Unstructured Forests of Octrees

Abstract—We present a parallel multigrid method for solving variable-coefficient elliptic partial differential equations on arbi-trary geometries using highly adapted meshes. Our method is designed for meshes that are built from an unstructured hexa-hedral macro mesh, in which each macro element is adaptively refined as an octree. This forest-of-octrees approach enables us to generate meshes for complex geometries with arbitrary levels of local refinement. We use geometric multigrid (GMG) for each of the octrees and algebraic multigrid (AMG) as the coarse grid solver. We designed our GMG sweeps to entirely avoid collectives, thus minimizing communication cost. We present weak and strong scaling results for the 3D variable-coefficient Poisson problem that demonstrate high parallel scal-ability. As a highlight, the largest problem we solve is on a non-uniform mesh with 100 billion unknowns on 262,144 cores of NCCS’s Cray XK6 “Jaguar”; in this solve we sustain 272 TFlops/s. I