The improvements of the generalized shift-splitting preconditioners for non-singular and singular saddle point problems

ABSTRACT

To solve the saddle point problems with symmetric positive definite (1,1) parts, the improved generalized shift-splitting (IGSS) preconditioner is established in this paper, which yields the IGSS iteration method. Theoretical analysis shows that the IGSS iteration method is convergent and semi-convergent unconditionally. The choices of the iteration parameters are discussed. Moreover, some spectral properties, including the eigenvalue and eigenvector distributions of the preconditioned matrix are also investigated. Finally, numerical results are presented to verify the robustness and the efficiency of the proposed iteration method and the corresponding preconditioner for solving the non-singular and singular saddle point problems.

Keywords:

• Saddle point problem
• improved generalized shift-splitting
• convergence
• semi-convergence
• spectral properties

2010 AMS Subject Classifications:

• 65F08
• 65F10

Acknowledgments

We would like to express our sincere thanks to the anonymous reviewers for their valuable suggestions and construct comments which greatly improved the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant number 11171273] and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University [grant number CX201628].