Title:

Asymptotic and numerical methods for highfrequency scattering problems

This thesis is concerned with the development, analysis and implementation of efficient and accurate numerical methods for solving highfrequency acoustic scattering problems. Classical boundary or finite element methods that are based on approximating the solution by polynomials can be effective for small and moderate frequencies. However, as the frequency increases, the solution to the scattering problem becomes more oscillatory and classical numerical methods cope very badly with high oscillation. For example, for twodimensional scattering problems, classical numerical methods require their number of degrees of freedom to grow at least linearly with frequency to capture the oscillatory behaviour of the solution accurately. Therefore, at large frequencies, classical numerical methods become essentially numerically intractable. In order to overcome the limitations of classical methods, one can seek to incorporate the known asymptotic behaviour of the solution in the numerical method. This involves using asymptotic theory to determine the oscillatory part of the solution and then using classical numerical methods to approximate the slowly varying remainder. Such methods are often referred to as hybrid numericalasymptotic methods. Determining the high frequency asymptotics of acoustic scattering problems is a classic problem in applied mathematics, with methods such as geometrical optics or the geometrical theory of diffraction providing asymptotic expansions of the solutions. Considerable amount of research has been directed towards both constructing these asymptotic expansions and proving error bounds for truncated asymptotic series of the solution, notably by Buslaev [23], Morawetz and Ludwig [78], and Melrose and Taylor [75], among others. Often, the oscillatory component of the solution can be determined explicitly from these asymptotic expansions. This can then be used in designing ecient hybrid methods. Furthermore, from the asymptotic expansions, frequencydependent bounds on the slowlyvarying remainder and its derivatives can be obtained (in some cases these follow directly from classical results, in other cases some additional work is required). The frequencydependent bounds are the key results used in the frequencyexplicit numerical error analysis of the approximation of the slowlyvarying remainder. This thesis presents a rigorous justification of one of the key result using only elementary techniques. Hybrid numericalasymptotic methods have been shown in theory to be substantially more efficient than classical numerical methods alone. For example, [40] presented a hybrid numericalasymptotic method in the context of boundary integral equations (BIEs) for solving the problem of highfrequency scattering by smooth, convex obstacles in two dimensions. It was proved in [40] that in order to maintain the accuracy as the frequency increases, the hybrid BIE method requires the number of degrees of freedom to grow slightly faster than k1=9, where k is a parameter proportional to the frequency. This is a substantial improvement from the classical boundary integral methods that require O(k) number of degrees of freedom to achieve the same accuracy for this problem. Despite this slow growth in the number of degrees of freedom, hybrid numericalasymptotic methods lead to stiffness matrices with entries that are highlyoscillatory singular integrals that can not be computed exactly. Thus, without efficient and accurate numerical treatment of these integrals, the hybrid numericalasymptotic methods, regardless of their attractive theoretical accuracy, can not be efficiently implemented in practice. In order to resolve this difficulty, this thesis develops a methodology for approximating the integrals arising from hybrid methods in the context of BIEs. The integrals are transformed under a change of variables into integrals amenable to Filontype quadratures. Filontype quadratures are designed to cope well with high oscillations in the integrands. Then, graded meshes are used to capture the singularities accurately. Along with kexplicit error bounds for the integration methods, this thesis derives kexplicit error bounds for the hybrid BIE methods that incorporate the error of the inexact approximation of the entries of the stiffness matrix. The error bounds suggest that, with an appropriate choice of parameters of Filon quadrature and mesh grading, the overall error of the hybrid method does not deteriorate due to inexact approximation of the stiffness matrix, therefore preserving its attractive theoretical convergence properties.
