Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2025-01-04T01:52:06.330Z Has data issue: false hasContentIssue false
  • English
  • Français

A Modified Relaxed Positive-Semidefinite and Skew-Hermitian Splitting Preconditioner for Generalized Saddle Point Problems

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Yang Cao ,
An Wang  and
Yu-Juan Chen
Yang Cao*
Affiliation:
School of Transportation, Nantong University, Nantong 226019, P.R. China Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing 210023, P.R. China
An Wang
Affiliation:
School of Sciences, Nantong University, Nantong 226019, P.R. China
Yu-Juan Chen
Affiliation:
School of Sciences, Nantong University, Nantong 226019, P.R. China
*
*Corresponding author. Email address:[email protected] (Y. Cao)
Get access

Abstract

Based on the relaxed factorization techniques studied recently and the idea of the simple-like preconditioner, a modified relaxed positive-semidefinite and skew-Hermitian splitting (MRPSS) preconditioner is proposed for generalized saddle point problems. Some properties, including the eigenvalue distribution, the eigenvector distribution and the minimal polynomial of the preconditioned matrix are studied. Numerical examples arising from the mixed finite element discretization of the Oseen equation are illustrated to show the efficiency of the new preconditioner.

Keywords

Generalized saddle point problemspositive-semidefinite and skew-Hermitian splittingpreconditioningKrylov subspace method

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F50: Sparse matrices
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 192 - 210
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bai, Z.-Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75, 791–815 (2006).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).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).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).Google Scholar
[5] Bai, Z.-Z. and Golub, G.H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27, 1–23 (2007).Google Scholar
[6] Bai, Z.-Z., Golub, G.H. and 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).Google Scholar
[7] Bai, Z.-Z., Golub, G.H., Lu, L.-Z. and Yin, J.-F., Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 26, 844–863 (2005).Google Scholar
[8] Bai, Z.-Z., Golub, G.H. and 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).Google Scholar
[9] Bai, Z.-Z. and Hadjidimos, A., Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix, Linear Algebra Appl., 463, 322–339 (2014).Google Scholar
[10] Bai, Z.-Z., Yin, J.-F. and Su, Y.-F., A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math., 24, 539–552 (2006).Google Scholar
[11] Benzi, M., Deparis, S., Grandperrin, G. and Quarteroni, A., Parameter estimates for the relaxed dimensional factorization preconditioner and application to hemodynamics, Comput. Methods Appl. Mech. Engrg., 300, 129–145 (2016).Google Scholar
[12] Benzi, M. and Golub, G.H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl. 26, 20–41 (2004).Google Scholar
[13] Benzi, M., Golub, G.H. and Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1–137 (2005).Google Scholar
[14] Benzi, M., Ng, M.K., Niu, Q. and Wang, Z., A relaxed dimensional fractorization preconditioner for the incompressible Navier-Stokes equations, J. Comput. Phys., 230, 6185–6202 (2011).Google Scholar
[15] Bergamaschi, L. and Martínez, Á., RMCP: Relaxed Mixed Constraint Preconditioners for saddle point linear systems arising in geomechanics, Comput. Methods Appl. Mech. Engrg., 221-222, 54–62 (2012).Google Scholar
[16] Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York (1991).Google Scholar
[17] Bui, T.-Q., Nguyen, M.N., Zhang, C.-Z. and Pham, D.A.K., An efficient meshfree method for analysis of two-dimensional piezoelectric structures, Smart. Mater. Struct., 20, 065016, 11 pages (2011).Google Scholar
[18] Cao, Y., Dong, J.-L. and 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).Google Scholar
[19] Cao, Y., Miao, S.-X. and Cui, Y.-S., A relaxed splitting preconditioner for generalized saddle point problems, Comput. Appl. Math., 34, 865–879 (2015).Google Scholar
[20] Cao, Y., Ren, Z.-R. and Shi, Q., A simplified HSS preconditioner for generalized saddle point problems, BIT Numer. Math., 56, 423–439 (2016).Google Scholar
[21] Cao, Y., Yao, L.-Q., Jiang, M.-Q. and Niu, Q., A relaxed HSS preconditioner for saddle point problems from meshfree discretization, J. Comput. Math., 31, 398–421 (2013).Google Scholar
[22] Elman, H.C., Silvester, D.J. and Wathen, A.J., Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics (2nd edn), Oxford University Press, Oxford (2014).Google Scholar
[23] Elman, H.C., Ramage, A. and Silvester, D.J., IFISS: a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softare, 33, Article 14 (2007).Google Scholar
[24] Fan, H.-T., Zheng, B. and Zhu, X.-Y., A relaxed positive semi-definite and skew-Hermitian splitting preconditioner for non-Hermitian generalized saddle point problems, Numer. Algor., 72, 813–834 (2016).Google Scholar
[25] Greif, C., Moulding, E. and Orban, D., Bounds on eigenvalues of matrices arising from interior-point methods, SIAM, J. Optim., 24, 49–83 (2014).Google Scholar
[26] Huang, Y.-M., A practical formula for computing optimal parameters in the HSS iteration methods, J. Comput. Appl. Math., 255, 142–149 (2014).Google Scholar
[27] Li, C. and Vuik, C., Eigenvalue analysis of the SIMPLE preconditioning for incompressible flow, Numer. Linear Algebra Appl., 11, 511–523 (2004).Google Scholar
[28] Liang, Z.-Z. and Zhang, G.-F., SIMPLE-like preconditioners for saddle point problems from the steady Navier-Stokes equations, J. Comput. Appl. Math., 302, 211–223 (2016).Google Scholar
[29] Pestana, J. and Wathen, A.J., Natural preconditioning and iterative methods for saddle point problems, SIAM Review, 57, 71–91 (2015).Google Scholar
[30] Pan, J.-Y., Ng, M.K. and Bai, Z.-Z., New preconditioners for saddle point problems, Appl. Math. Comput., 172, 762–771 (2006).Google Scholar
[31] Ren, Z.-R. and Cao, Y., An alternating positive-semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models, IMA J. Numer. Anal., 36, 922–946 (2016).Google Scholar
[32] Saad, Y., Iterative Methods for Sparse Linear Systems (2nd edn), SIAM: Philadelphia (2003).Google Scholar
[33] Shen, S.-Q., A note on PSS preconditioners for generalized saddle point problems, Appl. Math. Comput., 237, 723–729 (2014).Google Scholar