Fast Poisson solvers for spectral methods

TL;DR

This paper introduces fast spectral methods for solving Poisson's equation on various geometries, achieving optimal complexity and enabling solutions with up to 100 million degrees of freedom in minutes on a standard laptop.

Contribution

The paper develops novel spectral discretizations that exploit separated spectra properties, leading to efficient Poisson solvers with optimal complexity for multiple geometries.

Findings

Achieved solution of Poisson's equation with 100 million degrees of freedom in under two minutes.

Developed spectral methods with optimal complexity exploiting separated spectra.

Extended fast Poisson solvers to geometries like square, cylinder, sphere, and cube.

Abstract

Poisson's equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here, we derive spectral methods for solving Poisson's equation on a square, cylinder, solid sphere, and cube that have an optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom required to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of finite difference matrices, our solver exploits a separated spectra property that holds for our spectral discretizations. Without parallelization, we can solve Poisson's equation on a square with 100 million degrees of freedom in under two minutes on a standard laptop.