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Preconditioners with Symmetrized Techniques for Space Fractional Cahn-Hilliard Equations

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Abstract

In this paper, we study space fractional Cahn-Hilliard equations. A second-order stabilized finite difference scheme is exploited for the model equations. The resulting coefficient matrix is a nonsymmetric ill-conditioned Toeplitz-like matrix. Symmetrized strategies are proposed for the nonsymmetric system so that the conjugate gradient method can be utilized to derive the numerical solutions. Moreover, preconditioners based on the sine transform are designed to speed up the convergence rate of the proposed methods. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the interval (1/2, 3/2), which guarantees that the preconditioned conjugate gradient method converges linearly, within an iteration number independent of the matrix size. Numerical experiments are reported to show the effectiveness of the proposed methods.

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Acknowledgements

The authors are grateful to the anonymous referees for their constructive comments which benefit this paper a lot. This work is supported in part by research grants of the Science and Technology Development Fund, Macau SAR (file no. 0122/2020/A3), University of Macau (file no. MYRG2020-00224-FST), and the National Natural Science Foundation of China (Grand No. 11771162).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074, Wuhan, China

    Xin Huang & Dongfang Li

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, 430074, Wuhan, China

    Dongfang Li

  3. Department of Mathematics, University of Macau, Macao, China

    Hai-Wei Sun & Fan Zhang

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  1. Xin Huang
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  2. Dongfang Li
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Correspondence to Hai-Wei Sun.

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Huang, X., Li, D., Sun, HW. et al. Preconditioners with Symmetrized Techniques for Space Fractional Cahn-Hilliard Equations. J Sci Comput 92, 41 (2022). https://doi.org/10.1007/s10915-022-01900-0

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