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It is well known since 1960s that by exploring the tensor product structureof the discrete Laplacian on Cartesian meshes, one can develop a simple directPoisson solver with an $\mathcal O(N^{\frac{d+1}d})$ complexity in d-dimension,where N is the number of the total unknowns. The GPU acceleration ofnumerically solving PDEs has been explored successfully around fifteen yearsago and become more and more popular in the past decade, driven by significantadvancement in both hardware and software technologies, especially in therecent few years. We present in this paper a simple but extremely fast MATLABimplementation on a modern GPU, which can be easily reproduced, for solving 3DPoisson type equations using a spectral-element method. In particular, it costsless than one second on a Nvidia A100 for solving a Poisson equation with onebillion degree of freedoms. We also present applications of this fast solver tosolve a linear (time-independent) Schr\"odinger equation and a nonlinear(time-dependent) Cahn-Hilliard equation.

A simple GPU implementation of spectral-element methods for solving 3D Poisson type equations on rectangular domains and its applications