Skip to main content
SpringerLink
Log in

Preconditioning sparse grad-div/augmented Lagrangian stabilized saddle point systems

  • Published:
Computing and Visualization in Science

Abstract

This paper deals with the analysis of preconditioning techniques for a recently introduced sparse grad-div stabilization of the Oseen problem. The finite element discretization error for the Oseen problem can be reduced through the addition of a grad-div stabilization term to the momentum equation of the Oseen problem. Such a stabilization has an interesting effect on the properties of the discrete linear system of equations, in particular on the convergence properties of iterative solvers. Comparing to unstabilized systems, it swaps the levels of difficulties for solving the two main subproblems, i.e., solving for the first diagonal block and solving a Schur complement problem, that occur in preconditioners based on block triangular factorizations. In this paper we are concerned with a sparse variant of grad-div stabilization which has been shown to have a stabilization effect similar to the full grad-div stabilization while leading to a sparser system matrix. Our focus lies on the subsequent iterative solution of the discrete system of equations.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Benzi, M., Olshanskii, M.A.: An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28, 2095–2113 (2006)

  2. Benzi, M., Wang, Z.: Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 33, 2761–2784 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Benzi, M., Golub, G., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benzi, M., Olshanskii, M., Wang, Z.: Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 66, 486–508 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Börm, S., Le Borne, S.: \(\cal {H}\)-LU factorization in preconditioners for augmented Lagrangian and grad–div stabilized saddle point systems. Int. J. Numer. Methods Fluids 68, 83–98 (2012)

    Article  MATH  Google Scholar 

  6. Bowers, A., Le Borne, S., Rebholz, L.: Error analysis and iterative solvers for Navier–Stokes projection methods with standard and sparse grad–div stabilization. Comput. Methods Appl. Mech. Eng. 275, 1–19 (2014)

    Article  MATH  Google Scholar 

  7. de Niet, A., Wubs, F.: Two preconditioners for saddle point problems in fluid flows. Int. J. Numer. Methods Fluids 54, 355–377 (2007)

  8. Elman, H., Howle, V., Shadid, J., Shuttlerworth, R., Tumarino, R.: Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27, 1651–1668 (2005)

    Article  Google Scholar 

  9. Grasedyck, L., Kriemann, R., Le Borne, S.: Parallel black box \(\cal H\)-LU preconditioning for elliptic boundary value problems. Comput. Visual. Sci. 4–6, 273–291 (2008)

    Article  Google Scholar 

  10. Grasedyck, L., Kriemann, R., Le Borne, S.: Domain decomposition based \(\cal H\)-LU preconditioning. Numer. Math. 112, 565–600 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. He, X., Neytchava, M., Capizzano, S.: On an augmented Lagrangian-based preconditioning of Oseen type problems. BIT 51, 865–888 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Heister, T., Rapin, G.: Efficient augmented Lagrangian-type preconditioning for the Oseen problem using grad–div stabilization. Int. J. Numer. Methods Fluids 71, 118–134 (2013)

    Article  MathSciNet  Google Scholar 

  13. Kay, D., Loghin, D., Wathen, A.: A preconditioner for the steady-state Navier–Stokes equations. SIAM J. Sci. Comput. 24, 237–256 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Linke, A., Rebholz, L.: On a reduced sparsity stabilization of grad–div type for incompressible flow problems. Comput. Methods Appl. Mech. Eng. 261, 142–153 (2013)

    Article  MathSciNet  Google Scholar 

  15. Olshanskii, M.: A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods. Comput. Methods Appl. Mech. Eng. 191, 5515–5536 (2002)

    Article  MathSciNet  Google Scholar 

  16. Olshanskii, M., Reusken, A.: Grad–div stabilization for the stokes equations. Math. Comput. 73, 1699–1718 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Olshanskii, M., Vassilevski, Y.: Pressure Schur complement preconditioners for the discrete Oseen problem. SIAM J. Sci. Comput. 29, 2686–2704 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Roos, H., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations: Convection Diffusion and Flow Problems, Computational Mathematics, vol. 24. Springer, Berlin (1996)

    Google Scholar 

Download references

Acknowledgments

We dedicate this paper to Dietrich Braess on the occasion of his 75th birthday and to Wolfgang Hackbusch on the occasion of his 65th birthday.

Author information

Authors and Affiliations

  1. Hamburg University of Technology, Hamburg, Germany

    Sabine Le Borne

  2. Clemson University, Clemson, SC, USA

    Leo G. Rebholz

Authors
  1. Sabine Le Borne
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leo G. Rebholz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sabine Le Borne.

Additional information

Communicated by Gabriel Wittum.

L. G. Rebholz: Partially support by NSF grant DMS1112593.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Le Borne, S., Rebholz, L.G. Preconditioning sparse grad-div/augmented Lagrangian stabilized saddle point systems. Comput. Visual Sci. 16, 259–269 (2013). https://doi.org/10.1007/s00791-015-0236-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-015-0236-0

Keywords

Search

Navigation