The generalization of a two-dimensional spatial spectral volume integral equation to a three-dimensional spatial spectral integral equation formulation for electromagnetic scattering from dielectric objects in a stratified dielectric medium is explained. In the spectral domain, the Green function, contrast current density, and scattered electric field are represented on a complex integration manifold that evades the poles and branch cuts that are present in the Green function. In the spatial domain, the field-material interactions are reformulated by a normal-vector field approach, which obeys the Li factorization rules. Numerical evidence is shown that the computation time of this method scales as on the number of unknowns. The accuracy of the method for three numerical examples is compared to a finite element method reference.

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The marching-on-in-time electric field integral equation (MOT-EFIE) and the marchingon-in-time time differentiated electric field integral equation (MOT-TDEFIE) are based on Rao-Wilton-Glisson (RWG) spatial discretization. In both formulations we employ the Dirac-delta temporal testing functions; however, they differ in temporal basis functions, i.e., hat and quadratic spline basis functions. These schemes suffer from linear-in-time solution instability. We analyze the corresponding companion matrices using projection matrices and prove mathematically that each independent solenoidal current density corresponds to a Jordan block of size two. In combination with Lidskii-Vishik-Lyusternik perturbation theory we find that finite precision causes these Jordan block eigenvalues to split, and this is the root cause of the instability of both schemes. The split eigenvalues cause solutions with exponentially increasing magnitudes that are initially observed as constant and/or linear-in-time, yet these become exponentially increasing at discrete time steps beyond the inverse square root of the error due to finite precision, i.e., approximately after one hundred million discrete time steps in double precision arithmetic. We provide numerical evidence to further illustrate these findings.

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