When high-frequency acoustic or electromagnetic waves are incident upon an obstacle, the resulting scattered field is composed of rapidly oscillating waves. Conventional numerical methods for such problems use piecewise-polynomial approximation spaces which are not well-suited to capture the oscillatory solution. Hence these methods are prohibitively expensive in the high-frequency regime. Much work has been done in developing “hybrid numerical-asymptotic” (HNA) boundary element methods which utilise approximation spaces containing oscillatory functions carefully chosen to capture the high-frequency asymptotic behaviour of the solution. The computational cost of this approach is significantly smaller than that of conventional methods, and for many problems it is independent of the frequency. In this talk, I will outline the HNA method and discuss its extension to scattering by penetrable obstacles.
