A Fourth-Order Cut-cell Multigrid Method for Solving Elliptic Equations on Arbitrary Domains
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908769310474240 |
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| author | Liu, Jiyu Li, Zhixuan Yan, Jiatu Li, Zhiqi Zhang, Qinghai |
| author_facet | Liu, Jiyu Li, Zhixuan Yan, Jiatu Li, Zhiqi Zhang, Qinghai |
| contents | To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to two-dimensional constant-coefficient elliptic equations with both divergence and non-divergence forms and all types of boundary conditions, (2) the versatility of handling both regular and irregular domains with arbitrarily complex topology and geometry, (3) the fourth-order accuracy even at the presence of ${\cal C}^1$ discontinuities on the domain boundary, and (4) the optimal complexity of $O(h^{-2})$.Test results demonstrate the generality, accuracy, efficiency, robustness, and excellent conditioning of the proposed method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02975 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Fourth-Order Cut-cell Multigrid Method for Solving Elliptic Equations on Arbitrary Domains Liu, Jiyu Li, Zhixuan Yan, Jiatu Li, Zhiqi Zhang, Qinghai Numerical Analysis To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to two-dimensional constant-coefficient elliptic equations with both divergence and non-divergence forms and all types of boundary conditions, (2) the versatility of handling both regular and irregular domains with arbitrarily complex topology and geometry, (3) the fourth-order accuracy even at the presence of ${\cal C}^1$ discontinuities on the domain boundary, and (4) the optimal complexity of $O(h^{-2})$.Test results demonstrate the generality, accuracy, efficiency, robustness, and excellent conditioning of the proposed method. |
| title | A Fourth-Order Cut-cell Multigrid Method for Solving Elliptic Equations on Arbitrary Domains |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2601.02975 |