| Citation: |
Litao Zhang, Haochen Zhao, Guangxu Zhu, Xiaojing Zhang. THE MODIFIED DOUBLE SHIFT-SPLITTING PRECONDITIONER FOR NONSYMMETRIC GENERALIZED SADDLE POINT PROBLEMS FROM THE TIME-HARMONIC MAXWELL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1520-1535. doi: 10.11948/20240279
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THE MODIFIED DOUBLE SHIFT-SPLITTING PRECONDITIONER FOR NONSYMMETRIC GENERALIZED SADDLE POINT PROBLEMS FROM THE TIME-HARMONIC MAXWELL EQUATIONS
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Abstract
Recently, Fan, Zhu and Zheng [Computational and Applied Mathematics, 37(3), 3256-3266] proposed a generalized double shift-splitting (GDSS) preconditioner induced by a new matrix splitting method for nonsymmetric generalized saddle point problems, and gave the corresponding theoretical analysis and numerical experiments. In this paper, based on the generalized double shift-splitting (GDSS) preconditioner, we generalize the GDSS algorithms and further present the modified double shift-splitting (MDSS) preconditioner for nonsymmetric generalized saddle point problems having a nonsymmetric positive definite (1,1)-block and a positive definite (2,2)-block. Moreover, by similar theoretical analysis, we analyze the convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the MDSS preconditioned saddle point matrices. In final, one example is provided to confirm the effectiveness.
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Keywords:
- Modified double /
- shift-splitting /
- saddle point problem /
- convergence /
- preconditioner /
- eigenvalue
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References
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Litao Zhang, Haochen Zhao, Guangxu Zhu, Xiaojing Zhang. THE MODIFIED DOUBLE SHIFT-SPLITTING PRECONDITIONER FOR NONSYMMETRIC GENERALIZED SADDLE POINT PROBLEMS FROM THE TIME-HARMONIC MAXWELL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1520-1535. doi: 10.11948/20240279
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Figure 1.
A uniform mesh with
$ h=\frac{\sqrt{2}}{4} $ -
Figure 2.
The eigenvalue distribution for the MDSS iteration matrix
$ \Gamma=\mathcal {P}_{MDSS}^{-1}\mathcal {R}_{MDSS} $ $ \alpha=0.3, \beta=0.4, \gamma=1.4 $ $ \alpha=0.4, \beta=0.8, \gamma=1.1 $ $ \alpha=0.6, \beta=0.4, \gamma=1.6 $ $ \alpha=0.8, \beta=0.2, \gamma=1.3 $ $ h=\frac{\sqrt{2}}{8}. $ -
Figure 3.
The eigenvalue distribution for the GDSS iteration matrix
$ \Gamma=\mathcal {P}_{GDSS}^{-1}\mathcal {R}_{GDSS} $ $ \alpha=0.3, \beta=0.4 $ $ \alpha=0.4, \beta=0.8 $ $ \alpha=0.6, \beta=0.4 $ $ \alpha=0.8, \beta=0.2 $ $ h=\frac{\sqrt{2}}{8}. $ -
Figure 4.
The eigenvalue distribution for the MDSS iteration matrix
$ \Gamma=\mathcal {P}_{MDSS}^{-1}\mathcal {R}_{MDSS} $ $ \alpha=0.3, \beta=0.4, \gamma=1.4 $ $ \alpha=0.4, \beta=0.8, \gamma=1.1 $ $ \alpha=0.6, \beta=0.4, \gamma=1.6 $ $ \alpha=0.8, \beta=0.2, \gamma=1.3 $ $ h=\frac{\sqrt{2}}{12}. $ -
Figure 5.
The eigenvalue distribution for the GDSS iteration matrix
$ \Gamma=\mathcal {P}_{GDSS}^{-1}\mathcal {R}_{GDSS} $ $ \alpha=0.3, \beta=0.4 $ $ \alpha=0.4, \beta=0.8 $ $ \alpha=0.6, \beta=0.4 $ $ \alpha=0.8, \beta=0.2 $ $ h=\frac{\sqrt{2}}{12}. $ -
Figure 6.
The eigenvalue distribution for the MDSS iteration matrix
$ \Gamma=\mathcal {P}_{MDSS}^{-1}\mathcal {R}_{MDSS} $ $ \alpha=0.3, \beta=0.4, \gamma=1.4 $ $ \alpha=0.4, \beta=0.8, \gamma=1.1 $ $ \alpha=0.6, \beta=0.4, \gamma=1.6 $ $ \alpha=0.8, \beta=0.2, \gamma=1.3 $ $ h=\frac{\sqrt{2}}{18}. $ -
Figure 7.
The eigenvalue distribution for the GDSS iteration matrix
$ \Gamma=\mathcal {P}_{GDSS}^{-1}\mathcal {R}_{GDSS} $ $ \alpha=0.3, \beta=0.4 $ $ \alpha=0.4, \beta=0.8 $ $ \alpha=0.6, \beta=0.4 $ $ \alpha=0.8, \beta=0.2 $ $ h=\frac{\sqrt{2}}{18}. $