# Comparison Between Algebraic and Matrix-free Geometric Multigrid for a Stokes Problem on Adaptive Meshes with Variable Viscosity

## TL;DR

This paper compares algebraic and geometric multigrid methods for solving large-scale, adaptive mesh-based Stokes problems with variable viscosity, demonstrating the robustness and scalability of a matrix-free geometric multigrid approach.

## Contribution

It introduces a matrix-free geometric multigrid v-cycle for adaptive meshes and compares its performance and robustness against algebraic multigrid in large-scale mantle convection simulations.

## Key findings

- GMG shows greater robustness than AMG with increasing problem size.
- GMG scales efficiently up to 114,688 cores and 217 billion unknowns.
- The matrix-free GMG method outperforms AMG in adaptive, large-scale settings.

## Abstract

Problems arising in Earth's mantle convection involve finding the solution to Stokes systems with large viscosity contrasts. These systems contain localized features which, even with adaptive mesh refinement, result in linear systems that can be on the order of 10^9 or more unknowns. One common approach for preconditioning to the velocity block of these systems is to apply an Algebraic Multigrid (AMG) v-cycle (as is done in the ASPECT software, for example), however, with AMG, robustness can be difficult with respect to problem size and number of parallel processes. Additionally, we see an increase in iteration counts with adaptive refinement when using AMG. In contrast, the Geometric Multigrid (GMG) method, by using information about the geometry of the problem, should offer a more robust option. Here we present a matrix-free GMG v-cycle which works on adaptively refined, distributed meshes, and we will compare it against the current AMG preconditioner (Trilinos ML) used in the ASPECT software. We will demonstrate the robustness of GMG with respect to problem size and show scaling up to 114688 cores and $217$ billion unknowns. All computations are run using the open source, finite element library deal.ii.