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A numerical method for solving elliptic PDEs with variable coefficients ontwo-dimensional domains is presented. The method is based on high-ordercomposite spectral approximations and is designed for problems with smoothsolutions. The resulting system of linear equations is solved using a direct(as opposed to iterative) solver that has optimal O(N) complexity for allstages of the computation when applied to problems with non-oscillatorysolutions such as the Laplace and the Stokes equations. Numerical examplesdemonstrate that the scheme is capable of computing solutions with relativeaccuracy of $10^{-10}$ or better, even for challenging problems such as highlyoscillatory Helmholtz problems and convection-dominated convection diffusionequations. In terms of speed, it is demonstrated that a problem with anon-oscillatory solution that was discretized using $10^{8}$ nodes was solvedin 115 minutes on a personal work-station with two quad-core 3.3GHz CPUs. Sincethe solver is direct, and the "solution operator" fits in RAM, any solvesbeyond the first are very fast. In the example with $10^{8}$ unknowns, solvesrequire only 30 seconds.

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