Performant low-order matrix-free finite element kernels on GPU architectures

TL;DR

This paper develops and evaluates efficient low-order matrix-free finite element kernels optimized for GPU architectures, demonstrating competitive performance with matrix-based methods on modern GPUs.

Contribution

It introduces a strategy for implementing high-performance low-order matrix-free FEM kernels on GPUs, addressing applications with high heterogeneity and low-order accuracy needs.

Findings

Matrix-free kernels outperform matrix-based SpMV on GPUs

Performance achieved on V100, A100, MI250X GPUs

Applicable to Laplace and elasticity operators

Abstract

Numerical methods such as the Finite Element Method (FEM) have been successfully adapted to utilize the computational power of GPU accelerators. However, much of the effort around applying FEM to GPU's has been focused on high-order FEM due to higher arithmetic intensity and order of accuracy. For applications such as the simulation of subsurface processes, high levels of heterogeneity results in high-resolution grids characterized by highly discontinuous (cell-wise) material property fields. Moreover, due to the significant uncertainties in the characterization of the domain of interest, e.g. geologic reservoirs, the benefits of high order accuracy are reduced, and low-order methods are typically employed. In this study, we present a strategy for implementing highly performant low-order matrix-free FEM operator kernels in the context of the conjugate gradient (CG) method. Performance results of matrix-free Laplace and isotropic elasticity operator kernels are presented and are shown to compare favorably to matrix-based SpMV operators on V100, A100, and MI250X GPUs.