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

摘要

We present several versions of Regularized Combined Field Integral Equation(CFIER) formulations for the solution of three dimensional frequency domainelectromagnetic 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 electricdipole layer potentials that involves a composition of the latter type ofboundary layers with regularizing operators. The regularizing operators are oftwo types: (1) modified versions of electric field integral operators withcomplex wavenumbers, and (2) principal symbols of those operators in the senseof pseudodifferential operators. We show that the boundary integral operatorsthat enter these CFIER formulations are Fredholm of the second kind, andinvertible with bounded inverses in the classical trace spaces ofelectromagnetic scattering problems. We present a spectral analysis of CFIERoperators with regularizing operators that have purely imaginary wavenumbersfor spherical geometries. Under certain assumptions on the coupling constantsand the absolute values of the imaginary wavenumbers of the regularizingoperators, we show that the ensuing CFIER operators are coercive for sphericalgeometries. These properties allow us to derive wavenumber explicit bounds onthe condition numbers of certain CFIER operators that have been proposed in theliterature. When regularizing operators with complex wavenumbers with non-zeroreal parts are used, we show numerical evidence that those complex wavenumberscan be selected in a manner that leads to CFIER formulations whose conditionnumbers can be bounded independently of frequency for spherical geometries.