Large-scale Multigrid with Adaptive Galerkin Coarsening

TL;DR

This paper introduces an adaptive multigrid method that selectively applies Galerkin coarse-grid correction to efficiently solve large-scale PDEs with highly variable coefficients, reducing memory use while ensuring robust convergence.

Contribution

It presents a novel adaptive coarse-grid correction scheme combining uniform geometric coarsening with local Galerkin approximation, optimizing memory and convergence for complex PDEs.

Findings

Successfully solved problems with 10^10 degrees of freedom.

Achieved robust convergence with large coefficient jumps.

Demonstrated efficiency on massively parallel systems.

Abstract

We propose a robust, adaptive coarse-grid correction scheme for matrix-free geometric multigrid targeting PDEs with strongly varying coefficients. The method combines uniform geometric coarsening of the underlying grid with heterogeneous coarse-grid operators: Galerkin coarse grid approximation is applied locally in regions with large coefficient gradients, while lightweight, direct coarse grid approximation is used elsewhere. This selective application ensures that local Galerkin operators are computed and stored only where necessary, minimizing memory requirements while maintaining robust convergence. We demonstrate the method on a suite of sinker benchmark problems for the generalized Stokes equation, including grid-aligned and unaligned viscosity jumps, smoothly varying viscosity functions with large gradients, and different viscosity evaluation techniques. We analytically quantify the solver's memory consumption and demonstrate its efficiency by solving Stokes problems with $10^{10}$ degrees of freedom, viscosity jumps of $10^{6}$ magnitude, and more than 100{,}000 parallel processes.