Efficient Spectral Methods for Eigenvalue Problems of the Integral Fractional Laplacian on a Ball of Any Dimension
Year: 2024
Author: Suna Ma, Huiyuan Li, Zhimin Zhang, Hu Chen, Lizhen Chen
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 4 : pp. 1032–1062
Abstract
An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper. The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis. And a sharp estimate on the algebraic system’s condition number is established which behaves as $N^{4s}$ with respect to the polynomial degree $N,$ where $2s$ is the fractional derivative order. The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces. Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived. Meanwhile, rigorous error estimates of the eigenvalues and eigenvectors are obtained. Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2304-m2022-0243
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 4 : pp. 1032–1062
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 31
Keywords: Integral fractional Laplacian Spectral method Eigenvalue problem Regularity analysis Error estimate.