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A Grid-Overlay Finite Difference Method for Inhomogeneous Dirichlet Problems of the Fractional Laplacian on Arbitrary Bounded Domains

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Abstract

A grid-overlay finite difference method (GoFD) was proposed recently for the numerical solution of homogeneous Dirichlet boundary value problems of the fractional Laplacian on arbitrary bounded domains and shown to have advantages of both finite difference and finite element methods, including its efficient implementation through the fast Fourier transform and ability to work for complex domains and with mesh adaptation. The purpose of this work is to study GoFD for problems with inhomogeneous Dirichlet boundary conditions. The main challenge with inhomogeneous Dirichlet boundary conditions comes from the calculation of the fractional Laplacian of the exterior function that is defined in the whole space. In this work the whole space is truncated into a rectangular or cubic bounded domain and a finite difference approximation on a uniform grid is used to approximate the fractional Laplacian of the exterior function. This approximation can be carried out efficiently through the fast Fourier transform. Moreover, the error for the domain truncation is analyzed for exterior functions being bounded or with algebraic or exponential decay. The estimates are then used to determine the size of the truncated domain and analyze the complexity of computing the fractional Laplacian of the exterior function. Numerical results for a selection of one- and two-dimensional examples are presented to demonstrate the convergence of GoFD and effectiveness of mesh adaptation.

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Acknowledgements

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]. The authors are grateful to the anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Weizhang Huang

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

    Jinye Shen

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Huang, W., Shen, J. A Grid-Overlay Finite Difference Method for Inhomogeneous Dirichlet Problems of the Fractional Laplacian on Arbitrary Bounded Domains. J Sci Comput 102, 50 (2025). https://doi.org/10.1007/s10915-024-02782-0

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