A Highly Scalable Multilevel Schwarz Method with Boundary Geometry Preserving Coarse Spaces for 3D Elasticity Problems on Domains with Complex Geometry

Abstract

We consider overlapping Schwarz algorithms for solving linear and nonlinear systems of equations arising from the finite element discretization of elasticity problems on unstructured meshes in three dimensions. The parallel scalability of Schwarz methods is determined almost completely by how the coarse space is constructed. In this paper, we introduce a low cost, boundary geometry preserving coarse mesh that shares the same boundary geometry with the fine mesh but has a better scalability than the coarse mesh obtained by uniformly coarsening the fine mesh. A new coarsening algorithm and a partitioning strategy are developed. We numerically show that a multilevel Schwarz method with the new coarse spaces is highly scalable, in terms of the total compute time, for solving some three-dimensional linear and nonlinear elasticity equations discretized on unstructured meshes with several hundreds of millions of unknowns and on a supercomputer with over 10,000 processor cores.