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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.21913 |
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| _version_ | 1866908471121674240 |
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| author | Xiang, Chunzhi Wang, Bo Zhang, Wenzhong Cai, Wei |
| author_facet | Xiang, Chunzhi Wang, Bo Zhang, Wenzhong Cai, Wei |
| contents | In this paper, we present a fast multipole method (FMM) for solving the two-dimensional Laplace equation in a half-plane with Robin boundary conditions. The method is based on a novel expansion theory for the reaction component of the Green's function. By applying the Fourier transform, the reaction field component is obtained in a Sommerfeld-type integral form. We derive far-field approximations and corresponding shifting and translation operators from the Fourier integral representation. The FMM for the reaction component is then developed by using the new far-field approximations incorporated into the classic FMM framework in which the tree structure is constructed from the original and image charges. Combining this with the standard FMM for the free-space components, we develop a fast algorithm to compute the interaction of the half plane Laplace Green's function. We prove that the method exhibits exponential convergence, similar to the free-space FMM. Finally, numerical examples are presented to validate the theoretical results and demonstrate that the FMM achieves $O(N)$ computational complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast multipole method for the Laplace equation in half plane with Robin boundary condition Xiang, Chunzhi Wang, Bo Zhang, Wenzhong Cai, Wei Numerical Analysis Mathematical Physics 65D30, 65D32, 65R10, 41A60 G.1.8; F.2.1; I.6.4 In this paper, we present a fast multipole method (FMM) for solving the two-dimensional Laplace equation in a half-plane with Robin boundary conditions. The method is based on a novel expansion theory for the reaction component of the Green's function. By applying the Fourier transform, the reaction field component is obtained in a Sommerfeld-type integral form. We derive far-field approximations and corresponding shifting and translation operators from the Fourier integral representation. The FMM for the reaction component is then developed by using the new far-field approximations incorporated into the classic FMM framework in which the tree structure is constructed from the original and image charges. Combining this with the standard FMM for the free-space components, we develop a fast algorithm to compute the interaction of the half plane Laplace Green's function. We prove that the method exhibits exponential convergence, similar to the free-space FMM. Finally, numerical examples are presented to validate the theoretical results and demonstrate that the FMM achieves $O(N)$ computational complexity. |
| title | Fast multipole method for the Laplace equation in half plane with Robin boundary condition |
| topic | Numerical Analysis Mathematical Physics 65D30, 65D32, 65R10, 41A60 G.1.8; F.2.1; I.6.4 |
| url | https://arxiv.org/abs/2507.21913 |