Modified PHSS iterative methods for solving nonsingular and singular saddle point problems
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For large sparse saddle point problems, we establish a new version of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method, called the modified PHSS (MPHSS) method in this paper. Then, we theoretically study its convergence and semi-convergence properties and determine its optimal iteration parameter and corresponding optimal convergence factor. Furthermore, the spectral properties of the MPHSS preconditioned matrix are discussed in detail. Numerical experiments show that the MPHSS iteration method is effective and robust when it is used either as a solver or as a matrix splitting preconditioner for the generalized minimal residual (GMRES) method.
KeywordsSaddle point problem MPHSS method Convergence Semi-convergence Preconditioning
Mathematics Subject Classification (2010)65F10 65F50
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We would like to express our sincere thanks to the anonymous reviewer for his/her valuable suggestions and constructive comments which greatly improved the presentation of this paper.
This research was supported by the National Natural Science Foundation of China (No. 11171273) and sponsored by Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No. CX201628).
- 26.Huang, Z. -G., Wang, L. -G., Xu, Z., Cui, J.-J.: A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems. Numer Algorithms. https://doi.org/10.1007/s11075-017-0377-y (2017)