"Dendro: A parallel geometric multigrid method for finite element
calculations on octree meshes  with billions of unknowns"

Rahul Sampath
Ph.D. Candidate
Advisor: Professor George Biros
Mechanical Engineering and Applied Mechanics
University of Pennsylvania

Abstract
We present Dendro, a suite of parallel algorithms for the discretization and numerical solution of second-order elliptic partial differential equations using octree meshes. A characteristic of octree meshes is that they contain 'hanging' nodes. Dendro uses a novel technique to handle these 'hanging' nodes and construct conforming trilinear finite element discretizations. To our knowledge, Dendro is the first parallel, octree-based, matrix-free, geometric multigrid solver for finite element discretizations. We present fixed-size and iso-granular scalability results for solving a Poisson problem and a linear elastostatics problem on up to 4096 CPUs on the Cray XT3 ("BigBen'') at the Pittsburgh Supercomputing Center (PSC) and the Intel 64 system ("Abe'') at the National Center for Supercomputing Applications (NCSA).  We are able to solve problems with billions of elements on thousands of processors in less than 10 minutes. Although we do not discuss  adaptive mesh refinement here, the proposed method can be used  toward that goal as it has very low setup costs.