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Problems arising in Earth's mantle convection involve finding the solution toStokes systems with large viscosity contrasts. These systems contain localizedfeatures which, even with adaptive mesh refinement, result in linear systemsthat can be on the order of 10^9 or more unknowns. One common approach forpreconditioning to the velocity block of these systems is to apply an AlgebraicMultigrid (AMG) v-cycle (as is done in the ASPECT software, for example),however, with AMG, robustness can be difficult with respect to problem size andnumber of parallel processes. Additionally, we see an increase in iterationcounts with adaptive refinement when using AMG. In contrast, the GeometricMultigrid (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 AMGpreconditioner (Trilinos ML) used in the ASPECT software. We will demonstratethe robustness of GMG with respect to problem size and show scaling up to114688 cores and $217$ billion unknowns. All computations are run using theopen source, finite element library deal.ii.

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