CSDS seminar – First kind boundary integral equations and boundary element methods for acoustic scattering (Simon Chandler-Wilde)

We consider the so-called sound-soft problem in time-harmonic acoustic scattering, where the total field, a solution to the Helmholtz equation in the exterior of some obstacle, vanishes on the boundary of the obstacle.  Making an ansatz for the scattered field as a single-layer potential with some unknown density, a BIE for the density is obtained by applying this sound-soft boundary condition. This is an old formulation; it can be found, in the context of scattering by screens and apertures, already in 19th century texts [1]. In this talk we make a survey of previous results, and we explain in what sense this formulation applies when the obstacle is some arbitrary compact set. In the case when the scattering obstacle is the support of some Radon measure,  we establish conditions on this measure that ensure that a Galerkin boundary element method is convergent in which all integrals are with respect to this Radon measure [4]. We show computations for cases in which the obstacle is the fractal attractor of an iterated function system and the measure is d-dimensional Hausdorff measure, for some non-integer value of d [2,3].

[1] Lord Rayleigh, “Theory of Sound”, 2nd Ed., Vol. II, Macmillan, New York, 1896
[2] A. M. Caetano, S. N. Chandler-Wilde, A. Gibbs, D. P. Hewett and A. Moiola, A Hausdorff-measure boundary element method for acoustic scattering by fractal screens. Numer. Math., 156, 463-532 (2024)
[3] A. M. Caetano, S. N. Chandler-Wilde, X. Claeys, A. Gibbs, D. P. Hewett and A. Moiola, Integral equation methods for acoustic scattering by fractals. Proc. R. Soc. A., 481: 20230650 (2025)
[4] M. Hinz, S. N. Chandler-Wilde, and D.P. Hewett, Kernels of trace operators via fine continuity, arXiv:2507.04536 (2025)