On the Use of Schwarz Smoothing in AMG

Ulrike Meier Yang

Center for Applied Scientific Computing
Lawrence Livermore National Laboratory
Box 808, L-560
Livermore, CA 94551


The need to solve increasingly larger linear systems, with hundreds of millions or billions of unknowns, has necessitated the use of massively parallel computers and the investigation of scalable linear solvers, such as multigrid. For unstructured grid problems, an attractive choice is algebraic multigrid (AMG). Although AMG is a very effective method for many applications, however for some applications, e.g. structural elasticity problems where the governing equations are a system of PDEs with the unknowns being the displacements in each coordinate direction, the choice of conventional smoothers such as Jacobi or Gauss-Seidel within AMG is not sufficient to achieve good convergence.

In this talk, we investigate the use of the Schwarz alternating method as a smoother on the finest levels of AMG. Preliminary experiments show promising results for such problems as mentioned above. We consider both multiplicative and additive variants of the Schwarz method. The use of a multicoloring technique for parallelization of the multiplicative variant is considered. The choice of domains, efficiency and convergence behavior are discussed, and numerical results are presented.

*This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract number W-7405-Eng-48.