Computational Mathematics and Algorithms

Sandia National Laboratories

P.O. Box 969, MS 9217

Livermore, CA 94551

Jonathan Hu

Computational Mathematics and Algorithms

Sandia National Laboratories

P.O. Box 969, MS 9217

Livermore, CA 94551

Greg Newman

Sandia National Laboratories

Geophysical Technology

Sandia National Laboratories

P.O. Box 969, MS 9217

Livermore, CA 94551

Pavel B. Bochev

Computational Mathematics and Algorithms

Sandia National Laboratories

P.O. Box 5800, MS 1110

Albuquerque, NM 87185-1110

Abstract

We describe a parallel algebraic multigrid method for the solution of
Maxwell's equations in the frequency domain. The underlying formulation
leverages off of an algebraic multigrid scheme for real valued Maxwell
problems. This real valued method uses distributed relaxation
and a specialized grid transfer operator. The key to this multilevel
method is the proper representation of the (**curl**,**curl**) null
space on coarse meshes. This is achieved by maintaining certain commuting
properties of the inter-grid transfers.

To adapt the real value scheme to complex arithmetic, equivalent real forms are considered. The complex operator is written as a 2 x 2 real block matrix system. The real valued multigrid algorithm can then be used to generate grid transfers which are adapted to the equivalent real form of the problem. In order to complete the scheme a smoother must also be adapted to address the equivalent real form. We will show how application of the distributed relaxation idea on the equivalent real form matrix leads to a nice decoupling of the problem. To complete the method, complex polynomial smoothers are developed for use within the distributed relaxation process. While some care is required to develop the polynomials, they work well in parallel and avoid difficulties associated with parallel Gauss-Seidel.

Numerical experiments are presented for some 3D problems arising in geophysical subsurface imaging. The experiments illustrate the efficiency of the approach on various parallel machines in terms of both convergence and parallel speed-up.