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Numerical Algorithms

, Volume 72, Issue 3, pp 813–834 | Cite as

A relaxed positive semi-definite and skew-Hermitian splitting preconditioner for non-Hermitian generalized saddle point problems

  • Hong-tao Fan
  • Bing Zheng
  • Xin-yun Zhu
Original Paper

Abstract

For non-Hermitian saddle point linear systems, Pan, Ng and Bai presented a positive semi-definite and skew-Hermitian splitting (PSS) preconditioner (Pan et al. Appl. Math. Comput. 172, 762–771 2006), to accelerate the convergence rate of the Krylov subspace iteration methods like the GMRES method. In this paper, a relaxed positive semi-definite and skew-Hermitian (RPSS) splitting preconditioner based on the PSS preconditioner for the non-Hermitian generalized saddle point problems is considered. The distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed. Moreover, an upper bound on the degree of the minimal polynomial is also studied. Finally, numerical experiments of a model Navier-Stokes equation are presented to illustrate the efficiency of the RPSS preconditioner compared to the PSS preconditioner, the block diagonal preconditioner (BD), and the block triangular preconditioner (BT) in terms of the number of iteration and computational time.

Keywords

Non-Hermitian generalized saddle point problems Relaxed positive semi-definite and skew-Hermitian splitting Eigenvalue and eigenvector distribution Preconditioning Krylov subspace method 

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References

  1. 1.
    Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comp. 75, 791–815 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 23, 844–863 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bai, Z.-Z., Ng, M.K.: On inexact preconditioners for nonsymmetric matrices. SIAM J. Sci. Comput. 26, 1710–1724 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benzi, M., Gander, M., Golub, G.H.: Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT Numer. Math. 43, 881–900 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benzi, M., Golub, G.H.: A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl. 26, 20–41 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Benzi, M., Guo, X.-P.: A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations. Appl. Numer. Math. 61, 66–76 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. Academic Press, New York (1979)zbMATHGoogle Scholar
  12. 12.
    Betts, J.T.: Practical methods for optimal control using nonlinear programming. SIAM, Philadelphia (2001)zbMATHGoogle Scholar
  13. 13.
    Cao, Z.-H.: A note on constraint preconditioning for nonsymmetric indefinite matrices. SIAM J. Matrix Anal. Appl. 24, 121–125 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645–1661 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fan, H.-T., Zhu, X.-Y.: A generalized relaxed positive-definite and skew-Hermitian splitting preconditioner for non-Hermitian saddle point problems. Appl. Math. Comput. 258, 36–48 (2015)MathSciNetGoogle Scholar
  17. 17.
    Pan, J.-Y., Ng, M.K., Bai, Z.-Z.: New preconditioners for saddle point problems. Appl. Math. Comput. 172, 762–771 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Perugia, I., Simoncini, V.: Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer. Linear Algebra Appl. 7, 585–616 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Saad, Y.: Iterative methods for sparse linear systems. Publishing Company, Boston (1996)zbMATHGoogle Scholar
  20. 20.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shen, S.-Q.: A note on PSS preconditioners for generalized saddle point problems. Appl. Math. Comput. 237, 723–729 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Silvester, D.J., Elman, H.C., Ramage, A.: IFISS: Incompressible Flow Iterative Solution Software. http://www.manchester.ac.uk/ifiss
  23. 23.
    Simoncini, V.: Block triangular preconditioners for symmetric saddle-point problems. Appl. Numer. Math. 49, 63–80 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sturler, E.D., Liesen, J.: Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. SIAM J. Sci. Comput. 26, 1598–1619 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityGansuChina
  2. 2.Department of MathematicsUniversity of Texas of the Permian BasinOdessaUSA

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