A Hausdorff-measure boundary element method for acoustic scattering by fractal screens

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 inv...

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Vydáno v:Numerische Mathematik Ročník 156; číslo 2; s. 463 - 532
Hlavní autoři: Caetano, A. M., Chandler-Wilde, S. N., Gibbs, A., Hewett, D. P., Moiola, A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2024
Springer Nature B.V
Témata:
Acoustic scattering
Acoustic waves
Acoustics
Basis functions
Boundary element method
Boundary integral method
Convergence
Estimates
Fractals
Function space
Integral equations
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Regularity
Screens
Simulation
Theoretical
45A05
28A80
46E35
65R20
65N38
ISSN:0029-599X, 0945-3245
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Shrnutí: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 R n + 1 ( n = 1 , 2 ) when the scatterer, assumed to be a compact subset of R n × { 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 R 2 by Cantor sets, and in R 3 by Cantor dusts.
Bibliografie:ObjectType-Article-1
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-024-01399-7