Abstract
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.
Highlights
The potential for noninvasive electromagnetic detection of biological anomalies in the human body and of defects in structural artefacts has been recently noted by Banks et al in 1
Biological tissue is dispersive, and electromagnetic constitutive relationships represent this frequency dependence through “complex moduli” see, e.g., the data for grey and white brain tissue in 3, 4. These are manifested though hysteretic fading memory Volterra operators or as an equivalent set of evolution equations, 1, 5
The basic idea behind such a diagnostic technology is to compare the result of the transmission of electromagnetic waves through a patient’s tissue against a datum outcome from a healthy subject
Summary
The potential for noninvasive electromagnetic detection of biological anomalies in the human body and of defects in structural artefacts has been recently noted by Banks et al in 1. Our contribution lies in drawing an analogy between the Debye polarization model and the fading memory constitutive laws governing creep and relaxation in linear viscoelastic media This will allow us to apply known estimates from viscoelasticity theory and derive long-time stability estimates for the solutions to Maxwell’s equations Gronwall’s inequality is not used. It is worth noting that sharp stability estimates for the dual problem are often needed when deriving residual-based a posteriori error bounds for time dependent problems; see, for example, 10 In such cases it is essential to avoid the use of Gronwall’s inequality whenever possible. In both cases, long-time stability estimates Theorems 4.3 and 4.5 are derived without recourse to Gronwall’s inequality. W t − s v s ds, 1.7 and 1 p, q, r ∞ such that p−1 q−1 1 r−1
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