A fast, higher-order solver for scattering by penetrable bodies in three dimensions

Abstract

In this paper, we introduce a new fast, higher-order solver for scattering by inhomogeneous media in three dimensions. As in previously existing methods, the low complexity of our integral equation method, O(N log N) operations for an N point discretization, is obtained through extensive use of the fast Fourier transform (FFT) for the evaluation of convolutions. However, the present approach obtains significantly higher-order accuracy than these previous approaches, yielding, at worst, third-order far field accuracy (or substantially better for smooth scatterers), even for discontinuous and complex refractive index distributions (possibly containing severe geometric singularities such as corners and cusps). The increased order of convergence of our method results from (i) a partition of unity decomposition of the Green's function into a smooth part with unbounded support and a singular part with compact support, and (ii) replacement of the (possibly discontinuous) scatterer by an appropriate "Fourier smoothed" scatterer; the resulting convolutions can then be computed with higher-order accuracy by means of O(N log N) FFTs. We present a parallel implementation of our approach, and demonstrate the method's efficiency and accuracy through a variety of computational examples. For a very large scatterer considered earlier in the literature (with a volume of 3648λ3, where λ is the wavelength), using the same number of points per wavelength and in computing times comparable to those required by the previous approach, the present algorithm produces far-field values whose errors are two orders of magnitude smaller than those reported previously.

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