Near-field-free super-potential FFT method for the three-dimensional free-space Poisson equation

Abstract

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.