标题： 一种求解矩形域上三维Poisson型方程的谱元方法的简单GPU实现及其应用 显示英文标题

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

摘要： 自1960年代以来人们已经熟知，通过探索笛卡尔网格上离散拉普拉斯算子的张量积结构，可以构建一个简单的直接泊松求解器，在d维情况下具有 $\mathcal O(N^{\frac{d+1}d})$ 复杂度，其中N是总未知数的数量。 大约十五年前成功探索了用数值方法求解偏微分方程的GPU加速，并在过去十年间变得越来越流行，这得益于硬件和软件技术的重大进步，尤其是在最近几年。 本文提出了一种在现代GPU上的简单但非常快速的MATLAB实现，可以轻松重现，用于使用谱元法求解3D泊松型方程。 特别是，它在Nvidia A100上求解自由度为十亿的泊松方程耗时不到一秒。 我们还展示了此快速求解器在求解线性（稳态）薛定谔方程和非线性（瞬态）Cahn-Hilliard方程中的应用。 显示英文摘要

摘要： It is well known since 1960s that by exploring the tensor product structure of the discrete Laplacian on Cartesian meshes, one can develop a simple direct Poisson 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 of numerically solving PDEs has been explored successfully around fifteen years ago and become more and more popular in the past decade, driven by significant advancement in both hardware and software technologies, especially in the recent few years. We present in this paper a simple but extremely fast MATLAB implementation on a modern GPU, which can be easily reproduced, for solving 3D Poisson type equations using a spectral-element method. In particular, it costs less than one second on a Nvidia A100 for solving a Poisson equation with one billion degree of freedoms. We also present applications of this fast solver to solve a linear (time-independent) Schr\"odinger equation and a nonlinear (time-dependent) Cahn-Hilliard equation.