Recent Improvements in Parallel Algebraic Multigrid for Solving Maxwell's Equations

Pavel B. Bochev

Computational Mathematics and Algorithms
Sandia National Laboratories
P.O. Box 5800, MS 1110
Albuquerque, NM 87185-1110

Christopher J. Garasi

Computational Physics R&D
Sandia National Laboratories
P.O. Box 5800, MS 0819
Albuquerque, NM 87185-0819

Jonathan J. Hu

Computational Mathematics and Algorithms
Sandia National Laboratories
P.O. Box 969, MS 9217
Livermore, CA 94551

Ray S. Tuminaro

Computational Mathematics and Algorithms
Sandia National Laboratories
P.O. Box 969, MS 9217
Livermore, CA 94551

Allen C. Robinson
Computational Physics R&D
Sandia National Laboratories
P.O. Box 5800, MS 0819
Albuquerque, NM 87185-0819


Abstract

We describe our experiences with using parallel algebraic multigrid (AMG) for the solution of Maxwell's equations discretized via edge elements. A key difficulty is properly mapping the (curl,curl) operator's null space on to coarser grids via a prolongation operator that is constructed using only algebraic information (i.e. matrix coefficients and a minimal amount of element information). The AMG coarse grid correction scheme that we start with is based on the work of Stefan Reitzinger and Joachim Schöberl, and the smoother is a form of distributed relaxation. We describe modifications to the coarse grid correction and the smoother that result in improved convergence behavior.

The resulting parallel multilevel preconditioner is implemented within ML, a Sandia multilevel package that already contains similar techniques like smoothed aggregation. The resulting capability has been integrated within the Sandia supported Nevada code framework and applied to a 3D Arbitrary Lagrangian-Eulerian magnetohydrodynamics capability (ALEGRA/MHD). Repeated solution of the eddy current approximation to Maxwell's equations in a highly heterogeneous material properties environment is required in this application.

Numerical experiments are presented for various 3D model problems and an application of current interest: 3D simulations of Z-pinch implosions. The experiments illustrate the efficiency of the approach on various parallel machines in terms of both convergence and parallel speed-up.