A sparse discretisation for integral equation formulations of high frequency scattering problems

We consider two-dimensional scattering problems, formulated as an integral equation defined on the boundary of the scattering obstacle. The oscillatory nature of high-frequency scattering problems necessitates a large number of unknowns inclassical boundary element methods. In addition, the corresponding discretisation matrix of the integral equation is dense. We formulate a boundary element method with basis functions that incorporate the asymptotic behaviour of the solution at high frequencies. The method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns, but retains the accurate convergence of the boundary element method for lower frequencies. New in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals. As a result, we obtain a sparse discretisation matrix for the oscillatory problem. Moreover, the accuracy of a large part of the solution actually increases with increasing frequency.The sparse discretisation applies to problems where the phase of the solution can be predicted a priori, for example in the case of smooth and convex scatterers.