Fast Poisson solvers for spectral methods
This work provides the first fast Poisson solvers for spectral methods, addressing a long-standing bottleneck in computational science.
The authors derive spectral methods for solving Poisson's equation on various domains with optimal complexity, achieving a solution for 100 million degrees of freedom on a square in under two minutes on a laptop.
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.