Abstract. We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave.
Many acoustic and electromagnetic wave scattering problems can be formulated as the Helmholtz equation. Standard finite and boundary element method solution of these problems becomes expensive, as the frequency of incident wave increases. On going research has been devoted to finding methods that do not loose robustness when the wave number increases. Recently, Chandler-Wilde et al. have proposed a novel Galerkin boundary element method to solve the problem of acoustic scattering by a convex polygon with impedance boundary conditions. They applied approximation spaces consisting of piecewise polynomials supported on a graded mesh with smaller elements adjacent to the corners of the polygon and multiplied by plane wave basis functions. They demonstrated via rigorous error analysis that was supported by numerical experiments that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as frequency increases. In this paper, we discuss issues related to detail implementation of their numerical method.
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