Wavenumber explicit analysis for time-harmonic Maxwell equations in a spherical shell and spectral approximations

This article is devoted to wavenumber explicit analysis of the electric field satisfying the second-order time-harmonic Maxwell equations in a spherical shell and, hence, for variant scatterers with -perturbation of the inner ball radius. The spherical shell model is obtained by assuming that the forcing function is zero outside a circumscribing ball and replacing the radiation condition with a transparent boundary condition involving the capacity operator. Using the divergence-free vector spherical harmonic expansions for two components of the electric field, the Maxwell system is reduced to two sequences of decoupled one-dimensional boundary value problems in the radial direction. The reduced problems naturally allow for truncated vector spherical harmonic spectral approximation of the electric field and one-dimensional global polynomial approximation of the boundary value problems. We analyse the error in the resulting spectral approximation for the spherical shell model. Using a perturbation transformation, we generalize the approach for -perturbed nonspherical scatterers by representing the resulting field in -power series expansion with coefficients being spherical shell electric fields.