FEM on nonuniform meshes for nonlocal Laplacian: Semi-analytic Implementation in One Dimension

In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transformed domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically...

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Bibliographic Details
Published in:arXiv.org 2024-07
Main Authors: Chen, Hongbin, Sheng, Changtao, Li-Lian, Wang
Format: Article
Language:English
Subjects:
Dimensional analysis
Discretization
Stiffness matrix
Online Access:Get full text
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Summary:In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transformed domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically evaluate the entries for various interaction kernels. Moreover, the limiting cases of the nonlocal stiffness matrix when the interactional radius \(\delta\rightarrow0\) or \(\delta\rightarrow\infty\) automatically lead to integer and fractional FEM stiffness matrices, respectively, and the FEM discretisation is intrinsically compatible. We conduct ample numerical experiments to study and predict some of its properties and test on different types of nonlocal problems. To the best of our knowledge, such a semi-analytic approach has not been explored in literature even in the one-dimensional case.
ISSN:2331-8422