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Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian

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Abstract

A so-called grid-overlay finite difference method (GoFD) was proposed recently for the numerical solution of homogeneous Dirichlet boundary value problems (BVPs) of the fractional Laplacian on arbitrary bounded domains. It was shown to have advantages of both finite difference (FD) and finite element methods, including their efficient implementation through the fast Fourier transform (FFT) and the ability to work for complex domains and with mesh adaptation. The purpose of this work is to study GoFD in a meshfree setting, a key to which is to construct the data transfer matrix from a given point cloud to a uniform grid. Two approaches are proposed, one based on the moving least squares fitting and the other based on the Delaunay triangulation and piecewise linear interpolation. Numerical results obtained for examples with convex and concave domains and various types of point clouds are presented. They show that both approaches lead to comparable results. Moreover, the resulting meshfree GoFD converges in a similar order as GoFD with unstructured meshes and finite element approximation as the number of points in the cloud increases. Furthermore, numerical results show that the method is robust to random perturbations in the location of the points.

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Funding

W.H. was supported in part by the University of Kansas General Research Fund FY23 and the Simons Foundation through Grant MP-TSM-00002397. J.S. was supported in part by the National Natural Science Foundation of China through Grant 12101509.

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Author notes

  1. Bowen Shi and Weizhang Huang have contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, China

    Jinye Shen & Bowen Shi

  2. Department of Mathematics, University of Kansas, Jayhawk Boulevard, Lawrence, KS, 66045, USA

    Weizhang Huang

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  1. Jinye Shen
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  2. Bowen Shi
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Correspondence to Jinye Shen.

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Shen, J., Shi, B. & Huang, W. Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-024-00368-z

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  • DOI: https://doi.org/10.1007/s42967-024-00368-z

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