Dans ce travail, nous considérons le modèle de couches parfaitement adaptées, dit PML (Perfectly Matched Layers), introduit par Bérenger [3] pour la modélisation de frontières absorbantes en électromagnétisme. Nous menons une analyse mathématique de ce modèle, d'une part par une analyse modale par transformation de Fourier, d'autre part par des techniques énergétiques. Nous obtenons ainsi des résultats de stabilité uniforme en temps (qui précisent des résultats déjà connus dans la littérature) et établissons des résultats de décroissance d'énergie qui illustrent les propriétés d'absorption du modèle. Cette dernière technique permet aussi de démontrer la stabilité du schéma de Yee pour discrétiser les couches absorbantes.
In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This last technique allows us to prove the stability of the Yee's scheme for discretizing PML's.
Mots-clés : absorbing layers, PML, Maxwell's equations, stability, hyperbolic systems, Fourier analysis, energy techniques, Yee's scheme
@article{M2AN_2002__36_1_87_0,
author = {B\'ecache, Eliane and Joly, Patrick},
title = {On the analysis of {B\'erenger's} perfectly matched layers for {Maxwell's} equations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {87--119},
publisher = {EDP-Sciences},
volume = {36},
number = {1},
year = {2002},
doi = {10.1051/m2an:2002004},
zbl = {0992.78032},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2002004/}
}
TY - JOUR AU - Bécache, Eliane AU - Joly, Patrick TI - On the analysis of Bérenger's perfectly matched layers for Maxwell's equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2002 SP - 87 EP - 119 VL - 36 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2002004/ DO - 10.1051/m2an:2002004 LA - en ID - M2AN_2002__36_1_87_0 ER -
%0 Journal Article %A Bécache, Eliane %A Joly, Patrick %T On the analysis of Bérenger's perfectly matched layers for Maxwell's equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2002 %P 87-119 %V 36 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2002004/ %R 10.1051/m2an:2002004 %G en %F M2AN_2002__36_1_87_0
Bécache, Eliane; Joly, Patrick. On the analysis of Bérenger's perfectly matched layers for Maxwell's equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 1, pp. 87-119. doi : 10.1051/m2an:2002004. http://archive.numdam.org/articles/10.1051/m2an:2002004/
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