"Poisson Solvers for Exascale Platforms" by Paul Fischer

We discuss Poisson solver design and performance for exascale (here, GPU-based) platforms in the context of computational fluid dynamics, where factor once, solve many, applies. We are particularly focused on the range of 10³-10⁵ MPI ranks, where each rank is associated with a GPU capable of realizing ~ 1 TFLOPS. These parameters in uence design choices, particularly when running at the maximum-performance, strong-scale, limit. In this setting, we consider as examples high-order SEM/FEM discretizations involving E elements of order p (total number of grid points n~ Ep³) that are preconditioned either by sparse low-order FEM operators or with p-multigrid. In the latter context, the p = 1 coarse-grid problem has E = 10⁷-10⁹ unknowns and therefore is not small. Approaches for solving the unstructured coarse problem include algebraic
multigrid (e.g., using Hypre) or a two-level Schwarz method with a novel reduced coarse-space solve.