Method of Generalized Debye Sources in Problems of EM Scattering by Conducting Bodies

In 2010, Epstein and Greengard proposed an approach to solving time harmonic Maxwell's equations that is free both from spurious resonances and the low-frequency breakdown (C. L. Epstein and L. Greengard, "Debye sources and the numerical solution of the time harmonic Maxwell equations," Commun. Pure Appl. Math., vol. LXIII, pp. 0413–0463, 2010). Their approach generalizes the Debye potentials, which are traditionally used in spherically symmetric problems, to closed surfaces of arbitrary shape. The electromagnetic field is represented via two fictitious surface charges r and q (generalized Debye sources, scalars) and two fictitious surface currents j and m (vectors). The latter, in turn, are expressed via two other scalar functions, Ψ and Ψm. In the case of perfectly conducting scatterers, the problem reduces to a system of two Fredholm integral equations of the second kind, expressing the equality to zero of the normal component of the total magnetic field and of the surface divergence of the total tangential electric field on the scatterer's surface S, and two surface differential equations—Poisson's equations with the Laplace–Beltrami operator Δτ on S: ΔτΨ = ikr, ΔτΨm = -ikq, where k is the wavenumber.