Textbook efficiency: massively parallel matrix-free multigrid for the Stokes system

TL;DR

This paper develops a highly efficient, parallel multigrid solver for the Stokes system using hierarchical hybrid grids and matrix-free methods, achieving optimal performance and scalability for large-scale problems.

Contribution

It introduces a monolithic multigrid solver based on textbook multigrid efficiency principles, extended to higher-order finite elements and parallel computing, for the first time applied to the Stokes system.

Findings

Achieves textbook multigrid efficiency for Stokes problems.

Demonstrates scalability up to 147,456 processes.

Successfully solves systems with over 3.6 trillion unknowns.

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 (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.