High-Order Integral Equation Methods for Diffraction Problems Involving Screens and Apertures

This thesis presents a novel approach for the numerical solution of problems of diffraction by infinitely thin screens and apertures. The new methodology relies on combination of weighted versions of the classical operators associated with the Dirichlet and Neumann open-surface problems. In the two-dimensional case, a rigorous proof is presented, establishing that the new weighted formulations give rise to second-kind Fredholm integral equations, thus providing a generalization to open surfaces of the classical closed-surface Calderón formulae. High-order quadrature rules are introduced for the new weighted operators, both in the two-dimensional case as well as the scalar three-dimensional case. Used in conjunction with Krylov subspace iterative methods, these rules give rise to efficient and accurate numerical solvers which produce highly accurate solutions in small numbers of iterations, and whose performance is comparable to that arising from efficient high-order integral solvers recently introduced for closed-surface problems. Numerical results are presented for a wide range of frequencies and a variety of geometries in twoand three-dimensional space, including complex resonating structures as well as, for the first time, accurate numerical solutions of classical diffraction problems considered by the 19th-century pioneers: diffraction of highfrequency waves by the infinitely thin disc, the circular aperture, and the two-hole geometry inherent in Young’s experiment.