Local discontinuous Galerkin method for the integral fractional Laplacian
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918248069464064 |
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| author | Han, Rubing Wu, Shuonan Zhou, Hao |
| author_facet | Han, Rubing Wu, Shuonan Zhou, Hao |
| contents | We develop and analyze a local discontinuous Galerkin (LDG) method for solving integral fractional Laplacian problems on bounded Lipschitz domains. The method is based on a three-field mixed formulation involving the primal variable, its gradient, and the corresponding Riesz potential, yielding a flux-based structure well suited for LDG discretizations while retaining the intrinsic nonlocal interaction. A key ingredient of our analysis is a rigorous study of the weighted Hölder and Sobolev regularity of the Riesz potential, which enables accurate characterization of boundary singularities. Guided by these regularity results, we propose LDG schemes on quasi-uniform and graded meshes, with additional stabilization in the graded case to reconcile the discrepancy between the discrete spaces for the Riesz potential and flux fields. Optimal a priori error estimates are established, and numerical experiments corroborate the theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_12200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local discontinuous Galerkin method for the integral fractional Laplacian Han, Rubing Wu, Shuonan Zhou, Hao Numerical Analysis 65N15, 65N30, 35B65 We develop and analyze a local discontinuous Galerkin (LDG) method for solving integral fractional Laplacian problems on bounded Lipschitz domains. The method is based on a three-field mixed formulation involving the primal variable, its gradient, and the corresponding Riesz potential, yielding a flux-based structure well suited for LDG discretizations while retaining the intrinsic nonlocal interaction. A key ingredient of our analysis is a rigorous study of the weighted Hölder and Sobolev regularity of the Riesz potential, which enables accurate characterization of boundary singularities. Guided by these regularity results, we propose LDG schemes on quasi-uniform and graded meshes, with additional stabilization in the graded case to reconcile the discrepancy between the discrete spaces for the Riesz potential and flux fields. Optimal a priori error estimates are established, and numerical experiments corroborate the theoretical results. |
| title | Local discontinuous Galerkin method for the integral fractional Laplacian |
| topic | Numerical Analysis 65N15, 65N30, 35B65 |
| url | https://arxiv.org/abs/2512.12200 |