Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
Costabel, Martin
Formulating the scattering of electromagnetic waves by a penetrable object in the frequency domain via a volume integral equation is quite popular with physicists. There exist even widely used numerical codes based on this formulation (“discrete dipole approximation”), but the mathematical analysis is far from complete. The volume integral equation, sometimes called Lippmann-Schwinger equation, is a strongly singular integral equation, and it still poses interesting mathematical problems of a basic nature, even for the simple case of piecewise constant coefficients. In the talk, results about the essential spectrum of the volume integral operator will be presented, with emphasis on methods that work for non-smooth (Lipschitz) boundaries. It turns out that there is a way to transform the strongly singular integral equation into an equivalent coupled system of weakly singular volume integral equations and boundary integral equations such that the question of Fredholmness is reduced to that of well-known scalar boundary integral operators. A previous method of reduction to a coupled system of volume/boundary integral equations [1] led to a system of boundary integral equations that allowed a simple analysis only for smooth domains. [1] M. Costabel, E. Darrigrand, H. Sakly: On the essential spectrum of the volume integral operator in electromagnetic scattering. C. R. Acad. Sci. Paris, Ser. I bf 350 (2012) 193--197.