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Title: Finite element approximation of Maxwell’s equations with Debye memory
Authors: Shaw, S
Keywords: Debye relaxation;Maxwell's equations;Electromagnetism;Fading memory;Stability;Finite element method;Error bounds
Issue Date: 2010
Publisher: Hindawi Publishing Corporation
Citation: Advances in Numerical Analysis, Volume 2010, Article ID 923832
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 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 analagous 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 Jichun Li (in Comp. Meth. Appl. Mech. Eng., 196, (2007), pp. 3081–3094) this seems to be the the first time sharp constants have been given for this problem.
Description: Copyright © 2010 Simon Shaw. All rights reserved.
This article has been made available through the Brunel Open Access Publishing Fund.
Appears in Collections:Brunel OA Publishing Fund
Dept of Mathematics Research Papers
Mathematical Sciences

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