Well-posedness of domain integral equations for a dielectric object in homogeneous background

Abstract

An analysis of the mapping properties of three commonly used domain integro-differential operators for electromagnetic scattering by an inhomogeneous dielectric object embedded in a homogeneous background is presented in the Laplace domain. The corresponding three integro-differential equations are shown to be equivalent and well-posed under finite-energy conditions. The analysis allows for non-smooth changes, including edges and corners, in the dielectric properties. The results are obtained via the Riesz-Fredholm theory, in combination with the Helmholtz decomposition and the Sobolev embedding theorem.

Fluctuating volume-current formulation of electromagnetic fluctuations in inhomogeneous media: Incandescence and luminescence in arbitrary geometries

“…151 Specifically, basis functions must span the function space of the unknowns and testing functions must span the dual space of the range of the corresponding VIE operator. 152,153 DDA respects neither of these, and as a consequence its applicability is largely limited to situations involving scattering in structures with small index contrasts and weakly polarizable media, 145 beyond which it can lead to a number of severe convergence and accuracy problems. 134 (Note that DDA also makes a number of other approximations that break down in geometries involving wavelengthscale objects, cf.…”

Gaps in present discretization schemes for domain integral equations

“…For instance, by considering only smoothly varying dielectric profiles [1] or only lossy media [2,3,4,5]. In our research, we have been able to relax both these conditions for the case of a homogeneous background [6]. In particular, we have obtained well-posedness and equivalence results for three domain integral equations for 3D dielectric profiles with discontinuities, including a finite number of edges and corners.…”

SURFACE AND VOLUME INTEGRAL EQUATION METHODS FOR TIME-HARMONIC SOLUTIONS OF MAXWELL'S EQUATIONS (Invited Paper)

“…A wrong choice of these functions may deteriorate the accuracy and lead to non-converging solutions. In order to guarantee that the MoM solution converges in the norm of the numerical solution space, with the testing procedure of (46), requires finding the basis functions from the finite dimensional subspace of the domain of an integral operator and the testing functions from the dual of the range space of the operator [91,92]. This leads to conforming discretization strategies.…”

“…To guarantee that the volume formulations are well-posed, the material parameters should be bounded from both below and above [91,92]. For example, the electric permittivity should satisfy…”

“…There is both theoretical [91,92] and numerical [57,100] evidence that the VIEs, in particular the one written in terms of the volume currents, JMVIE, when discretized correctly, gives stable and accurate solutions on a wide range of material parameters and frequencies, and is also free of the dense-mesh breakdown. Additional regularization, e.g., with Helmholtz decomposition [100], may still be needed to reduce the number of iterations for high contrast materials.…”