Analysis of convergent integral equation methods for high-frequency scattering

Abstract

The main aim of this thesis is to devise numerical methods for the solution of high-frequency scattering problems in 2 dimensional settings by utilizing geometrical optics ansatz and asymptotic properties of solutions for convex obstacles (see [1]). To this end, we formulate the sound soft scattering problem as a well-posed boundary integral equation. Among the numerical methods (Nyström, collocation, Galerkin, two-grid and multi-grid) appropriate for solving integral equations, we focus on the classical but e cacious ones, namely the two- and multi-grid methods. We rst portray the defect correction principle for integral equations of the second kind which constitutes a basis for the two- and multi-grid methods, then we de ne both methods over the defect correction iteration. We also set up these methods to compute the scattering return by the unit circle numerically and compare theoretical and numerical results. By virtue of the geometrical optics ansatz, which expresses the normal derivative of the total eld as a highly oscillating complex exponential modulated by a slowly oscillating amplitude, we construct a new Galerkin method well adapted to the slowly oscillating nature of the unknown function which we approximate by polynomials. We hereby eliminate the serious drawbacks arising from high oscillations for approximating the solutions. As our main convergence result will display, our new algorithm entails that it su ces to increase the degrees of freedom proportional to k (for any > 0) in order to preserve a given accuracy. In contrast with the previous e orts on the problem, we construct our local approximation spaces with particular emphasis on the transition regions to capture the boundary layers around shadow boundaries and utilize approximation spaces in the deep shadow region to incorporate the e ects of grazing rays.