Current methods rely on iterative linear system solution methods such as Conjugate Gradients. For some types of scatterers, including inlets, with resonance-regime features, the iteration count required for the solution to converge becomes large, leading to high computational cost. Some researchers have proposed techniques to overcome this, such as multigrid or near neighbor preconditioning, which perform well for other classes of PDEs.

Dr. Warnick's current research into the spectrum of surface integral operators shows that the difficulty in solving the scattering problem grows as the scatterer becomes large due to long-range coupling effects, which causes local methods such as near-neighbor preconditioning and multigrid to break down for Maxwell's equations. A new type of spectral multigrid is currently being developped. This multigrid preconditions the long-range interactions inherent in the surface integral operator, with the intent of developing an optimal surface integral equation solver for which computational cost grows linearly with problem size.

The first Figure compares the convergence history of GMRES in work units to that of spectral multigrid for TM scattering from an inlet with a depth of about 20 wavelengths.

The second Figure shows the cost in arbitrary units required to solve scattering from random polygonal scatterers to a fixed residual error using spectral multigrid. The linear trend shows that at least in the range of problem sizes shown, overall cost scales as N.

The above work is a collaboration between Dr. Karl Warnick and Prof. Weng Cho Chew. Please send suggestions, comments, and inquiries to: warnick@uiuc.edu.