A GPU accelerated and error-controlled solver for the unbounded Poisson equation in three dimensions

TL;DR

This paper introduces a GPU-accelerated, error-controlled solver for the 3D free-space Poisson equation using finite Fourier series, achieving spectral accuracy and efficient computation suitable for large-scale problems.

Contribution

The paper presents a novel GPU-compatible solver based on Fourier series with fully controllable error, optimized for spectral accuracy and computational efficiency.

Findings

Achieves spectral accuracy for smooth, fast-decaying densities.

Scales with O(N log N) operations, ideal for GPU implementation.

Validated through numerical tests on CPU and GPU, demonstrating efficiency.

Abstract

An efficient solver for the three dimensional free-space Poisson equation is presented. The underlying numerical method is based on finite Fourier series approximation. While the error of all involved approximations can be fully controlled, the overall computation error is driven by the convergence of the finite Fourier series of the density. For smooth and fast-decaying densities the proposed method will be spectral accurate. The method scales with $\mathcal{O}(N\log N)$ operations, where $N$ is the total number of discretization points in the Cartesian grid. The majority of the computational costs come from fast Fourier transforms (FFT), which makes it ideal for GPU computation. Several numerical computations on CPU and GPU validate the method and show efficiency and convergence behavior. Tests are performed using the Vienna Scientific Cluster 3 (VSC3). A free MATLAB implementation for CPU and GPU is provided to the interested community.