A FFT-based GMRES for fast solving of Poisson equation in concatenated geometry
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916974138753024 |
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| author | Jiang, Zichao Lian, Jiacheng Wang, Zhuolin |
| author_facet | Jiang, Zichao Lian, Jiacheng Wang, Zhuolin |
| contents | Fast Fourier Transform (FFT)-based solvers for the Poisson equation are highly efficient, exhibiting $O(N\log N)$ computational complexity and excellent parallelism. However, their application is typically restricted to simple, regular geometries due to the separability requirement of the underlying discrete operators. This paper introduces a novel domain decomposition method that extends the applicability of FFT-based solvers to complex composite domains geometries constructed from multiple sub-regions. The method transforms the global problem into a system of sub-problems coupled through Schur complements at the interfaces. A key challenge is that the Schur complement disrupts the matrix structure required for direct FFT-based inversion. To overcome this, we develop a FFT-based preconditioner to accelerate the Generalized Minimal Residual (GMRES) method for the interface system. The central innovation is a novel preconditioner based on the inverse of the block operator without the Schur complement, which can be applied efficiently using the FFT-based solver. The resulting preconditioned iteration retains an optimal complexity for each step. Numerical experiments on a cross-shaped domain demonstrate that the proposed solver achieves the expected second-order accuracy of the underlying finite difference scheme. Furthermore, it exhibits significantly improved computational performance compared to a classic sparse GMRES solver based on Eigen libeary. For a problem with $10^5$ grid points, our method achieves a speedup of over 40 times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23180 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A FFT-based GMRES for fast solving of Poisson equation in concatenated geometry Jiang, Zichao Lian, Jiacheng Wang, Zhuolin Numerical Analysis Computational Physics Fast Fourier Transform (FFT)-based solvers for the Poisson equation are highly efficient, exhibiting $O(N\log N)$ computational complexity and excellent parallelism. However, their application is typically restricted to simple, regular geometries due to the separability requirement of the underlying discrete operators. This paper introduces a novel domain decomposition method that extends the applicability of FFT-based solvers to complex composite domains geometries constructed from multiple sub-regions. The method transforms the global problem into a system of sub-problems coupled through Schur complements at the interfaces. A key challenge is that the Schur complement disrupts the matrix structure required for direct FFT-based inversion. To overcome this, we develop a FFT-based preconditioner to accelerate the Generalized Minimal Residual (GMRES) method for the interface system. The central innovation is a novel preconditioner based on the inverse of the block operator without the Schur complement, which can be applied efficiently using the FFT-based solver. The resulting preconditioned iteration retains an optimal complexity for each step. Numerical experiments on a cross-shaped domain demonstrate that the proposed solver achieves the expected second-order accuracy of the underlying finite difference scheme. Furthermore, it exhibits significantly improved computational performance compared to a classic sparse GMRES solver based on Eigen libeary. For a problem with $10^5$ grid points, our method achieves a speedup of over 40 times. |
| title | A FFT-based GMRES for fast solving of Poisson equation in concatenated geometry |
| topic | Numerical Analysis Computational Physics |
| url | https://arxiv.org/abs/2509.23180 |