Brunel University Research Archive (BURA) >
Schools >
School of Information Systems, Computing and Mathematics >
School of Information Systems, Computing and Mathematics Research Papers >

Please use this identifier to cite or link to this item: http://bura.brunel.ac.uk/handle/2438/5003

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
Publication 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.
Sponsorship: This article is available through the Brunel Open Access Publishing Fund.
URI: http://bura.brunel.ac.uk/handle/2438/5003
DOI: http://dx.doi.org/10.1155/2010/923832
Appears in Collections:School of Information Systems, Computing and Mathematics Research Papers
Mathematical Science
Brunel OA Publishing Fund

Files in This Item:

File Description SizeFormat
Fulltext.pdf643.17 kBAdobe PDFView/Open

Items in BURA are protected by copyright, with all rights reserved, unless otherwise indicated.

 


Library (c) Brunel University.    Powered By: DSpace
Send us your
Feedback. Last Updated: September 14, 2010.
Managed by:
Hassan Bhuiyan