Smoothed Aggregation for the Helmholtz Problem

Our electromagnetics research has focused on scattering problems governed by the Helmholtz equation in 2D with non reflecting boundary conditions. In addition to the challenges posed by the discretization, the Helmholtz problem typically generates matrices that are indefinite and complex-valued and that have oscillatory algebraically smooth modes. Standard multigrid techniques are not designed to handle such matrices. The goal of the research is to develop effective multigrid preconditioners for discontinuous Galerkin (DG) discretizations of the Helmholtz problem that overcome these difficulties. We design our solvers to be algebraic and to generate a typical smoothed aggregation hierarchy of just prolongation, restriction and coarse level operators. We also use a fixed moderate amount of smoothing on each level and target poorly- to well-resolved problems.

Our work so far has dealt with DG discretizations using linear basis functions and has been fairly successful in achieving our goals. Here is an overview of our prolongation smoothing strategy. In the future, we plan to develop solvers for higher-order DG discretizations of the Helmholtz problem by leveraging our work on higher-order solvers for DG discretizations of the Poisson problem. Higher-order discretizations are more desirable for the Helmholtz problem, because phase error is greatly mitigated, when compared to lower-order discretizations.

Some of the contributions of this research are described in the publication, Smoothed Aggregation for Helmholtz Problems, and the presentation, Smoothed Aggregation Multigrid for Helmholtz Problems.

In the below image, a sample coarse annulus domain is shown. A wave is incident to the interior boundary, i.e. scatterer, and then scatters throughout the domain until hitting the exterior non reflecting boundary.

In the below image, a sample low energy planewave mode of a discrete operator is shown. One of the primary difficulties of this problem is to capture these low energy, but oscillatory modes in the coarse space. This difficulty lead to our development of an algebraic multiple coarsening strategy that allows for incorporating the rich wavelike near nullspace of the discrete operators into the coarse space.



Project supported by NSF DMS 06-12448.

Last updated September, 2009