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A Fast Iterative Solver for Multidimensional Spatial Fractional Cahn-Hilliard Equations

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Abstract

This paper is concerned with the fast algorithm for solving multidimensional spatial fractional Cahn-Hilliard equations. The equations are discretized by a linear and energy-stable finite difference scheme. It gives a system of linear equations with a \(2\times 2\) indefinite ill-conditioned block matrix. We construct a positive definite block preconditioner based on the sine transform for the minimal residual method to solve the indefinite system. Theoretically, we prove that all the eigenvalues of the preconditioned matrix are located in the intervals \([-\frac{3}{2},-\frac{1}{2\sqrt{2}}]\cup [\frac{1}{2\sqrt{2}},\frac{3}{2}]\) without outliers. Thus, the preconditioned MINRES method has a linear convergence rate within an iteration number independent of the matrix size. A fast implementation is presented for the preconditioned matrix–vector multiplication, which reduces the computation complexity significantly. The matrix-size independent convergence rate and the fast implementation guarantee a linearithmic (nearly optimal) complexity of the proposed solver. Numerical examples are given to demonstrate the effectiveness of the proposed method.

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Data Availability

The data of natural images and codes involved in this paper are available from the corresponding author on reasonable request.

Notes

  1. a matrix diagonalizable by (multi-dimension) discrete sine transform.

  2. see Lemma 7

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Funding

This work is supported by the National Natural Science Foundation of China (Grant nos. 11771162, 12231003, 12301480), research grants of the Science and Technology Development Fund, Macau SAR (file no. 0122/2020/A3), University of Macau (file no. MYRG-GRG2023–00085-FST-UMDF).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541006, China

    Xin Huang

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

    Dongfang Li

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

    Dongfang Li

  4. School of Science, Harbin Institute of Technology, Shenzhen, 518055, China

    Xuelei Lin

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

    Hai-Wei Sun

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  1. Xin Huang
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Correspondence to Xuelei Lin.

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This work is supported by the National Natural Science Foundation of China (Grant nos. 11771162, 12231003, 12301480), China Postdoctoral Science Foundation (Grant 2023M731205), research grants of the Science and Technology Development Fund, Macau SAR (file no. 0122/2020/A3), University of Macau (file no. MYRG-GRG2023–00085-FST-UMDF).

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Huang, X., Li, D., Lin, X. et al. A Fast Iterative Solver for Multidimensional Spatial Fractional Cahn-Hilliard Equations. J Sci Comput 102, 58 (2025). https://doi.org/10.1007/s10915-025-02795-3

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