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We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM)for high frequency scattering by two-dimensional screens and apertures, whosecomputational cost to achieve any prescribed accuracy remains bounded withincreasing frequency. Our method is a collocation implementation of the highorder hp HNA approximation space of Hewett et al. IMA J. Numer. Anal. 35(2015), pp.1698- 1728, where a Galerkin implementation was studied.  An advantage of the current collocation scheme is that the one-dimensionalhighly oscillatory singular integrals appearing in the BEM matrix entries aresignificantly easier to evaluate than the two-dimensional integrals appearingin the Galerkin case, which leads to much faster computation times. Here wecompute the required integrals at frequency-independent cost using thenumerical method of steepest descent, which involves complex contourdeformation.  The change from Galerkin to collocation is nontrivial because naivecollocation implementations based on square linear systems suffer from severenumerical instabilities associated with the numerical redundancy of the HNAbasis, which produces highly ill-conditioned BEM matrices. In this paper weshow how these instabilities can be removed by oversampling, and solving theresulting overdetermined collocation system in a weighted least-squares senseusing a truncated singular value decomposition. On the basis of our numericalexperiments, the amount of oversampling required to stabilise the method ismodest (around 25% typically suffices) and independent of frequency. As anapplication of our method we present numerical results for high frequencyscattering by prefractal approximations to the middle-third Cantor set.

Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation