Class of Uzawa-NPHSS iteration method for solving nonsingular and singular saddle point problems

ABSTRACT

Based on the non-alternating preconditioned HSS iteration scheme, a class of the Uzawa-NPHSS iteration method for solving nonsingular and singular non-Hermitian saddle point problems with the (1,1) part of the coefficient matrix being non-Hermitian positive definite is established. The convergence properties for the nonsingular saddle point problems and the semi-convergence properties for the singular ones of the proposed method are carefully discussed under suitable conditions. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analysed in detail. Additionally, the parameter selection strategy for the Uzawa-NPHSS iteration method is provided. Numerical experiments are implemented to confirm the theoretical results, which show the feasibility and effectiveness of the proposed method.

Keywords:

• Saddle point problem
• Uzawa method
• non-alternating preconditioned HSS iteration
• convergence
• semi-convergence

2010 Mathematics Subject Classifications:

• 65F10
• 65F15
• 65F50

Acknowledgments

The authors sincerely thank the anonymous reviewers for their valuable comments that have led to the present improved version of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant number 11671318] and Shangluo Universities Key Disciplines Project, Discipline name: Mathematics.