High-order, fast, and well-conditioned algorithms for the solution of three-dimensional acoustic and electromagnetic scattering problems
We present a novel computational methodology aimed at overcoming the aforementioned difficulties. At the heart of our approach are integral equation formulations that exhibit excellent spectral properties. In the case of scattering from perfectly conducting structures, and just as the classical Combined Field Integral Equation (CFIE), our equations result from representations of the scattered fields as a combination of magnetic- and electric-dipole distributions on the surface of the scatterer. In contrast with the classical equations, however, our electric-dipole operators involve use of certain types of regularizing operators whose design is based on the pseudodifferential calculus on manifolds. We call the resulting equations Regularized Combined Field Integral Equations (CFIE-R). Unlike the CFIE, the CFIE-R are well-conditioned equations; careful selection of coupling parameters, further, yields CFIE-R operators with excellent spectral distributions---with closely clustered eigenvalues---so that small numbers of iterations suffice to solve the corresponding equations by means of Krylov subspace iterative solvers such as GMRES. We present a high-order Nystrom approach based on use of partitions of unity and high-order integration schemes that produces high-order algorithms for acoustic and electromagnetic scattering problems. A variety of numerical results demonstrate that, for a given accuracy, the new equations can give rise to order-of-magnitude reductions in computational costs over those resulting from previous approaches.