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

Randolph R. Settgast, Yohann Dudouit, Nicola Castelletto, William R. Tobin, Benjamin C. Corbett, Sergey Klevtsov
Atmospheric, Earth, and Energy Division, Lawrence Livermore National Laboratory, Livermore, CA, United States
arXiv:2308.09839 [math.NA], (22 Aug 2023)

@misc{settgast2023performant,

   title={Performant low-order matrix-free finite element kernels on GPU architectures},

   author={Randolph R. Settgast and Yohann Dudouit and Nicola Castelletto and William R. Tobin and Benjamin C. Corbett and Sergey Klevtsov},

   year={2023},

   eprint={2308.09839},

   archivePrefix={arXiv},

   primaryClass={math.NA}

}

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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.
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