### Abstract

Recently developed integral equation methods for surface scattering simulations ---that combine the advantages of rigorous solvers (error controllability) with those of asymptotic methods (frequency-independent discretizations)--- have displayed the potential of delivering scattering returns within a prescribed error-tolerance in times that do not depend on the wavenumber k. The near-optimal characteristics of these novel schemes, pioneered by Bruno, Geuzaine, Monro and Reitich, have rapidly generated great interest and significant number of work in recent years.

In single-scattering configurations, an actual proof that provides a rigorous upper bound for the operation count of O(k^{1/9}) in the case of circular/spherical boundaries was recently established by Dominguez, Graham and Smyshlyaev for a p-version boundary element implementation of a similar approach. As in the original algorithm, they take profound advantage of the exponential decay (with increasing wavenumber k) of the corresponding physical density in the deep shadow region, and approximate this quantity by zero there.

In this talk, we present two improved Galerkin schemes for the solution of single-scattering problems, and we show that, within the framework of our first scheme, the error in best approximation of the surface current grows at most at O(k^a) (for any a>0) over the "entire" boundary. Moreover, as we show, our second approach based on a novel change of variables around the "transition regions" reduces this dependency to O(log(k)).