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Journal of Computational Mathematics ›› 2019, Vol. 37 ›› Issue (1): 18-32.DOI: 10.4208/jcm.1707-m2016-0730

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ON THE GENERALIZED DETERIORATED POSITIVE SEMI-DEFINITE AND SKEW-HERMITIAN SPLITTING PRECONDITIONER

Davod Hezari1, Vahid Edalatpour1, Hadi Feyzollahzadeh2, Davod Khojasteh Salkuyeh1   

  1. 1. Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran;
    2. Department of Mathematical and Computer Science, Technical Faculty, University of Bonab, Bonab, Iran
  • Received:2016-08-28 Revised:2017-05-31 Online:2019-01-15 Published:2019-01-15
  • Supported by:

    The authors are grateful to the anonymous referees and the editor of the journal for their valuable comments and suggestions which improved the quality of this paper. The work of Davod Khojasteh Salkuyeh is partially supported by University of Guilan.

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Davod Hezari, Vahid Edalatpour, Hadi Feyzollahzadeh, Davod Khojasteh Salkuyeh. ON THE GENERALIZED DETERIORATED POSITIVE SEMI-DEFINITE AND SKEW-HERMITIAN SPLITTING PRECONDITIONER[J]. Journal of Computational Mathematics, 2019, 37(1): 18-32.

For nonsymmetric saddle point problems, Huang et al. in[Numer. Algor. 75 (2017), pp. 1161-1191] established a generalized variant of the deteriorated positive semi-definite and skew-Hermitian splitting (GVDPSS) preconditioner to expedite the convergence speed of the Krylov subspace iteration methods like the GMRES method. In this paper, some new convergence properties as well as some new numerical results are presented to validate the theoretical results.
Key words: Saddle point problem, Preconditioner, Nonsymmetric, Symmetric, Positive definite, Krylov subspace method

CLC Number: 

[1] Z.Z. Bai, Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems, Appl. Math. Comput., 109(2000), 273-285.

[2] Z.Z. Bai, G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting methods for saddlepoint problems, IMA J. Numer. Anal., 27(2007), 1-23.

[3] Z.Z. Bai, G.H. Golub, L.Z. Lu, J.F. Yin, Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 26(2004), 844-863.

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

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

[6] Z.Z. Bai, J.F. Yin, Y.F. Su, A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math., 24(2006), 539-552.

[7] M. Benzi, G.H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26(2004), 20-41.

[8] M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14(2005), 1-137.

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

[10] Y. Cao, J. Du, Q. Niu, Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math., 272(2014), 239-250.

[11] Y. Cao, L.Q. Yao, M.Q. Jiang, Q. Niu, A relaxed HSS preconditioner for saddle point problems from mesh free discretization, J. Comput Math. 31(2013), 398-421.

[12] H.C. Elman, A. Ramage, D.J. Silvester, IFISS:A Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Software., 33(2007), Article 14.

[13] Z.G Huang, L.G Wang, Z. Xu, J.J Cui, A generalized variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems, Numr. Algor. 75(2017) 1161-1191.

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

[15] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM:Philadelphia, 2003.

[16] D.K. Salkuyeh, M. Masoudi, A new relaxed HSS preconditioner for saddle point problems, Numer. Algor., 74(2017) 781-795.

[17] E.D. Sturler, J. Liesen, Block-diagonal and constraint preconditioners for nonsymmetric indenite linear systems, SIAM, J. Sci. Comput., 26(2005), 1598-619.

[18] J. Zhang, C. Gu, A variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems, BIT Numer. Math., 2016, doi:10.1007/s10543-015-0590-9.
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