Hybrid numerical-asymptotic boundary element methods for high frequency wave scattering

There has been considerable interest in recent years in the development of numerical methods for time-harmonic acoustic and electromagnetic wave scattering problems that can efficiently resolve the scattered field at high frequencies. Standard finite or boundary element methods (FEMs and BEMs), with piecewise polynomial approximation spaces, suffer from the restriction that a fixed number of degrees of freedom is required per wavelength in order to represent the oscillatory solution, leading to excessive computational cost when the scatterer is large compared to the wavelength. The hybrid numerical-asymptotic (HNA) approach aims to reduce the number of degrees of freedom required, by enriching the numerical approximation space with oscillatory functions, chosen using partial knowledge of the high frequency (short wavelength) asymptotic behaviour of the solution. The BEM setting is particularly attractive for such an approach, since knowledge of the high frequency asymptotics is required only on the boundary of the scatterer; for a recent review of the HNA methodology in the BEM context see [3]. In this setting one takes the relevant boundary value problem, which in the acoustic case involves the Helmholtz equation