Well-conditioned boundary integral equation formulations for the solution of high-frequency electromagnetic scattering problems

Abstract

We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations for the solution of three dimensional frequency domain electromagnetic scattering problems with Perfectly Electric Conducting (PEC) boundary conditions. Just as in the Combined Field Integral Equations (CFIE), we seek the scattered fields in the form of a combined magnetic and electric dipole layer potentials that involves a composition of the latter type of boundary layers with regularizing operators. The regularizing operators are of two types: (1) modified versions of electric field integral operators with complex wavenumbers, and (2) principal symbols of those operators in the sense of pseudodifferential operators. We show that the boundary integral operators that enter these CFIER formulations are Fredholm of the second kind, and invertible with bounded inverses in the classical trace spaces of electromagnetic scattering problems. We present a spectral analysis of CFIER operators with regularizing operators that have purely imaginary wavenumbers for spherical geometries—we refer to these operators as Calderón–Ikawa CFIER. Under certain assumptions on the coupling constants and the absolute values of the imaginary wavenumbers of the regularizing operators, we show that the ensuing Calderón–Ikawa CFIER operators are coercive for spherical geometries. These properties allow us to derive wavenumber explicit bounds on the condition numbers of Calderón–Ikawa CFIER operators. When regularizing operators with complex wavenumbers with non-zero real parts are used—we refer to these operators as Calderón-Complex CFIER, we show numerical evidence that those complex wavenumbers can be selected in a manner that leads to CFIER formulations whose condition numbers can be bounded independently of frequency for spherical geometries. In addition, the Calderón-Complex CFIER operators possess excellent spectral properties in the high-frequency regime for both convex and non-convex scatterers. We provide numerical evidence that our solvers based on fast, high-order Nyström discretization of these equations converge in very small numbers of GMRES iterations, and the iteration counts are virtually independent of frequency for several smooth scatterers with slowly varying curvatures.