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Abstract
We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modeling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a wellposedness result and a frequencyexplicit bound on the solution operator that are valid for sufficiently large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction and that are only assumed to be bounded (i.e., L ^{∞}) in the other spatial directions. This class of coefficients therefore includes coefficients modeling transmission by penetrable obstacles with a (potentially large) number of layers (in 2d) or fibers (in 3d). Importantly, the frequencyexplicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly oscillatory coefficients and study the limiting behavior when the period of oscillations goes to zero. In particular, we bound the H ^{1} error committed by the firstorder 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 the best of 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.
Original language  English 

Pages (fromto)  13191363 
Number of pages  45 
Journal  Siam Journal on Mathematical Analysis 
Volume  55 
Issue number  2 
Early online date  27 Apr 2023 
DOIs  
Publication status  Published  31 Dec 2023 
Bibliographical note
Funding:The work of the second author was supported by the EPSRC grant EP/R005591/1.
Keywords
 Helmholtz equation
 high frequency
 homogenization
 transmission problem
ASJC Scopus subject areas
 Computational Mathematics
 Analysis
 Applied Mathematics
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Dive into the research topics of 'Scattering by finelylayered obstacles: frequencyexplicit bounds and homogenization'. Together they form a unique fingerprint.Projects
 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council