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

TL;DR

This paper introduces a highly efficient GPU implementation of spectral-element methods for solving 3D Poisson equations on rectangular domains, achieving near-instant solutions for billion-scale problems and demonstrating applications to Schrödinger and Cahn-Hilliard equations.

Contribution

It provides a simple, reproducible MATLAB GPU implementation of a spectral-element solver for 3D Poisson equations with unprecedented speed and scalability.

Findings

Solves 3D Poisson equations with 1 billion degrees of freedom in under a second.

Demonstrates application to Schrödinger and Cahn-Hilliard equations.

Achieves high scalability and efficiency on modern GPU hardware.

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) Schr\"odinger equation and a nonlinear (time-dependent) Cahn-Hilliard equation.