Parallel Multigrid Solvers for 3D Unstructured Finite Element Problems in Large Deformation Elasticity and Plasticity

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) as a mechanism 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 parallel algorithms, based on geometric heuristics, to optimize the quality of MISs and the meshes constructed from them, for use in multigrid solvers for 3D unstructured problems. We discuss parallel issues of our algorithms, multigrid solvers in general, and the parallel nite element application that we have developed to test our solver on challenging problems. We show that our solver, and parallel nite element architecture, does indeed scale well, with test problems in 3D large deformation elasticity and plasticity, with 40 million degree of freedom problem on 240 IBM 4-way SMP PowerPC nodes.