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Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization

Abstract : We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibres (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly-oscillatory coefficients and study the limiting behaviour when the period of oscillations goes to zero. In particular, we bound the $H^1$ error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small.
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https://hal.inria.fr/hal-03354770
Contributor : Théophile Chaumont-Frelet Connect in order to contact the contributor
Submitted on : Sunday, September 26, 2021 - 12:27:57 PM
Last modification on : Thursday, August 4, 2022 - 5:00:05 PM
Long-term archiving on: : Monday, December 27, 2021 - 6:07:08 PM

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  • HAL Id : hal-03354770, version 1
  • ARXIV : 2109.11267

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Théophile Chaumont-Frelet, Euan A Spence. Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization. 2021. ⟨hal-03354770⟩

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