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.