Result: A Simple GPU Implementation of Spectral-Element Methods for Solving 3D Poisson Type Equations on Rectangular Domains and Its Applications: A simple GPU implementation of spectral-element methods for solving 3D Poisson type equations on rectangular domains and its applications

Title:
A Simple GPU Implementation of Spectral-Element Methods for Solving 3D Poisson Type Equations on Rectangular Domains and Its Applications: A simple GPU implementation of spectral-element methods for solving 3D Poisson type equations on rectangular domains and its applications
Authors:
Liu, Xinyu, Shen, Jie, Zhang, Xiangxiong
Source:
Communications in Computational Physics. 36:1157-1185
Publication Status:
Preprint
Publisher Information:
Global Science Press, 2024.
Publication Year:
2024
Subject Terms:
Numerical computation of matrix exponential and similar matrix functions, Finite difference methods for boundary value problems involving PDEs, FOS: Computer and information sciences, tensor matrix multiplication, rectangular domain, GPU, Parallel numerical computation, spectral element methods, Numerical Analysis (math.NA), Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Direct numerical methods for linear systems and matrix inversion, direct solver, Computer Science - Distributed, Parallel, and Cluster Computing, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Spectral, collocation and related methods for boundary value problems involving PDEs, Mathematics - Numerical Analysis, Distributed, Parallel, and Cluster Computing (cs.DC), 3D Poisson equation, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Document Type:
Academic journal Article
File Description:
application/xml
ISSN:
1991-7120
1815-2406
DOI:
10.4208/cicp.oa-2024-0072
DOI:
10.48550/arxiv.2310.00226
Access URL:
http://arxiv.org/abs/2310.00226
Rights:
arXiv Non-Exclusive Distribution
Accession Number:
edsair.doi.dedup.....06dd4b47c4eab1a5f4c19fd2ef40aab3
Database:
OpenAIRE

Further Information

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.