Parallel geometric multigrid preconditioner for 3D FEM in NuscaS software package

Abstract

Multigrid methods are among the fastest numerical algorithms for solving large sparse linear systems. The Conjugate Gradient method with Multigrid as a preconditioner (MGCG) features a good convergence even when the Multigrid solver itself is not efficient. The parallel FEM package NuscaS allows us to solve adaptive FEM problems with 3D unstructured meshes on parallel computers such as PC-clusters. The parallel version of the library is based on the geometric decomposition applied for computing nodes of a parallel system; the distributed-memory architecture and message-passing model of parallel programming are assumed. In our previous works, we extend the NuscaS functionality by introducing parallel adaptation of tetrahedral FEM meshes and dynamic load balancing capabilities. In this work we focus on efficient implementation of Geometric Multigrid as a parallel preconditioner for the Conjugate Gradient iterative solver used in the NuscaS package. Based on the geometric decomposition, for each level of Multigrid, meshes are partitioned and assigned to processors of a parallel architecture. Fine-grid levels are constructed by subdivision of mesh elements using the parallel 8-tetrahedra longest-edge refinement mesh algorithm, where every process keeps the assigned part of mesh on each level of Multigrid. The efficiency of the proposed implementation is investigated experimentally. © 2014 Springer-Verlag.