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Semi-convergence analysis of preconditioned deteriorated PSS iteration method for singular saddle point problems

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Abstract

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.

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Acknowledgements

We would like to express our sincere thanks to the unknown reviewers for their careful reading of the manuscript. Their useful comments and valuable suggestions greatly improve the quality of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Zhao-Zheng Liang & Guo-Feng Zhang

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  1. Zhao-Zheng Liang
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  2. Guo-Feng Zhang
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Correspondence to Zhao-Zheng Liang or Guo-Feng Zhang.

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This work was supported by the National Natural Science Foundation of China (No. 11271174).

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Liang, ZZ., Zhang, GF. Semi-convergence analysis of preconditioned deteriorated PSS iteration method for singular saddle point problems. Numer Algor 78, 379–404 (2018). https://doi.org/10.1007/s11075-017-0380-3

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  • DOI: https://doi.org/10.1007/s11075-017-0380-3

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