Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

ON A SPLITTING PRECONDITIONER FOR SADDLE POINT PROBLEMS

  • Received : 2018.01.19
  • Accepted : 2018.05.02
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Cao et al. in (Numer. Linear. Algebra Appl. 18 (2011) 875-895) proposed a splitting method for saddle point problems which unconditionally converges to the solution of the system. It was shown that a Krylov subspace method like GMRES in conjunction with the induced preconditioner is very effective for the saddle point problems. In this paper we first modify the iterative method, discuss its convergence properties and apply the induced preconditioner to the problem. Numerical experiments of the corresponding preconditioner are compared to the primitive one to show the superiority of our method.

Keywords

Acknowledgement

Supported by : University of Guilan

References

  1. Z.Z. Bai, Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl. 16 (2009), 447-479. https://doi.org/10.1002/nla.626
  2. Z.Z. Bai and G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting methods for saddle-point problems, IMA J. Numer. Anal. 27 (2007), 1-23. https://doi.org/10.1093/imanum/drl017
  3. Z.Z. Bai and Z.Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008), 2900-2932. https://doi.org/10.1016/j.laa.2008.01.018
  4. Z.Z. Bai, G.H. Golub and M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003), 603-626. https://doi.org/10.1137/S0895479801395458
  5. Z.Z. Bai, J.F. Yin and Y.F. Su, A shift-splitting preconditioner for non-Hermitian positive definite matrics, J. Comput. Math. 24 (2006), 539-552.
  6. M. Benzi and X.P. Guo, A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations, Appl. Numer. Math. 61 (2011), 66-76. https://doi.org/10.1016/j.apnum.2010.08.005
  7. M. Benzi, M.K. Ng, Q. Niu and Z. Wang, A relaxed dimensional fractorization preconditioner for the incompressible Navier-Stokes equations, J. Comput. Phys. 230 (2011), 6185-6202. https://doi.org/10.1016/j.jcp.2011.04.001
  8. M. Benzi, G.H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numerica 14 (2005), 1-137. https://doi.org/10.1017/S0962492904000212
  9. M. Benzi and G.H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl. 26 (2004), 20-41. https://doi.org/10.1137/S0895479802417106
  10. H. Bramble, J.E. Pasciak and A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997), 1072-1092. https://doi.org/10.1137/S0036142994273343
  11. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.
  12. Y. Cao, J. Du and Q. Niu, Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math. 272 (2014), 239-250. https://doi.org/10.1016/j.cam.2014.05.017
  13. Y. Cao, M.Q. Jiang and Y.L. Zheng, A splitting preconditioner for saddle point problems, Numer. Linear Algebra Appl. 18 (2011), 875-895. https://doi.org/10.1002/nla.772
  14. F. Chen and Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008), 765-771.
  15. C. Chen and C. Ma, A generalized shift-splitting preconditioner for saddle point problems, Appl. Math. Lett. 43 (2015), 49-55. https://doi.org/10.1016/j.aml.2014.12.001
  16. H.C. Elman, D.J. Silvester and A.J. Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, Oxford, 2003.
  17. H.C. Elman, A. Ramage and D.J. Silvester, A Matlab toolbox for modelling incompressible How, ACM Trans. Math. Software. 33 (2007), Article 14.
  18. G.H. Golub and C. Greif, On solving block-structured indefinite linear systems, SIAM J. Sci. Comput. 24 (2003), 2076-2092. https://doi.org/10.1137/S1064827500375096
  19. D. Hezari, V. Edalatpour, H. Feyzollahzadeh and D.K. Salkuyeh, On the generalized deteriorated positive semi-definite and skew-hermitian spliting preconditioner, Journal of Computational Mathematics, to appear, 2018.
  20. Q. Hu and J. Zou,An iterative method with variable relaxation parameters for saddle-point problems, SIAM J. Matrix Anal. Appl. 23 (2001), 317-338. https://doi.org/10.1137/S0895479899364064
  21. Y. Saad and M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (1986), 856-869. https://doi.org/10.1137/0907058
  22. Y. Saad, A exible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput. 14 (1993), 461-469. https://doi.org/10.1137/0914028
  23. Y. Saad, Iterative methods for sparse linear systems, Second edition PWS, New York, 1995.
  24. D.K. Salkuyeh and M. Masoudi, A new relaxed HSS preconditioner for saddle point problems, Numer. Algor. 74 (2017), 781795.
  25. D.K. Salkuyeh, M. Masoudi and D. Hezari, On the generalized shift-splitting preconditioner for saddle point problems, Appl. Math. Lett. 48 (2015), 55-61. https://doi.org/10.1016/j.aml.2015.02.026