{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:44:22Z","timestamp":1776843862079,"version":"3.51.2"},"reference-count":39,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,12,27]],"date-time":"2010-12-27T00:00:00Z","timestamp":1293408000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Numerical Analysis"],"published-print":{"date-parts":[[2010,12,27]]},"abstract":"<jats:p>Maxwell's equations in a bounded Debye medium are formulated in terms of the standard\npartial differential equations of electromagnetism with a Volterra-type history dependence\nof the polarization on the electric field intensity. This leads to Maxwell's equations with\nmemory. We make a correspondence between this type of constitutive law and the hereditary\nintegral 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\nshow long-time stability by shunning Gronwall's lemma and estimating the history kernels\nmore carefully by appeal to the underlying physical fading memory. We also give a fully\ndiscrete scheme for the electric field wave equation and derive stability bounds which are\nexactly analogous to those for the continuous problem, thus providing a foundation for\nlong-time numerical integration. We finish by also providing error bounds for which the\nconstant grows, at worst, linearly in time (excluding the time dependence in the norms\nof the exact solution). Although the first (mixed) finite element error analysis for the\nDebye problem was given by Li (2007), this seems to be the first time sharp constants have been given for\nthis problem.<\/jats:p>","DOI":"10.1155\/2010\/923832","type":"journal-article","created":{"date-parts":[[2010,12,27]],"date-time":"2010-12-27T19:38:50Z","timestamp":1293478730000},"page":"1-28","source":"Crossref","is-referenced-by-count":14,"title":["Finite Element Approximation of Maxwell's Equations with Debye Memory"],"prefix":"10.1155","volume":"2010","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1406-7225","authenticated-orcid":true,"given":"Simon","family":"Shaw","sequence":"first","affiliation":[{"name":"BICOM, Institute of Computational Mathematics, Brunel University, Uxbridge, Middlesex UB8 3PH, 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