A GPU parallelized spectral method for elliptic equations in rectangular domains

We design and implement the first polynomial-based spectral method on graphic processing units (GPUs). The key to success lies in the seamless integration of the matrix diagonalization technique and new generation CUDA tools. The method is applicable to elliptic equations with general boundary conditions in both 2-D and 3-D cases. We show remarkable speedups of more than 10 times in the 2-D case and more than 30 times in the 3-D case. The GPU-accelerated spectral method is applied to the coupled system of Navier– Stokes equation and Allen–Cahn equation, demonstrating that the new algorithm is particularly suitable for simulations with finer meshes and longer run-times.