 
Summary: Parallel Multigrid Solver for 3D Unstructured Finite Element
Problems
Mark Adams \Lambda James W. Demmel y
Abstract
Multigrid is a popular solution method for the system of linear algebraic equations that arise from PDEs discretized
with the finite element method. The application of multigrid to unstructured grid problems, however, is not well de
veloped. We discuss a method, that uses many of the same techniques as the finite element method itself, to apply
standard multigrid algorithms to unstructured finite element problems. We use maximal independent sets (MISs) as a
mechanism to automatically coarsen unstructured grids; the inherent flexibility in the selection of an MIS allows for
the use of heuristics to improve their effectiveness for a multigrid solver. We present parallel algorithms, based on ge
ometric heuristics, to optimize the quality of MISs and the meshes constructed from them, for use in multigrid solvers
for 3D unstructured problems. We conduct scalability studies that demonstrate the effectiveness of our methods on a
problem in large deformation elasticity and plasticity of up to 40 million degrees of freedom on 960 processor IBM
PowerPC 4way SMP cluster with about 60% parallel efficiency.
Key words: unstructured multigrid, parallel sparse solvers, parallel maximal independent sets
1 Introduction
This work is motivated by the success of the finite element method in effectively simulating complex physical systems
in science and engineering, coupled with the wide spread availability of ever more powerful parallel computers, which
has lead to the need for efficient equation solvers for implicit finite element applications. Finite element matrices are
often poorly conditioned this fact has made the use of direct solvers popular as their solve time is unaffected by the
