Parallel Multilevel Solvers for 3D Problems

Michael Jung and Michael Thess

Fakultät für Mathematik Technische Universität Chemnitz-Zwickau

D–09107 Chemnitz

            e–mail: dr.michael.jung@mathematik.tu-chemnitz.de m.thess@mathematik.tu-chemnitz.de

We present a uniform parallelization strategy for iterative solvers, as e.g., preconditioned conjugate gradient methods with hierarchical preconditioners (Yserentant, BPX) and multigrid methods. The parallelization concept is based on a non-overlapping domain decomposition data structure such that our implementations are well-suited for MIMD parallel machines under message-passing. We discuss in detail the parallelization of a multigrid algorithm. Especially, we propose well-parallelizable smoothers of Gauss-Seidel type. We apply the Yserentant preconditioner to the elasticity problem of smooth, thin shells of an arbitrary form. The problem is formulated by using Koiter’s equations of equilibrium for a homogeneous and isotropic thin shell, clamped on a part of its boundary and loaded by a resultant on its middle surface. We describe an implementation of the BPX preconditioner for the same mechanical problem. This implementation is based on a transformation of an existing hierarchical list into a corresponding BPX list. Finally, we compare the different parallel iterative solvers applied to 2D, 2\frac{1}{2}D, and 3D problems, and we show some advantages and disadvantages of these methods.