A Simple GPU Implementation of Spectral-Element Methods for Solving 3D Poisson Type Equations on Rectangular Domains and Its Applications
Year: 2024
Author: Xinyu Liu, Jie Shen, Xiangxiong Zhang
Communications in Computational Physics, Vol. 36 (2024), Iss. 5 : pp. 1157–1185
Abstract
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) Schrödinger equation and a nonlinear (time-dependent) Cahn-Hilliard equation.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2024-0072
Communications in Computational Physics, Vol. 36 (2024), Iss. 5 : pp. 1157–1185
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Keywords: 3D Poisson equation direct solver spectral element methods rectangular domain GPU tensor matrix multiplication.