Skip to main content
SpringerLink
Log in

A simplified HSS preconditioner for generalized saddle point problems

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

For generalized saddle point problems, we propose a simplified Hermitian and skew-Hermitian splitting (SHSS) preconditioner which is much closer to the generalized saddle point matrix than the HSS preconditioner. It is proved that all eigenvalues of the SHSS preconditioned matrix are real and nonunit eigenvalues are located in a positive interval. We also study the eigenvector distribution and the degree of the minimal polynomial of the preconditioned matrix. Numerical examples of a model Stokes problem show the effectiveness of the SHSS preconditioner.

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

Similar content being viewed by others

References

  1. Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z.-Z.: Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl. 16, 447–479 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.-Z.: Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models. Numer. Linear Algebra Appl. 19, 914–936 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-Z.: Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks. J. Comput. Appl. Math. 237, 295–306 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27, 1–23 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bai, Z.-Z., Golub, G.H., Li, C.-K.: Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput. 28, 583–603 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bai, Z.-Z., Golub, G.H., Li, C.-K.: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76, 287–298 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bai, Z.-Z., Ng, M.K.: On inexact preconditioners for nonsymmetric matrices. SIAM J. Sci. Comput. 26, 1710–1724 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bai, Z.-Z., Ng, M.K., Wang, Z.-Q.: Constraint preconditioners for symmetric indefinite matrices. SIAM J. Matrix Anal. Appl. 31, 410–433 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Math. Eng. 37, 229–256 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Benzi, M., Gander, M.J., Golub, G.H.: Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT Numer. Math. 43, 881–900 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Benzi, M., Golub, G.H.: A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl. 26, 20–41 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Benzi, M., Guo, X.-P.: A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations. Appl. Numer. Math. 61, 66–76 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Benzi, M., Ng, M.K., Niu, Q., Wang, Z.: A relaxed dimensional fractorization preconditioner for the incompressible Navier-Stokes equations. J. Comput. Phys. 230, 6185–6202 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  19. Cao, Y., Dong, J.-L., Wang, Y.-M.: A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier–Stokes equation. J. Comput. Appl. Math. 273, 41–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cao, Y., Jiang, M.-Q., Zheng, Y.-L.: A splitting preconditioner for saddle point problems. Numer. Linear Algebra Appl. 18, 875–895 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cao, Y., Yao, L.-Q., Jiang, M.-Q.: A modified dimensional split preconditioner for generalized saddle point problems. J. Comput. Appl. Math. 250, 70–82 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cao, Y., Yao, L.-Q., Jiang, M.-Q., Niu, Q.: A relaxed HSS preconditioner for saddle point problems from meshfree discretization. J. Comput. Math. 31, 398–421 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cao, Y., Yao, L.-Q., Yi, S.-C.: A weighted nodal-radial point interpolation meshless method for 2D solid problems. Eng. Anal. Bound. Elem. 39, 88–100 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chan, L.C., Ng, M.K., Tsing, N.K.: Spectral analysis for HSS preconditioners. Numer. Math. Theor. Math. Appl. 1, 57–77 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2014)

    Book  MATH  Google Scholar 

  26. Golub, G.H., Greif, C.: On solving block-structured indefinite linear systems. SIAM J. Sci. Comput. 24, 2076–2092 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huang, T.-Z., Wu, S.-L., Li, C.-X.: The spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for generalized saddle point problems. J. Comput. Appl. Math. 229, 37–46 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Huang, Y.-M.: A practical formula for computing optimal parameters in the HSS iteration methods. J. Comput. Appl. Math. 255, 142–149 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jiang, M.-Q., Cao, Y., Yao, L.-Q.: On parameterized block triangular preconditioners for generalized saddle point problems. Appl. Math. Comput. 216, 1777–1789 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Academic Press, New York (1979)

    MATH  Google Scholar 

  31. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  32. Pan, J.-Y., Ng, M.K., Bai, Z.-Z.: New preconditioners for saddle point problems. Appl. Math. Comput. 172, 762–771 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Ren, Z.-R., Cao, Y.: An alternating positive-semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models. IMA J. Numer. Anal. (2015). doi:10.1093/imanum/drv014

    MathSciNet  Google Scholar 

  34. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  35. Silvester, D.J., Kechkar, N.: Stablized bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem. Comput. Methods Appl. Mech. Eng. 79, 71–86 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. Simoncini, V., Benzi, M.: Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems. SIAM J. Matrix Anal. Appl. 26, 377–389 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ventura, G.: An augmented Lagrangian approach to essential boundary conditions in meshless methods. Int. J. Numer. Math. Eng. 53, 825–842 (2002)

    Article  Google Scholar 

  38. Zhang, G.-F., Ren, Z.-R., Zhou, Y.-Y.: On HSS-based constraint preconditioners for generalized saddle-point problems. Numer. Algorithms 57, 273–287 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Transportation, Nantong University, 226019, Nantong, People’s Republic of China

    Yang Cao & Quan Shi

  2. School of Statistics and Mathematics, Central University of Finance and Economics, 100081, Beijing, People’s Republic of China

    Zhi-Ru Ren

Authors
  1. Yang Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-Ru Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Quan Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-Ru Ren.

Additional information

Communicated by Zhong-Zhi Bai.

This work is supported by the National Natural Science Foundation of China (Nos. 11301290, 11301521, 61171132).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, Y., Ren, ZR. & Shi, Q. A simplified HSS preconditioner for generalized saddle point problems. Bit Numer Math 56, 423–439 (2016). https://doi.org/10.1007/s10543-015-0588-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-015-0588-3

Keywords

Mathematics Subject Classification

Search

Navigation