Multigrid preconditioning for time-harmonic
                          Maxwell's equations in 3D

                        D. A. Aruliah and U. M. Ascher
                       Department of Computer Science,
                       University of British Columbia,
                           Vancouver, BC, V6T 1Z4,
                                   Canada.
                    dhavide@cs.ubc.ca and ascher@cs.ubc.ca

                                   Abstract

We consider the rapid simulation of three-dimensional electromagnetic problems
in parameter regimes where the quasi-static approximation applies, the
permeability is constant, the conductivity may vary significantly, and the
range of frequencies is moderate.  The difficulties encountered include
handling solution discontinuities across interfaces and accelerating
convergence of traditional iterative methods for the solution of the linear
systems of algebraic equations that arise when discretizing Maxwell's
equations in the frequency domain.

In previous works, we used a potential-current formulation combined with a
pressure-Poisson-like differentiation in order to pose the differential system
in a weakly-coupled elliptic form.  A finite-volume discretization on a
staggered grid was subsequently employed to arrive at a large, sparse, linear
system of equations with a block-diagonally dominant structure.  A
preconditioned BiCGStab iteration, where the diagonal blocks were
approximately inverted by an ILU-decomposition, was found to be particularly
efficient for solving the latter system.

In this paper, we first provide a Fourier analysis to show that inverting the
diagonal blocks of the reformulated system indeed yields an efficient
preconditioner.  We then consider replacing the full inversion of these
diagonal blocks by just one multigrid cycle, and extend the Fourier analysis
to show that the condition number of the resulting preconditioned matrix is
still bounded independent of the grid.

Finally, we present numerical examples for more realistic situations involving
large variations in conductivity (i.e.  jump discontinuities).
Block-preconditioning with one multigrid cycle using J. Dendy's BOXMG solver
is found to yield convergence in very few iterations, apparently independent
of the grid size.  The experiments show that the somewhat restrictive
assumptions of the Fourier analysis do not prohibit it from describing the
essential local behaviour of the preconditioned operator under consideration.
A very efficient, practical solver is obtained.

(Submitted Oct. 16th, 2000)