Maxwell's equations are a system of partial differential equations which, coupled with constitutive laws that model the response of a given medium to the electromagnetic field, govern electromagnetic wave propagation in the given medium. In this talk we will focus primarily on the Debye model which governs the evolution of the polarization forced by the electric field in a simple relaxing dielectric. The combined Maxwell-Debye (MD) model is asymptotically singularly perturbed and displays complex dispersive and diffusive wave behavior.
We present a dispersion Analysis of the hp-finite element method (hp-FEM) applied to the MD model in 1 spatial dimension. We derive an analytic form for the numerical dispersion error of hp-FEM applied to the MD system and compute both h and
p convergence tests for a broad range of frequencies of interest. Our work indicates that h or p refinement alone will not have satisfactory convergence properties for the MD problem.