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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. The GPU acceleration of numerically solving PDEs has beenexplored successfully around fifteen years ago and become more and more popularin the past decade, driven by significant advancement in both hardware andsoftware technologies, especially in the recent few years. We present in thispaper a simple but extremely fast MATLAB implementation on a modern GPU, whichcan be easily reproduced, for solving 3D Poisson type equations using aspectral-element method. In particular, it costs less than one second on aNvidia A100 for solving a Poisson equation with one billion degree of freedoms.

A simple GPU implementation of spectral-element methods for solving 3D Poisson type equations on cartesian meshes