TY - JOUR
T1 - A Simple GPU Implementation of Spectral-Element Methods for Solving 3D Poisson Type Equations on Rectangular Domains and Its Applications
AU - Liu , Xinyu
AU - Shen , Jie
AU - Zhang , Xiangxiong
JO - Communications in Computational Physics
VL - 5
SP - 1157
EP - 1185
PY - 2024
DA - 2024/12
SN - 36
DO - http://doi.org/10.4208/cicp.OA-2024-0072
UR - https://global-sci.org/intro/article_detail/cicp/23606.html
KW - 3D Poisson equation, direct solver, spectral element methods, rectangular domain, GPU, tensor matrix multiplication.
AB - <p style="text-align: justify;">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.</p>