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Fast, High-Order Methods for Scattering by Inhomogeneous Media

In this thesis, we introduce a new, fast, high-order method for scattering by inhomogeneous media in three dimensions. As in previously existing methods, the low (O(N log N)) complexity of our integral equation method is obtained through extensive use of the fast Fourier transform (FFT) in evaluating the required convolutions. Unlike previous FFT-based methods, however, this method yields high-order accuracy, even for scatterers containing geometric singularities such as discontinuities, corners, and cusps. We begin our discussion with a thorough theoretical analysis of an efficient, high-order method recently introduced by Bruno and Sei (IEEE Trans. in Antenn. Propag., 2000), which motivated the present work. This two-dimensional method is based on a Fourier approximation of the integral equation in polar coordinates and a related, generally low-order, Fourier smoothing of the scatterer. The claim that use of this low-order approximation of the scatterer leads to a high-order accurate numerical method generated considerable controversy. Our proofs establish that this method indeed yields high-order accurate solutions. We also introduce several substantial improvements to the numerical implementation of this two-dimensional algorithm, which lead to increased numerical stability with decreased computational cost. We then present our new, fast, high-order method in three dimensions. An immediate generalization of the polar coordinate approach in two dimensions to a spherical coordinate approach in three dimensions appears less advantageous than our chosen approach: Fourier approximation and integration in Cartesian coordinates. To obtain smooth and periodic functons (which are approximated to high-order via Fourier series), we 1) decompose the Green's function into a smooth part with infinite support and a singular part with compact support; and 2) replace, as in the two-dimensional approach, the (possibly discontinuous) scatterer with its truncated Cartesian Fourier series. The accuracy of our three-dimensional method is approximately equal to that of the two-dimensional method mentioned above and, interestingly, is actually much simpler than the two-dimensional approach. In addition to our theoretical discussion of these high-order methods, we present a parallel implementation of our three-dimensional Cartesian approach. The efficiency, high-order accuracy, and overall performance of both the polar and Cartesian methods are demonstrated through several computational examples.