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.

翻译：数值方法如有限元法（FEM）已成功适配以利用GPU加速器的计算能力。然而，当前将FEM应用于GPU的研究主要聚焦于高阶FEM，因其具有更高的算术强度和精度阶次。对于地下过程模拟等应用场景，高度异质性导致需要采用具有高度非连续（单元级）材料属性场的高分辨率网格。此外，由于目标区域（如地质储层）表征存在显著不确定性，高阶精度的优势有所减弱，因此通常采用低阶方法。本研究提出了一种在共轭梯度（CG）法框架下实现高性能低阶无矩阵FEM算子内核的策略。我们给出了无矩阵拉普拉斯算子和各向同性弹性算子内核的性能结果，并证明其在V100、A100和MI250X GPU上相比基于矩阵的SpMV算子具有更优表现。