Parallel algebraic multigrid based on subdomain blocking

Abstract

The algebraic multigrid (AMG) approach provides a purely algebraic means to tackle the efficient solution of systems of equations posed on large unstructured grids, in 2D and 3D. While sequential AMG has been used for increasingly large problems (with several million unknowns), its application to even larger applications requires a parallel version. Since, in contrast to geometric multigrid, the hierarchy of coarser levels and the related operators develop dynamically during the setup phase of AMG, a direct parallelization is very complicated. Moreover, a “naive” parallelization would, in general, require unpredictable and highly complex communication patterns which seriously limit the achievable scalability, in particular of the costly setup phase. In this paper, we consider a classical AMG variant which has turned out be highly robust and efficient in solving large systems of equations corresponding to elliptic PDEs, discretized by finite differences or finite volumes. Based on a straightforward partitioning of variables (using one of the available algebraic partitioning tools such as Metis), a parallelization approach is proposed which minimizes the communication without sacrificing convergence in complex situations. Results will be presented for industrial CFD and oil-reservoir simulation applications on distributed memory machines, including PC-clusters.