Year: 2022
Author: Daxin Nie, Jing Sun, Weihua Deng, Daxin Nie
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 3 : pp. 744–767
Abstract
We make the split of the integral fractional Laplacian as $$(−∆)^su = (−∆)(−∆)^{s−1}u,$$ where $s ∈ (0,\frac{1}{2}) ∪ (\frac{1}{2}, 1).$ Based on this splitting, we respectively discretize the one- and two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation estimate. Moreover, the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an $\mathcal{O}(h ^{1+α−2s})$ convergence rate is obtained when the solution $u ∈ C ^{1,α}(\overline{Ω}^ δ_n),$ where $n$ is the dimension of the space, $α ∈ ({\rm max}(0, 2s − 1), 1], δ$ is a fixed positive constant, and $h$ denotes mesh size. Finally, the performed numerical experiments confirm the theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2021-0173
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 3 : pp. 744–767
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: One- and two-dimensional integral fractional Laplacian Lagrange interpolation operator splitting finite difference the inhomogeneous fractional Dirichlet problem error estimates.
Author Details
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Anomalous and nonergodic multiscale modeling, analyses and algorithms
Weihua, Deng
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https://doi.org/10.1360/SSM-2023-0046 [Citations: 1]