Numerical methods for high frequency scattering by multiple obstacles

For problems of time harmonic wave scattering, standard numerical methods (using piecewise polynomial approximation spaces) require the computational cost to grow with the frequency of the problem, in order to maintain a fixed accuracy. This can make many problems of practical interest difficult or impossible to solve at high frequencies. The Hybrid Numerical Asymptotic Boundary Element Method (HNA BEM) overcomes this by enriching the approximation space with oscillatory basis functions, in such a way that accuracy may be maintained with a computational cost that grows only modestly with frequency. HNA methods have previously been developed for a range of problems, including screens in two and three dimensions, also convex, non-convex and penetrable polygons in two dimensions. To date, all HNA methods have been developed for problems of plane wave scattering by a single obstacle. The key aim of this thesis is to extend the HNA method to multiple obstacles. A range of extensions to the HNA method are made in this thesis. Previous HNA methods for convex polygons use an approximation space on two overlapping meshes, here we use HNA on a single mesh. This single-mesh approach is easier to implement, and we prove that the frequency-dependence of the size of the approximation space is the same as for the overlapping mesh. We generalise HNA theory to provide a priori error estimates for a broader range of incident fields than just the plane wave, including point sources, beam sources, and Herglotz-type incidence. We also extend the HNA ansatz to include multiple obstacles. In addition to the development of HNA methods, we also consider other ideas and developments related to multiple scattering problems. This includes the first (to the best knowledge of the author) mesh and frequency explicit condition for wellposedness of Galerkin BEM for multiple scattering. We investigate numerical implementation of Embedding Formulae, which provide the far-field pattern for any incident plane wave, given the far-field patterns induced by a small (frequency independent) number of plane waves. We establish points at which a naive implementation of the theory can cause numerical instabilities and present alternative, numerically stable Embedding Formulae. We also extend the Embedding Formulae to produce the far-field pattern of any Herglotz-type wave. The recently developed Tmatrom method, a numerically stable T-matrix method, is explored as an alternative means of extending the HNA method from single to multiple obstacles. Tmatrom typically requires a number of single scattering problems to be solved, and this number grows (more than) linearly with the frequency of the problem. Using the numerically stable Embedding Formulae, we show that Tmatrom can be applied by solving a number of problems that depends only on the geometry of the obstacle, and not the frequency of the incident wave