Textbook Efficiency: Massively Parallel Matrix-Free Multigrid for the Stokes System
Nils Kohl, Ulrich Rüde
Abstract
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, which is extended to higher-order finite element discretizations, and a corresponding matrix-free implementation. The computational cost of the full multigrid iteration is quantified, and the solver is applied to multiple benchmark problems. Through a parameter study, we suggest configurations that achieve theoretical TME for both stabilized equal order and Taylor--Hood discretizations. We then identify and quantify the gaps to achieve TME for parallel multigrid implementations in practice. 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.