Analysis of convergent integral equation methods for high-frequency scattering

The main aim of this thesis is to devise numerical methods for the solution ofhigh-frequency scattering problems in 2 dimensional settings by utilizing geometricaloptics ansatz and asymptotic properties of solutions for convex obstacles (see [1]). Tothis end, we formulate the sound soft scattering problem as a well-posed boundary integralequation. Among the numerical methods (Nyström, collocation, Galerkin, two-gridand multi-grid) appropriate for solving integral equations, we focus on the classical butefficacious ones, namely the two- and multi-grid methods. We first portray the defectcorrection principle for integral equations of the second kind which constitutes a basisfor the two- and multi-grid methods, then we define both methods over the defect correctioniteration. We also set up these methods to compute the scattering return by theunit circle numerically and compare theoretical and numerical results. By virtue of thegeometrical optics ansatz, which expresses the normal derivative of the total field asa highly oscillating complex exponential modulated by a slowly oscillating amplitude,we construct a new Galerkin method well adapted to the slowly oscillating nature ofthe unknown function which we approximate by polynomials. We hereby eliminate theserious drawbacks arising from high oscillations for approximating the solutions. Asour main convergence result will display, our new algorithm entails that it suffices toincrease the degrees of freedom proportional to k^epsilon (for any epsilon >0) in order to preserve agiven accuracy. In contrast with the previous efforts on the problem, we construct ourlocal approximation spaces with particular emphasis on the transition regions to capturethe boundary layers around shadow boundaries and utilize approximation spacesin the deep shadow region to incorporate the effects of grazing rays.