  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.
