论文标题
教科书效率：斯托克斯系统的大规模平行基矩阵多机
Textbook efficiency: massively parallel matrix-free multigrid for the Stokes system
论文作者
论文摘要
正如Achi Brandt介绍的那样，我们采用教科书Multigrid效率（TME）来为Stokes System构建一个渐近最佳的单片Multigrid求解器。几何多机求解器建立在层次混合电网（HHG）的概念上，该层次均扩展到高阶有限元离散化和相应的无基质实现。量化了完整的跨部（FMG）迭代的计算成本，并将求解器应用于多个基准问题。通过一项参数研究，我们提出了可以实现TME，稳定的等序和泰勒 - 霍德离散化的配置。相关计算内核的出色节点级性能是通过屋顶分析提出的。最后，我们证明了最多$ 147,456 $并行流程的薄弱和强大可扩展性，并解决了$ 3.6 \ times 10^{12} $（万亿）未知的Stokes Systems。
We employ textbook multigrid efficiency (TME), as introduced by Achi Brandt, to construct an asymptotically optimal monolithic multigrid solver for the Stokes system. The geometric multigrid solver builds upon the concept of hierarchical hybrid grids (HHG), which is extended to higher-order finite-element discretizations, and a corresponding matrix-free implementation. The computational cost of the full multigrid (FMG) iteration is quantified, and the solver is applied to multiple benchmark problems. Through a parameter study, we suggest configurations that achieve TME for both, stabilized equal-order, and Taylor-Hood discretizations. The excellent node-level performance of the relevant compute kernels is presented via a roofline analysis. Finally, we demonstrate the weak and strong scalability to up to $147,456$ parallel processes and solve Stokes systems with more than $3.6 \times 10^{12}$ (trillion) unknowns.
