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.propagation of waves across large numbers of volumetric elements [9]. An important computational alternative to finite-difference and finite-element approaches is found in boundary integral methods. Numerical methods based on integral formulations of scattering problems enjoy a number of attractive properties as they formulate the problems on lower-dimensional, bounded computational domains and capture intrinsically the outgoing character of scattered waves. Thus, on account of the dimensional reduction and associated small discretizations (significantly smaller than the discretizations required by volumetric finite-element or finite-difference approximations), in conjunction with available fast solvers [12, 15-17, 19, 20, 26, 27, 50, 51, 55], numerical algor...

A Calderón multiplicative preconditioner for the electromagnetic Poincaré–Steklov operator of a heterogeneous domain with scattering applications

“…This implies that T k regularizes P, as indicated by the following corollary, in correspondence with the same property for the PS operator of a homogeneous domain [5,17].…”

“…, with Π ∇Γ and Π−→ curlΓ the orthogonal projectors associated to the Helmholtz decomposition of H − 1 2 (div Γ , Γ) and defined in [17].…”

“…This has the additional advantage that C is also a preconditioner for T, which will be exploited in the construction of an efficient preconditioner for the proposed hybrid formulation in Section 4. In line with [14] and with contributions on similar preconditioning strategies for other operators [16][17][18][19][20][21]49, 50], we call C a Calderón multiplicative preconditioner. Similar to [14], we choose…”

“…Then, we show that P is regularized by the so-called electric field integral operator T k , further defined in (10), meaning that their product can be written as a compact perturbation of a well-conditioned operator. The self-regularizing property of T k itself, i.e., the fact that T 2 k is a compact perturbation of the identity operator (up to a constant multiplicative factor) on domains with smooth boundaries, was the source of inspiration for efficient preconditioners in BEMs for various integral equations, such as the electric field integral equation (EFIE) [13][14][15], the combined field integral equations of [16], the regularized combined field integral equations (CFIER) of [17], a single source CFIE for dielectric scattering [18], the Poggio-MillerChan-Harrington-Wu-Tsai (PMCHWT) equation [19], an electric current formulation [20] and the EFIE in a layered medium [21]. We emphasize that the cited BEMs are restricted to piecewise homogeneous scatterers, and are not applicable to general heterogeneous domains.…”

“…This paper is structured as follows: in Section 2 we introduce the electromagnetic Poincaré-Steklov operator P of a heterogeneous domain and, in line with a similar and well-known result for the electromagnetic PS operator of a homogeneous domain [5,17], we derive a decomposition in terms of T k , which implies that T k regularizes P, in a Sobolev space framework, as already remarked in [17]. Inspired by these properties, we construct a Calderón multiplicative preconditioner (CMP) for a Schur complement finite element discretization of P and investigate its effect on the singular value distribution of the discretized PS operator in Section 3.…”