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Maxwell's equations in a bounded Debye medium are formulated in terms of the standardpartial differential equations of electromagnetism with a Volterra-type history dependenceof the polarization on the electric field intensity. This leads to Maxwell's equations withmemory. We make a correspondence between this type of constitutive law and the hereditaryintegral constitutive laws from linear viscoelasticity, and we are then able to apply known results from viscoelasticity theory to this Maxwell system. In particular, we canshow long-time stability by shunning Gronwall's lemma and estimating the history kernelsmore carefully by appeal to the underlying physical fading memory. We also give a fullydiscrete scheme for the electric field wave equation and derive stability bounds which areexactly analogous to those for the continuous problem, thus providing a foundation forlong-time numerical integration. We finish by also providing error bounds for which theconstant grows, at worst, linearly in time (excluding the time dependence in the normsof the exact solution). Although the first (mixed) finite element error analysis for theDebye problem was given by Li (2007), this seems to be the first time sharp constants have been given forthis problem

Finite Element Approximation of Maxwell's Equations with Debye Memory