Convergence of Multigrid Applied to the PML-Electromagnetic Scattering Problem

Joseph E. Pasciak

Email: pasciak@tamu.edu
Postal address: Texas A+M University, College Station TX 77845

In this talk, I will consider the behavior of multigrid methods applied to the iterative solution of a system of equations which result when a PML truncation technique is applied to discretize a time harmonic electromagnetic scattering problem. The PML technique introduces a "smooth'' complex tensor coefficient into the problem. To start the investigation, we first consider the case of a smooth coefficient in divergence form.  In this case, we are able to do a perturbation analysis (comparing with the constant coefficient operator) and prove uniform rates of iterative convergence for multigrid algorithms with point smoothing.  This is also verified experimentally.  In contrast, the PML problem introduces strong non-diagonal interaction into the differential operator.  As a model, we consider a two-dimensional problem which results from a radially symmetric problem in three dimensions. Here, the multigrid algorithm with point smoothing fails dismally. In contrast, a multigrid algorithm with line smoothing along the remaining angular variable is quite effective when accelerated with GMRES.