A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method

TL;DR

This paper introduces a high-order spectral collocation method with an O(N) direct solver for variable coefficient elliptic PDEs, enabling fast, accurate solutions even for large-scale, complex problems.

Contribution

The paper presents a novel direct solver with optimal linear complexity for high-order spectral discretizations of elliptic PDEs with variable coefficients.

Findings

Achieves relative accuracy of 10^{-10} for complex problems

Solves large-scale problems with 10^8 nodes in under 2 hours

First solve takes about 115 minutes; subsequent solves are very fast

Abstract

A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The resulting system of linear equations is solved using a direct (as opposed to iterative) solver that has optimal O(N) complexity for all stages of the computation when applied to problems with non-oscillatory solutions such as the Laplace and the Stokes equations. Numerical examples demonstrate that the scheme is capable of computing solutions with relative accuracy of $10^{-10}$ or better, even for challenging problems such as highly oscillatory Helmholtz problems and convection-dominated convection diffusion equations. In terms of speed, it is demonstrated that a problem with a non-oscillatory solution that was discretized using $10^{8}$ nodes was solved in 115 minutes on a personal work-station with two quad-core 3.3GHz CPUs. Since the solver is direct, and the "solution operator" fits in RAM, any solves beyond the first are very fast. In the example with $10^{8}$ unknowns, solves require only 30 seconds.