On the analysis of Bérenger's perfectly matched layers for Maxwell's equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 1, p. 87-119
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.
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.
DOI : https://doi.org/10.1051/m2an:2002004
Classification:  65M06,  65M12,  35L05,  35L40,  78M20,  35B35
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {1},
     year = {2002},
     pages = {87-119},
     doi = {10.1051/m2an:2002004},
     zbl = {0992.78032},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2002__36_1_87_0}
}
Bécache, Eliane; Joly, Patrick. On the analysis of Bérenger's perfectly matched layers for Maxwell's equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 1, pp. 87-119. doi : 10.1051/m2an:2002004. http://www.numdam.org/item/M2AN_2002__36_1_87_0/

[1] S. Abarbanel and D. Gottlieb, A mathematical analysis of the PML method. J. Comput. Phys. 134 (1997) 357-363. | Zbl 0887.65122

[2] S. Abarbanel and D. Gottlieb, On the construction and analysis of absorbing layers in CEM. Appl. Numer. Math. 27 (1998) 331-340. | Zbl 0924.35160

[3] J.P. Bérenger, A Perfectly Matched Layer for the Absorption of Electromagnetic Waves. J. Comput. Phys. 114 (1994) 185-200. | Zbl 0814.65129

[4] F. Collino and P. Monk, Conditions et couches absorbantes pour les équations de Maxwell, in G. Cohen and P. Joly, Aspects récents en méthodes numériques pour les équations de Maxwell, Eds. École des Ondes, Chapter 4, INRIA, Rocquencourt (1998).

[5] J.W. Goodrich and T. Hagstrom, A comparison of two accurate boundary treatments for computational aeroacoustics. AIAA Paper-1585 (1997).

[6] J.S. Hesthaven, On the Analysis and Construction of Perfectly Matched Layers for the Linearized Euler Equations. J. Comput. Phys. 142 (1998) 129-147. | Zbl 0933.76063

[7] F.Q. Hu, On absorbing boundary conditions for linearized euler equations by a perfectly matched layer. J. Comput. Phys. 129 (1996) 201-219. | Zbl 0879.76084

[8] T. Kato, Perturbation Theory for Linear Operators. Springer (1995). | MR 1335452 | Zbl 0836.47009

[9] H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations, in Pure Appl. Math. 136, Academic Press, Boston, USA (1989). | MR 998379 | Zbl 0689.35001

[10] J. Métral and O. Vacus, Caractère bien posé du problème de Cauchy pour le système de Bérenger. C.R. Acad. Sci. I Math. 10 (1999) 847-852. | Zbl 0928.35176

[11] P.G. Petropoulos, L. Zhao and A.C. Cangellaris, A reflectionless sponge layer absorbing boundary condition for the solution of Maxwell's equations with high-order staggered finite difference schemes. J. Comput. Phys. 139 (1998) 184-208. | Zbl 0915.65123

[12] A.N. Rahmouni, Des modèles PML bien posés pour divers problèmes hyperboliques. Ph.D. thesis, Université Paris Nord-Paris XIII (2000).

[13] Allen Taflove, Computational electrodynamics: the finite-difference time-domain method. Artech House (1995). | MR 1338377 | Zbl 0840.65126

[14] E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27 (1998) 533-557. | Zbl 0933.35188

[15] L. Zhao and A.C. Cangellaris, A General Approach for the Development of Unsplit-Field Time-Domain Implementations of Perfectly Matched Layers for FDTD Grid Truncation. IEEE Microwave and Guided Letters 6 May 1996.

[16] R.W. Ziolkowski, Time-derivative lorentz material model-based absorbing boundary condition. IEEE Trans. Antennas Propagation 45 (1997) 1530-1535.