FEM on nonuniform meshes for nonlocal Laplacian: Semi-analytic Implementation in One Dimension
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866916320442843136 |
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| author | Chen, Hongbin Sheng, Changtao Wang, Li-Lian |
| author_facet | Chen, Hongbin Sheng, Changtao Wang, Li-Lian |
| contents | 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 $δ\rightarrow0$ or $δ\rightarrow\infty$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08988 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | FEM on nonuniform meshes for nonlocal Laplacian: Semi-analytic Implementation in One Dimension Chen, Hongbin Sheng, Changtao Wang, Li-Lian Numerical Analysis 65L60, 65N30, 65N50 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 $δ\rightarrow0$ or $δ\rightarrow\infty$ 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. |
| title | FEM on nonuniform meshes for nonlocal Laplacian: Semi-analytic Implementation in One Dimension |
| topic | Numerical Analysis 65L60, 65N30, 65N50 |
| url | https://arxiv.org/abs/2407.08988 |