A Parallel Geometric Multigrid Method for Finite Elements on Octree Meshes

In this article, we present a parallel geometric multigrid algorithm for solving variable-coefficient elliptic partial differential equations (PDEs) on the unit box (with Dirichlet or Neumann boundary conditions) using highly nonuniform, octree-based, conforming finite element discretizations. Our octrees are 2:1 balanced, that is, we allow no more than one octree-level difference between octree-nodes that share a face, edge, or vertex. We describe a parallel algorithm whose input is an arbitrary 2:1 balanced fine-grid octree and whose output is a set of coarser 2:1 balanced octrees that are required in the multigrid scheme. Also, we derive matrix-free schemes for the discretized finite element operators and the intergrid transfer operations. The overall scheme is second-order accurate, for sufficiently smooth right-hand sides, and its complexity, for nearly uniform trees, is O( N np log N np ) + O(np lognp), where N is the number of octree-nodes, and np is the number of processors. Our implementation uses the Message Passing Interface (MPI) standard. We present numerical experiments for the Laplace and Navier (linear elasticity) operators that demonstrate portability and scalability of our method on a variety of NSF TeraGrid platforms: on the Cray XT3 MPP system “Bigben” at the Pittsburgh Supercomputing Center (PSC), the Intel 64 Linux Cluster “Abe” at the National Center for Supercomputing Applications (NCSA), and the Sun Constellation Linux Cluster “Ranger” at the Texas Advanced Computing Center (TACC). Our largest run was a highly-nonuniform, 8billion-unknown, elasticity calculation on 32,000 processors. Our implementation is publically available in the Dendro library, which is built on top of the PETSc library from Argonne National Laboratory.