Fast Poisson solvers for spectral methods

Poisson’s equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference (FD) 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 optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom used to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of FD matrices, our solver exploits a separated spectra property that holds for our carefully designed spectral discretizations. Without parallelization we can solve Poisson’s equation on a square with 100 million degrees of freedom in under 2 min on a standard laptop.