  We present a novel methodology for the numerical solution of problems of
diffraction by infinitely thin screens in three dimensional space. Our approach
relies on new integral formulations as well as associated high-order quadrature
rules. The new integral formulations involve weighted versions of the classical
integral operators associated with the thin-screen Dirichlet and Neumann
problems as well as a generalization to the open surface problem of the
classical Calderon formulae. The high-order quadrature rules we introduce for
these operators, in turn, resolve the multiple Green function and edge
singularities (which occur at arbitrarily close distances from each other, and
which include weakly singular as well as hypersingular kernels) and thus give
rise to super-algebraically fast convergence as the discretization sizes are
increased. When used in conjunction with Krylov-subspace linear algebra solvers
such as GMRES, the resulting solvers produce results of high accuracy in small
numbers of iterations for low and high frequencies alike. We demonstrate our
methodology with a variety of numerical results for screen and aperture
problems at high frequencies---including simulation of classical experiments
such as the diffraction by a circular disc (including observation of the famous
Poisson spot), interference fringes resulting from diffraction across two
nearby circular apertures, as well as more complex geometries consisting of
multiple scatterers and cavities.
