High frequency BEM formulations for scattering problems often represent the solution on the surface of the obstacle as a weighted sum of basis functions which each comprise an envelope function multiplied by an oscillatory function. The envelope functions partition the solution in space; these are typically slowly varying with respect to wavelength and have support which is large with respect to wavelength. In contrast, the oscillatory functions are usually chosen to oscillate at or around the wavenumber in the medium. The combination of these, along with the oscillatory component of the Green’s function, means the quantity to be integrated numerically may oscillate with a wavenumber as high as 2k, and that the integration domain is two dimensional and covers many wavelengths. Computing such an integral numerically by a standard approach will therefore require O(k^2) operations. This is a major computational bottleneck, even if the enriched approximation space proves to be more efficient than a standard one (i.e. if the inclusion of the oscillatory components reduces the number of basis functions required to achieve a small error).

This talk will describe a contour integral transform valid for certain types of envelope and oscillatory functions. Since the resulting integral involves line integrals instead of surface integrals, the computational cost of evaluation scales with a more favourable O(k). Such approaches have long been used for evaluating the self-interaction integrals which arise in BEM, in which case the challenge is overcoming the singularity in the Green’s function. Here a more general approach is drawn from an equivalent problem in aperture diffraction, which also applies to non-self interactions with oscillatory basis functions. The transform will be presented and the restrictions it places on the choices of oscillatory and envelope functions, and on surface geometry, will be discussed.