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Associate Prof. Andrea Moiola | A Hausdorff-measure BEM for acoustic scattering by fractal screens..
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Speaker(s): Associate Professor Andrea Moiola (Università degli Studi di Pavia)
Date: 18 April 2023 - 11:00 to 11:45
Venue: INI Seminar Room 1
Session Title: A Hausdorff-measure BEM for acoustic scattering by fractal screens - part 1
Event: [MWSW03] Computational methods for multiple scattering
Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. We formulate such scattering problems as singular integral equations and we approximate them using the boundary element method (BEM). 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 when the scatterer is a compact $d$-set for some suitable Hausdorff dimension $d$. For a class of fractals that are attractors of iterated function systems (IFS), 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. Quadrature rules and numerical results will be presented in the second part of the presentation.
Date: 18 April 2023 - 11:00 to 11:45
Venue: INI Seminar Room 1
Session Title: A Hausdorff-measure BEM for acoustic scattering by fractal screens - part 1
Event: [MWSW03] Computational methods for multiple scattering
Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. We formulate such scattering problems as singular integral equations and we approximate them using the boundary element method (BEM). 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 when the scatterer is a compact $d$-set for some suitable Hausdorff dimension $d$. For a class of fractals that are attractors of iterated function systems (IFS), 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. Quadrature rules and numerical results will be presented in the second part of the presentation.
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