In this short paper we have outlined the basic ideas behind some recent approaches to the problem of approximating the solution of high frequency acoustic scattering problems in two dimensions by bounded sound soft convex obstacles. By incorporating appropriate oscillatory functions into the approximation space, it is possible to demonstrate via a rigorous numerical analysis that the number of degrees of freedom need grow at a substantially sublinear rate with respect to an increase in the frequency of the incident field or the size of the obstacle, for the case of smooth convex obstacles and for the case of convex polygons. For the case of convex curvilinear polygons, recent numerical results support the conjecture that a similar level of performance can be achieved. For full details of each of these schemes we refer to the relevant references detailed below, and for a full description of the recent developments in this field we refer to the recent survey paper by Chandler-Wilde and Graham.
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