 
Summary: Heuristics for the Automatic Construction of Coarse Grids in
Multigrid Solvers for Finite Element Problems in Solid Mechanics
Mark Adams \Lambda
April 20, 1999
Abstract
Multigrid is a popular solution method for the set 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 developed. 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), like many ``algebraic'' multigrid methods, as a heuristic 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 heuristics and
algorithms to optimize the quality of MISs, and the meshes constructed from them, for use in multigrid
solvers for unstructured problems in solid mechanics. We present numerical results that demonstrate the
effectiveness of the our methods on several model problems in linear elasticity.
Key words: maximal independent sets, multigrid, unstructured meshes, parallel solvers
1 Introduction
This work is motivated by the success of the finite element method in simulating complex physical systems
in science and engineering, coupled with the wide spread availability of ever more powerful computers, which
has lead to the need for efficient equation solvers for implicit finite element applications. Finite element
