Year: 2024
Author: Changtao Sheng, Li-Lian Wang, Hongbin Chen, Huiyuan Li
Communications in Computational Physics, Vol. 36 (2024), Iss. 3 : pp. 673–710
Abstract
We show that the entries of the stiffness matrix, associated with the $C^0$-piecewise linear finite element discretization of the hyper-singular integral fractional Laplacian (IFL) on rectangular meshes, can be simply expressed as one-dimensional integrals on a finite interval. Particularly, the FEM stiffness matrix on uniform meshes has a block-Toeplitz structure, so the matrix-vector multiplication can be implemented by FFT efficiently. The analytic integral representations not only allow for accurate evaluation of the entries, but also facilitate the study of some intrinsic properties of the stiffness matrix. For instance, we can obtain the asymptotic decay rate of the entries, so the “dense” stiffness matrix turns out to be “sparse” with an $\mathcal{O}(h^3)$ cutoff. We provide ample numerical examples of PDEs involving the IFL on rectangular or $L$-shaped domains to demonstrate the optimal convergence and efficiency of this semi-analytical approach. With this, we can also offer some benchmarks for the FEM on general meshes implemented by other means (e.g., for accuracy check and comparison when triangulation reduces to rectangular meshes).
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2023-0011
Communications in Computational Physics, Vol. 36 (2024), Iss. 3 : pp. 673–710
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 38
Keywords: Integral fractional Laplacian nonlocal/singular operators FEM on rectangular meshes stiffness matrix with Toeplitz structure.
Author Details
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FEM on nonuniform meshes for nonlocal Laplacian: Semi-analytic implementation in one dimension
Chen, Hongbin
Sheng, Changtao
Wang, Li-Lian
Calcolo, Vol. 62 (2025), Iss. 1
https://doi.org/10.1007/s10092-024-00632-x [Citations: 0]