Fast Maxwell Solvers Based on Exact Discrete Eigen-Decompositions I. Two-Dimensional Case
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916651399643136 |
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| author | Wang, Lixiu Jia, Lueling Cao, Zijian Li, Huiyuan Zhang, Zhimin |
| author_facet | Wang, Lixiu Jia, Lueling Cao, Zijian Li, Huiyuan Zhang, Zhimin |
| contents | In this paper, we propose fast solvers for Maxwell's equations in rectangular domains. We first discretize the simplified Maxwell's eigenvalue problems by employing the lowest-order rectangular Nédélec elements and derive the discrete eigen-solutions explicitly, providing a Hodge-Helmholtz decomposition framework at the discrete level. Based on exact eigen-decompositions, we further design fast solvers for various Maxwell's source problems, guaranteeing either the divergence-free constraint or the Gauss's law at the discrete level. With the help of fast sine/cosine transforms, the computational time grows asymptotically as $\mathcal{O}(n^2\log n)$ with $n$ being the number of grids in each direction. Our fast Maxwell solvers outperform other existing Maxwell solvers in the literature and fully rival fast scalar Poisson/Helmholtz solvers based on trigonometric transforms in either efficiency, robustness, or storage complexity. It is also utilized to perform an efficient pre-conditioning for solving Maxwell's source problems with variable coefficients. Finally, numerical experiments are carried out to illustrate the effectiveness and efficiency of the proposed fast solver. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast Maxwell Solvers Based on Exact Discrete Eigen-Decompositions I. Two-Dimensional Case Wang, Lixiu Jia, Lueling Cao, Zijian Li, Huiyuan Zhang, Zhimin Numerical Analysis 65N30, 35Q61, 65F05, 68W40 In this paper, we propose fast solvers for Maxwell's equations in rectangular domains. We first discretize the simplified Maxwell's eigenvalue problems by employing the lowest-order rectangular Nédélec elements and derive the discrete eigen-solutions explicitly, providing a Hodge-Helmholtz decomposition framework at the discrete level. Based on exact eigen-decompositions, we further design fast solvers for various Maxwell's source problems, guaranteeing either the divergence-free constraint or the Gauss's law at the discrete level. With the help of fast sine/cosine transforms, the computational time grows asymptotically as $\mathcal{O}(n^2\log n)$ with $n$ being the number of grids in each direction. Our fast Maxwell solvers outperform other existing Maxwell solvers in the literature and fully rival fast scalar Poisson/Helmholtz solvers based on trigonometric transforms in either efficiency, robustness, or storage complexity. It is also utilized to perform an efficient pre-conditioning for solving Maxwell's source problems with variable coefficients. Finally, numerical experiments are carried out to illustrate the effectiveness and efficiency of the proposed fast solver. |
| title | Fast Maxwell Solvers Based on Exact Discrete Eigen-Decompositions I. Two-Dimensional Case |
| topic | Numerical Analysis 65N30, 35Q61, 65F05, 68W40 |
| url | https://arxiv.org/abs/2503.09933 |