Bérenger/Maxwell with Discontinous Absorptions : Existence, Perfection, and No Loss
Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exposé no. 10, 20 p.

We analyse Bérenger’s split algorithm applied to the system version of the two dimensional wave equation with absorptions equal to Heaviside functions of x j , j=1,2. The methods form the core of the analysis [11] for three dimensional Maxwell equations with absorptions not necessarily piecewise constant. The split problem is well posed, has no loss of derivatives (for divergence free data in the case of Maxwell), and is perfectly matched.

DOI : 10.5802/slsedp.38
Halpern, Laurence 1 ; Rauch, Jeffrey 2

1 LAGA, UMR 7539 CNRS Université Paris 13 93430 Villetaneuse France
2 Department of Mathematics University of Michigan Ann Arbor 48109 MI USA
@article{SLSEDP_2012-2013____A10_0,
     author = {Halpern, Laurence and Rauch, Jeffrey},
     title = {B\'erenger/Maxwell with {Discontinous} {Absorptions~:} {Existence,} {Perfection,} and {No} {Loss}},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:10},
     pages = {1--20},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2012-2013},
     doi = {10.5802/slsedp.38},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/slsedp.38/}
}
TY  - JOUR
AU  - Halpern, Laurence
AU  - Rauch, Jeffrey
TI  - Bérenger/Maxwell with Discontinous Absorptions : Existence, Perfection, and No Loss
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:10
PY  - 2012-2013
SP  - 1
EP  - 20
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/articles/10.5802/slsedp.38/
DO  - 10.5802/slsedp.38
LA  - en
ID  - SLSEDP_2012-2013____A10_0
ER  - 
%0 Journal Article
%A Halpern, Laurence
%A Rauch, Jeffrey
%T Bérenger/Maxwell with Discontinous Absorptions : Existence, Perfection, and No Loss
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:10
%D 2012-2013
%P 1-20
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://www.numdam.org/articles/10.5802/slsedp.38/
%R 10.5802/slsedp.38
%G en
%F SLSEDP_2012-2013____A10_0
Halpern, Laurence; Rauch, Jeffrey. Bérenger/Maxwell with Discontinous Absorptions : Existence, Perfection, and No Loss. Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exposé no. 10, 20 p. doi : 10.5802/slsedp.38. http://www.numdam.org/articles/10.5802/slsedp.38/

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

[2] D. Appelo, T. Hagstrom, and G. Kreiss, Perfectly matched layers for hyperbolic systems: General formulation, well-posedness and stability , SIAM J. Appl. Math., 67, no. 1, 1–23, 2007. | MR | Zbl

[3] A. Bamberger, P. Joly, J.E. Roberts, Second order absorbing boundary conditions for the wave equation: a solution for the corner problem, INRIA Report RR-0644, 1987. | MR | Zbl

[4] E. Bécache, S. Fauqueux, and P. Joly, Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys., 188, 399-433, 2003. | MR | Zbl

[5] E. Bécache. and P. Joly, On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations, M2AN Math. Model. Numer. Anal., 36, 87-119, 2002. | Numdam | MR | Zbl

[6] J.-P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves , J. Comput. Phys., 114, 185-200 , 1994. | MR | Zbl

[7] J.-P. Bérenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 127, 363-379, 1996. | MR | Zbl

[8] J.-P. Bérenger, Perfectly matched layers (PML) for computational electromagnetics, Synthesis lectures on computational electromagnetics, Morgan and Claypool, 2007.

[9] J. Diaz and P. Joly, A time domain analysis of PML models in acoustics, Computer Methods in Applied Mechanics and Engineering 195, 29-32, 3820-3853, 2006. | MR | Zbl

[10] L. Halpern, S. Petit-Bergez, and J. Rauch The analysis of matched layers, Confluentes Math., 3 no. 2, 159-236, 2011. | MR

[11] L. Halpern and J. Rauch, in preparation.

[12] P. Joly, S. Lohrengel, O. Vacus, Un résultat d’existence et d’unicité pour l’équation de Helmholtz avec conditions aux limites absorbantes d’ordre 2, C. R. Acad. Sci. Paris Sér. 1 Math. 329 (3) (1999) 193-198. | MR | Zbl

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

[14] S. Petit-Bergez, Problèmes faiblement bien posés : discrétisation et applications, Thèse de l’Université Paris 13, 2006.

Cité par Sources :