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Error analysis of a collocation method on graded meshes for a fractional Laplacian problem

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Abstract

The numerical solution of a 1D fractional Laplacian boundary value problem is studied. Although the fractional Laplacian is one of the most important and prominent nonlocal operators, its numerical analysis is challenging, partly because the problem’s solution has in general a weak singularity at the boundary of the domain. To solve the problem numerically, we use piecewise linear collocation on a mesh that is graded to handle the boundary singularity. A rigorous analysis yields a bound on the maximum nodal error which shows how the order of convergence of the method depends on the grading of the mesh; hence, one can determine the optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.

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Funding

The work of the first author was supported by the Science Fund for Distinguished Young Scholars of Gansu Province under Grant No. 23JRRA1020 and the Fundamental Research Funds for the Central Universities under grant lzujbky-2023-06. This work was supported by the National Natural Science Foundation of China under Grant Nos. 12225107 and 12071195, and the Innovative Groups of Basic Research in Gansu Province under Grant No. 22JR5RA391. The work of Martin Stynes is supported in part by the National Natural Science Foundation of China under grants 12171025 and NSAF-U2230402.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Minghua Chen, Weihua Deng, Chao Min & Jiankang Shi

  2. Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, 100094, China

    Martin Stynes

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  1. Minghua Chen
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  2. Weihua Deng
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  3. Chao Min
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  4. Jiankang Shi
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  5. Martin Stynes
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Correspondence to Martin Stynes.

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Communicated by: Bangti Jin

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Chen, M., Deng, W., Min, C. et al. Error analysis of a collocation method on graded meshes for a fractional Laplacian problem. Adv Comput Math 50, 49 (2024). https://doi.org/10.1007/s10444-024-10146-3

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