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The modified PAHSS–PU and modified PPHSS-SOR iterative methods for saddle point problems

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Abstract

Recently, by combining the preconditioned accelerated Hermitian and skew-Hermitian splitting (PAHSS) and the parameterized Uzawa (PU) methods, Zheng and Ma (Appl Math Comput 273, 217–225, 2016b) presented the PAHSS–PU method for saddle point problems. By adding a block lower triangular matrix to the coefficient matrix on two sides of the first equation of the PAHSS–PU iterative scheme, the modified PAHSS–PU (MPAHSS–PU) iteration method is proposed in this paper, which has a faster convergence rate than the PAHSS–PU one. Furthermore, changing the position of the parameters in the MPAHSS–PU method, we develop another new method referred to as the modified PPHSS-SOR (MPPHSS-SOR) iteration method for solving saddle point problems. We provide the convergence properties of the MPAHSS–PU and the MPPHSS-SOR iteration methods, which show that the new methods are convergent if the related parameters satisfy suitable conditions. Meanwhile, practical ways to choose iteration parameters for the proposed methods are developed. Finally, numerical experiments demonstrate that the MPAHSS–PU and the MPPHSS-SOR methods outperform some existing ones both on the number of iterations and the computational times.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, People’s Republic of China

    Zheng-Ge Huang, Li-Gong Wang, Zhong Xu & Jing-Jing Cui

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  1. Zheng-Ge Huang
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  2. Li-Gong Wang
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  3. Zhong Xu
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  4. Jing-Jing Cui
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Corresponding author

Correspondence to Li-Gong Wang.

Additional information

Communicated by Andreas Fischer.

This research was supported by the National Natural Science Foundation of China, the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JM1032) and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (no. CX201628).

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Huang, ZG., Wang, LG., Xu, Z. et al. The modified PAHSS–PU and modified PPHSS-SOR iterative methods for saddle point problems. Comp. Appl. Math. 37, 6076–6107 (2018). https://doi.org/10.1007/s40314-018-0680-9

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  • DOI: https://doi.org/10.1007/s40314-018-0680-9

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