A collocation method for high-frequency scattering by convex polygons
Journal of Computational and Applied Mathematics
A high-order tangential basis algorithm for electromagnetic scattering by curved surfaces
Journal of Computational Physics
An integration scheme for electromagnetic scattering using plane wave edge elements
Advances in Engineering Software
Convergence analysis of a multigrid algorithm for the acoustic single layer equation
Applied Numerical Mathematics
Wavenumber-Explicit $hp$-BEM for High Frequency Scattering
SIAM Journal on Numerical Analysis
Journal of Computational Physics
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In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is ${\cal O}(N^{-(\nu+1)} \log^{1/2}N)$, where the number of degrees of freedom is proportional to $N\log N$. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.