# A high-wavenumber boundary-element method for an acoustic scattering problem

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Chandler-Wilde, S. N., Langdon, S. and Ritter, L. (2004) A high-wavenumber boundary-element method for an acoustic scattering problem. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 362 (1816). pp. 647-671. ISSN 1364-503X

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To link to this article DOI: 10.1098/rsta.2003.1339

## Abstract/Summary

In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval $[a,b]$, which only requires the discretization of $[a,b]$, we show theoretically and experimentally that the $L_2$ error in computing the acoustic field on $[a,b]$ is ${\cal O}(\log^{\nu+3/2}|k(b-a)| M^{-(\nu+1)})$, where $M$ is the number of degrees of freedom and $k$ is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems.

Item Type: Article Yes Faculty of Science > School of Mathematical and Physical Sciences > Department of Mathematics and Statistics 4948 high–frequency scattering; Galerkin boundary–element method; outdoor sound propagation Royal Society Publishing

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