Abstract
Spectral methods are useful for applications that benefit from high-order precisions. However, if the same number of degrees of freedom is used, the computational cost of a spectral method is considerably higher than that of a general finite difference or finite element method. After the investigation in Chen et al. (J Comput Phys 250:555---564, 2013), we provide for the first time a framework of graphics processing units (GPU)-accelerated spectral methods for systems of coupled elliptic equations. The involved dense matrix computations, as the main obstacle for fast spectral methods on a traditional CPU, turns out to be an opportunity for high speedups on a many-core GPU. We obtain an order-of-magnitude speedup for solving 2-D and 3-D systems using a Kepler 20 GPU over a high-end multi-core processor, with two popular spectral methods, namely, the spectral collocation method and the spectral-Galerkin method. The new framework is applicable to systems of $$L$$ L coupled second-order equations with general boundary conditions, where $$L$$ L is an integer of moderate size. The ultimate goal is to apply the developed solver to complex and nonlinear time-dependent problems. As two interesting examples, a 2-D FitzHugh-Nagumo equation is solved with the spectral collocation method and a 3-D Cahn-Hilliard equation is solved with the spectral-Galerkin method. We thus demonstrate a practical solution for demanding problems that utilize high-order spatial resolution and longer run-times.
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