Comparison between algebraic and matrix-free geometric multigrid for a Stokes problem on adaptive meshes with variable viscosity

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, we find that AMG is lacking robustness with respect to problem size and number of parallel processes. Additionally, we see an increase in iteration counts with 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(1) software. We will demonstrate the robustness of GMG with respect to problem size and show scaling up to 114,688 cores and 217 billion unknowns. All computations are run using the open-source, finite element library deal.II.(2)