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）已成功适配以利用 GPU 加速器的计算能力。然而，将有限元法应用于 GPU 的研究工作大多集中于高阶有限元，因其具有更高的算术强度和精度阶数。对于诸如地下过程模拟等应用场景，高度的非均匀性会导致高分辨率网格中呈现高度间断（单元级）的材料属性场。此外，由于目标区域（例如地质储层）表征存在显著不确定性，高阶精度的优势有所降低，因此通常采用低阶方法。在本研究中，我们提出了一种在共轭梯度法框架下实现高性能低阶无矩阵有限元算子核心的策略。本文展示了无矩阵拉普拉斯算子和各向同性弹性算子的性能结果，结果表明，在 V100、A100 和 MI250X GPU 上，其性能优于基于矩阵的 SpMV 算子。