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.

中文翻译：

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教科书效率：Stokes 系统的大规模并行无矩阵多重网格

我们采用 Achi Brandt 介绍的教科书多重网格效率 (TME) 为斯托克斯系统构建渐近最优的整体多重网格求解器。几何多重网格求解器建立在分层混合网格 (HHG) 的概念之上，该概念扩展到高阶有限元离散化，以及相应的无矩阵实现。量化全多重网格 (FMG) 迭代的计算成本，并将求解器应用于多个基准问题。通过参数研究，我们建议实现稳定等阶和 Taylor-Hood 离散化的 TME 的配置。相关计算内核的出色节点级性能通过屋顶线分析呈现。最后，我们展示了高达 147 美元的弱和强可扩展性，