Maxwell's equations in a bounded Debye medium are formulated in terms of the standard
partial differential equations of electromagnetism with a Volterra-type history dependence
of the polarization on the electric field intensity. This leads to Maxwell's equations with
memory. We make a correspondence between this type of constitutive law and the hereditary
integral constitutive laws from linear viscoelasticity, and we are then able to apply known results from viscoelasticity theory to this Maxwell system. In particular, we can
show long-time stability by shunning Gronwall's lemma and estimating the history kernels
more carefully by appeal to the underlying physical fading memory. We also give a fully
discrete scheme for the electric field wave equation and derive stability bounds which are
exactly analogous to those for the continuous problem, thus providing a foundation for
long-time numerical integration. We finish by also providing error bounds for which the
constant grows, at worst, linearly in time (excluding the time dependence in the norms
of the exact solution). Although the first (mixed) finite element error analysis for the
Debye problem was given by Li (2007), this seems to be the first time sharp constants have been given for
this problem.
Superconvergence and Extrapolation Analysis of a Nonconforming Mixed Finite Element Approximation for Time-Harmonic Maxwell’s Equations
