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Preconditioning Technique Based on Sine Transformation for Nonlocal Helmholtz Equations with Fractional Laplacian

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Abstract

We propose two preconditioners based on the fast sine transformation for solving linear systems with ill-conditioned multilevel Toeplitz structure. These matrices are generated by discretizing the two-dimensional nonlocal Helmholtz equations with fractional Laplacian operators via the finite difference method. For complex wave numbers with nonnegative real parts, we give the spectral analysis of the preconditioned matrices. Additionally, we extend the proposed preconditioners and algorithm to address the general cases of the nonlocal Helmholtz equations that feature negative, complex with negative real parts, and variable wave numbers. Numerical experiments also indicate that the proposed preconditioners outperform the existing preconditioners.

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Funding

This work is supported in part by research grants of the Science and Technology Development Fund, Macau SAR (File no. 0122/2020/A3), and University of Macau (File no. MYRG2020-00224-FST).

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Authors and Affiliations

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

    Tian-Yi Li, Hai-Wei Sun & Tao Sun

  2. School of Applied Science, Beijing Information Science and Technology University, Beijing, China

    Fang Chen

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  1. Tian-Yi Li
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  4. Tao Sun
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Correspondence to Hai-Wei Sun.

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Li, TY., Chen, F., Sun, HW. et al. Preconditioning Technique Based on Sine Transformation for Nonlocal Helmholtz Equations with Fractional Laplacian. J Sci Comput 97, 17 (2023). https://doi.org/10.1007/s10915-023-02332-0

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