An optimal complexity spectral solver for the Poisson equation

TL;DR

This paper introduces an efficient spectral solver for the Poisson equation on square domains that achieves optimal complexity, handles general boundary conditions, and is scalable to large problems with millions of unknowns.

Contribution

The proposed method combines ultraspherical spectral and ADI techniques to improve efficiency and flexibility over existing spectral solvers for the Poisson equation.

Findings

Achieves optimal $ ext{O}(n^2)$ complexity for smooth solutions.

Eliminates basis conversion between Chebyshev and Legendre.

Resolves large-scale problems with millions of unknowns in under a minute.

Abstract

We propose a spectral solver for the Poisson equation on a square domain, achieving optimal complexity through the ultraspherical spectral method and the alternating direction implicit (ADI) method. Compared with the state-of-the-art spectral solver for the Poisson equation \cite{for}, our method not only eliminates the need for conversions between Chebyshev and Legendre bases but also is applicable to more general boundary conditions while maintaining spectral accuracy. We prove that, for solutions with sufficient smoothness, a fixed number of ADI iterations suffices to meet a specified tolerance, yielding an optimal complexity of $\mathcal{O}(n^2)$. The solver can also be extended to other equations as long as they can be split into two one-dimensional operators with nearly real and disjoint spectra. Numerical experiments demonstrate that our algorithm can resolve solutions with millions of unknowns in under a minute, with significant speedups when leveraging low-rank approximations.