Heuristics for the Automatic Construction of Coarse Grids in Multigrid Solvers for Finite Element Problems in Solid Mechanics

Multigrid is a popular solution method for the set of linear algebraic equations that arise from PDEs discretized with the nite 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 nite element method itself, to apply standard multigrid algorithms to unstructured nite element problems. We use maximal independent sets (MISs), like many \algebraic" multigrid methods, as a heuristic to automatically coarsen unstructured grids. The inherent exibility in the selection of an MIS allows for the use of heuristics to improve their eeectiveness 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 eeectiveness of the our methods on several model problems in linear elasticity.