The fast multipole method (FMM) for electromagnetic scattering problems

The fast multipole method (FMM) developed by V. Rokhlin (1990) to efficiently solve acoustic scattering problems is modified and adapted to the second-kind-integral-equation formulation of electromagnetic scattering problems in two dimensions. The present implementation treats the exterior Dirichlet problem for two-dimensional, closed, conducting objects of arbitrary geometry. The FMM reduces the operation count for solving the second-kind integral equation from O(n³) for Gaussian elimination to O(n^4/3) per conjugate-gradient iteration, where n is the number of sample points on the boundary of the scatterer. A sample technique for accelerating convergence of the iterative method, termed complexifying k, the wavenumber, is also presented. This has the effect of bounding the condition number of the discrete system; consequently, the operation count of the entire FMM (all iterations) becomes O(n^4/3). Computational results for moderate values of ka, where a is the characteristic size of the scatterer, are given