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

TL;DR

This paper introduces a spectrally accurate, FFT-based super-potential method for solving the 3D free-space Poisson equation efficiently, avoiding near-field corrections and achieving high accuracy with reduced computational cost.

Contribution

It presents a novel super-potential formulation using Gaussian-sum approximation that improves regularity and efficiency over existing methods for 3D Poisson problems with smooth sources.

Findings

Achieves double-precision accuracy on benchmark problems.

Reduces runtime and error compared to previous schemes.

Eliminates near-field corrections through improved kernel regularity.

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.