A Direct Solver with $O(N)$ Complexity for Variable Coefficient Elliptic PDEs Discretized via a High-Order Composite Spectral Collocation Method

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 nonoscillatory solutions such as the Laplace and the Stokes equations. Numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of $10^{-10}$ or better 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 nonoscillatory solution that was discretized using $10^{8}$ nodes can be solved in 115 minutes on a personal workstation with two quad-core 3.3 GHz CPUs. Since the solver is direct, and the “solution ...