Efficient Distributed Matrix-free Multigrid Methods on Locally Refined Meshes for FEM Computations

Abstract

This work studies three multigrid variants for matrix-free finite-element computations on locally refined meshes: geometric local smoothing, geometric global coarsening (both h-multigrid), and polynomial global coarsening (a variant of p-multigrid). We have integrated the algorithms into the same framework—the open source finite-element library deal.II—, which allows us to make fair comparisons regarding their implementation complexity, computational efficiency, and parallel scalability as well as to compare the measurements with theoretically derived performance metrics. Serial simulations and parallel weak and strong scaling on up to 147,456 CPU cores on 3,072 compute nodes are presented. The results obtained indicate that global-coarsening algorithms show a better parallel behavior for comparable smoothers due to the better load balance, particularly on the expensive fine levels. In the serial case, the costs of applying hanging-node constraints might be significant, leading to advantages of local smoothing, even though the number of solver iterations needed is slightly higher. When using p- and h-multigrid in sequence (hp-multigrid), the results indicate that it makes sense to decrease the degree of the elements first from a performance point of view due to the cheaper transfer.