Near-field-free super-potential FFT method for the three-dimensional free-space Poisson equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908547606904832 |
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| author | Exl, Lukas Schaffer, Sebastian |
| author_facet | Exl, Lukas Schaffer, Sebastian |
| contents | We present a spectrally accurate, efficient FFT-based method for the three-dimensional free-space Poisson equation with smooth, compactly supported sources. The method adopts a super-potential formulation: we first compute the convolution with the biharmonic Green's function, then recover the potential by spectral differentiation, applying the Laplacian in Fourier space. A separable Gaussian-sum (GS) approximation enables efficient precomputation and quasi-linear, FFT-based convolution. Owing to the biharmonic kernel's improved regularity, the GS cutoff error is fourth-order, uniform for all target points, eliminating the near-field corrections and Taylor expansions required in standard GS/Ewald-type methods. Benchmarks on Gaussian, oscillatory, and compactly supported densities reach the double-precision limit and, at matched accuracy on the same hardware, reduce both error and per-solve runtime relative to our original GS-based scheme. The resulting method is simple, reproducible, and efficient for three-dimensional free-space Poisson problems with smooth sources on uniform grids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04489 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Near-field-free super-potential FFT method for the three-dimensional free-space Poisson equation Exl, Lukas Schaffer, Sebastian Computational Physics Numerical Analysis 65N35, 65T50, 35J05 We present a spectrally accurate, efficient FFT-based method for the three-dimensional free-space Poisson equation with smooth, compactly supported sources. The method adopts a super-potential formulation: we first compute the convolution with the biharmonic Green's function, then recover the potential by spectral differentiation, applying the Laplacian in Fourier space. A separable Gaussian-sum (GS) approximation enables efficient precomputation and quasi-linear, FFT-based convolution. Owing to the biharmonic kernel's improved regularity, the GS cutoff error is fourth-order, uniform for all target points, eliminating the near-field corrections and Taylor expansions required in standard GS/Ewald-type methods. Benchmarks on Gaussian, oscillatory, and compactly supported densities reach the double-precision limit and, at matched accuracy on the same hardware, reduce both error and per-solve runtime relative to our original GS-based scheme. The resulting method is simple, reproducible, and efficient for three-dimensional free-space Poisson problems with smooth sources on uniform grids. |
| title | Near-field-free super-potential FFT method for the three-dimensional free-space Poisson equation |
| topic | Computational Physics Numerical Analysis 65N35, 65T50, 35J05 |
| url | https://arxiv.org/abs/2506.04489 |