support recent stability results for coupled algebraic multigrid. Comparison of the proposed multigrid preconditioner with other methods shows its robust behaviour even for very elongated geometries, where the pressure mass matrix is no longer a good preconditioner for the pressure Schur complement.

Keywords

Stokes Multigrid Smoothed aggregation Preconditioner 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams M.F. (2004). Algebraic multigrid methods for constrained linear systems with applications to contact problems in solid mechanics. Numer. Linear Algebra Appl. 11: 141–153 CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Braess D. and Sarazin R. (1997). An efficient smoother for the Stokes problem. Appl. Numer. Math. 23: 3–19 CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Brezina M. and Vaněk P. (1999). A black-box iterative solver based on a two-level Schwarz method. Computing 63(3): 233–263 CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Cui M.R. (2004). Analysis of iterative algorithms of Uzawa type for saddle point problems. Appl. Numer. Math. 50: 133–146 CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Dobrowolski M. (2003). On the LBB constant on stretched domains. Math. Nachr. 254–255: 64–67 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Elman, H.C.: Preconditioning strategies for models of incompressible flow. Research report CS-TR no.4543/ UMIACS TR no.2003-111, University of Maryland, November (2003)Google Scholar
  7. 7.
    Elman H.C., Silvester D.J. and Wathen A.J. (2002). Performance and analysis of saddle point preconditioners for the discrete steady-state Navier–Stokes equations. Numer. Math. 90: 665–688 CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Elman H.C., Howle V.E., Shadid J.N. and Tuminaro R.S. (2003). A parallel block multi-level preconditioner for the 3D incompressible Navier–Stokes equations. J. Comput. Phys. 187: 504–523 CrossRefMATHGoogle Scholar
  9. 9.
    Franca L. and Stenberg R. (1991). Error analysis of some GLS methods for elasticity equations. SIAM J. Numer. Anal. 28: 1680–1697 CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Griebel M., Neunhoffer T. and Regler H. (1998). Algebraic multigrid methods for the solution of the Navier–Stokes equations in complicated geometries. Int. J. Numer. Methods Fluids 26: 281–301 CrossRefMATHGoogle Scholar
  11. 11.
    Loghin D. and Wathen A.J. (2003). Schur complement preconditioning for elliptic systems of partial differential equations. Numer. Linear Algebra Appl. 10: 423–443 CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Picasso M. and Rappaz J. (2001). Stability of time-splitting schemes for the Stokes problem with stabilized finite elements. Numer. Methods Partial Differ. Equ. 17(6): 632–656 CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Powell, C., Silvester, D.: Black-box preconditioning for mixed formulation of self-adjoint elliptic PDEs, challenges in scientific computing—CISC 2002, 268–285, Lecture Notes in Computer Science Engineering, vol. 35. Springer, Berlin (2003)Google Scholar
  14. 14.
    Silvester D., Elman H., Kay D. and Wathen A. (2001). Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow. J. Comput. Appl. Math. 128: 261–279 CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Schöberl J. and Zulehner W. (2003). On Schwarz-type smoothers for saddle point problems. Numer. Math. 95: 377–399 CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Stüben K. (2001). A review of algebraic multigrid. Comput. Appl. Math. 128: 281–309 CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Vaněk P., Brezina M. and Mandel J. (2001). Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 88(3): 559–579 CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Verfürth R. (1984). Error estimates for a mixed finite element approximation of the Stokes problem. RAIRO Anal. Numer. 18: 175–182 MathSciNetMATHGoogle Scholar
  19. 19.
    Wabro M. (2004). Coupled algebraic multigrid methods for the Oseen problem. Comput. Vis. Sci. 7: 141–151 MathSciNetMATHGoogle Scholar
  20. 20.
    Webster R. (1994). An algebraic multigrid solver for Navier–Stokes problems. Int. J. Numer. Methods Fluids 18: 761–780 CrossRefMATHGoogle Scholar
  21. 21.
    Wesseling P. and Oosterlee C.W. (2001). Geometric multigrid with applications to computational fluid dynamics. J. Comput. Appl. Math. 128: 311–334 CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute of Analysis and Scientific ComputingEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

Personalised recommendations