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.
(In press).
Preview |
Text
Hewett_A Hausdorff-measure boundary element method for acoustic scattering by fractal screens_AOP.pdf Download (4MB) | Preview |
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.
| Type: | Article |
|---|---|
| Title: | A Hausdorff-measure boundary element method for acoustic scattering by fractal screens |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00211-024-01399-7 |
| Publisher version: | http://dx.doi.org/10.1007/s00211-024-01399-7 |
| Language: | English |
| Additional information: | © 2024 Springer Nature. This article is licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/). |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10188926 |
Archive Staff Only
 |
View Item |
