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

arXiv: https://arxiv.org/abs/2010.13513 · HTML: https://mlsanity.com/paper/2010.13513/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2020.

## Built on by (most cited first, 1 of 1)

- [Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms](https://mlsanity.com/paper/2603.17641/) (2026)

## Abstract (arXiv)

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 × 1012 (trillion) unknowns.