Wavenumber explicit analysis for time-harmonic Maxwell equations in a spherical shell and spectral approximations. (22nd May 2017) - Ma, Lina; Shen, Jie; Wang, Li-Lian; Yang, Zhiguo
Abstract: 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 $ε$ -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 $ε$ -perturbed nonspherical scatterers by representing the resulting field in $ε$ -power series expansion with coefficients being spherical shell electric fields.