2013/07/10 by Adrianna Gillman, Per‐Gunnar Martinsson, Gillman, A. +1 · 4 citations
Engineering · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #Electromagnetic Scattering and Analysis
paper · pdf · doi:10.48550/arxiv.1307.2665
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 108 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 108 unknowns, solves require only 30 seconds.