eprintid: 10188926
rev_number: 9
eprint_status: archive
userid: 699
dir: disk0/10/18/89/26
datestamp: 2024-03-13 15:15:21
lastmod: 2024-03-13 15:15:21
status_changed: 2024-03-13 15:15:21
type: article
metadata_visibility: show
sword_depositor: 699
creators_name: Caetano, AM
creators_name: Chandler-Wilde, SN
creators_name: Gibbs, A
creators_name: Hewett, DP
creators_name: Moiola, A
title: A Hausdorff-measure boundary element method for acoustic scattering by fractal screens
ispublished: inpress
divisions: UCL
divisions: B04
divisions: C06
divisions: F59
note: © 2024 Springer Nature. This article is licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).
abstract: Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in Rn+1 (n=1,2) when the scatterer, assumed to be a compact subset of Rn×{0}, is a d-set for some d∈(n-1,n], so that, in particular, the scatterer has Hausdorff dimension d. For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in R2 by Cantor sets, and in R3 by Cantor dusts.
date: 2024-02-26
date_type: published
publisher: Springer Science and Business Media LLC
official_url: http://dx.doi.org/10.1007/s00211-024-01399-7
oa_status: green
full_text_type: pub
language: eng
primo: open
primo_central: open_green
verified: verified_manual
elements_id: 2256437
doi: 10.1007/s00211-024-01399-7
lyricists_name: Hewett, David
lyricists_id: DHEWE35
actors_name: Hewett, David
actors_id: DHEWE35
actors_role: owner
full_text_status: public
publication: Numerische Mathematik
citation:        Caetano, AM;    Chandler-Wilde, SN;    Gibbs, A;    Hewett, DP;    Moiola, A;      (2024)    A Hausdorff-measure boundary element method for acoustic scattering by fractal screens.                   Numerische Mathematik        10.1007/s00211-024-01399-7 <https://doi.org/10.1007/s00211-024-01399-7>.    (In press).    Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10188926/2/Hewett_A%20Hausdorff-measure%20boundary%20element%20method%20for%20acoustic%20scattering%20by%20fractal%20screens_AOP.pdf