A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of Bérenger. We prove that the Bérenger and closely related layers define well-posed transmission problems in great generality. When the Bérenger method or one of its close relatives is well-posed, perfect matching is proved. The proofs use the energy method, Fourier–Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the Bérenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in Bérenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of Bérenger's algorithm for Maxwell's equations, our favorite choice leads to a strongly well-posed augmented system and is both perfect and nonamplifying in great generality. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has leading reflection coefficient equal to zero at all angles of incidence. Open problems are indicated throughout.
@article{CML_2011__3_2_159_0,
author = {Laurence Halpern and Sabrina Petit-Bergez and Jeffrey B. Rauch},
title = {The analysis of matched layers},
journal = {Confluentes Mathematici},
pages = {159--236},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {2},
year = {2011},
doi = {10.1142/S1793744211000291},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000291/}
}
TY - JOUR AU - Laurence Halpern AU - Sabrina Petit-Bergez AU - Jeffrey B. Rauch TI - The analysis of matched layers JO - Confluentes Mathematici PY - 2011 SP - 159 EP - 236 VL - 3 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000291/ DO - 10.1142/S1793744211000291 LA - en ID - CML_2011__3_2_159_0 ER -
%0 Journal Article %A Laurence Halpern %A Sabrina Petit-Bergez %A Jeffrey B. Rauch %T The analysis of matched layers %J Confluentes Mathematici %D 2011 %P 159-236 %V 3 %N 2 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000291/ %R 10.1142/S1793744211000291 %G en %F CML_2011__3_2_159_0
Laurence Halpern; Sabrina Petit-Bergez; Jeffrey B. Rauch. The analysis of matched layers. Confluentes Mathematici, Tome 3 (2011) no. 2, pp. 159-236. doi : 10.1142/S1793744211000291. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000291/
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