On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 2: A Heterogeneous Multiscale Method for Maxwell's equations

Patrick Ciarlet 1 Sonia Fliss 1 Christian Stohrer 2
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : In the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these difficulties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell’s equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell’s equations in a periodic setting and give numerical experiments in accordance to the stated results.
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Computers and Mathematics with Applications, Elsevier, 2017, 73 (9), pp.1900-1919. 〈10.1016/j.camwa.2017.02.043〉
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Patrick Ciarlet, Sonia Fliss, Christian Stohrer. On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 2: A Heterogeneous Multiscale Method for Maxwell's equations. Computers and Mathematics with Applications, Elsevier, 2017, 73 (9), pp.1900-1919. 〈10.1016/j.camwa.2017.02.043〉. 〈hal-01364782v2〉

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