In this paper we develop a class of efficient Galerkin boundary element methods for the solution of two-dimensional exterior single-scattering problems. Our approach is based upon construction of Galerkin approximation spaces confined to the asymptotic behaviour of the solution through a certain direct sum of appropriate function spaces weighted by the oscillations in the incident field of radiation. Specifically, the function spaces in the illuminated/shadow regions and the shadow boundaries are simply algebraic polynomials whereas those in the transition regions are generated utilizing novel, yet simple, emph{frequency dependent changes of variables perfectly matched with the boundary layers of the amplitude} in these regions. While, on the one hand, we rigorously verify for smooth convex obstacles that these methods require only an $mathcal{O}left( k^{epsilon} right)$ increase in the number of degrees of freedom to maintain any given accuracy independent of frequency, and on the other hand, remaining in the realm of smooth obstacles they are applicable in more general single-scattering configurations. The most distinctive property of our algorithms is their emph{remarkable success} in approximating the solution in the shadow region when compared with the algorithms available in the literature.
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