Boundary element method for solving high frequency scattering problems for obstacles with no corners

Paper presented at the 3rd Strathmore International Mathematics Conference (SIMC 2015), 3 - 7 August 2015, Strathmore University, Nairobi, Kenya.We consider scattering of a time-harmonic acoustic incident plane wave by a sound soft smooth object with Lipschitz boundary. The application of conventional boundary or finite element methods, have computational cost that grows linearly respect to the frequency of the incident wave. Recent research has been devoted in finding methods which does not loose robustness as frequency of the incident wave increases. Arden, Chandler-Wilde and Langdon proposed a collocation method to solve a high frequency scattering by convex polygons. They use a boundary element method, and incorporating products of plane wave basis functions with piecewise polynomials supported on a graded mesh into approximation space. They demonstrated via numerical experiments the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to frequency. Here we proposed a collocation method for high frequency scattering by smooth objects (objects with no corners, e.g. a circle).We applied same approximations as theirs, but employing uniform mesh. We demonstrate through numerical experiments the logarithmical grow of the solutions as frequency increases, with much reduced computational cost