Higher-Order Fourier Approximation in Scattering by Two-Dimensional, Inhomogeneous Media

This paper provides a theoretical analysis of a higher-order, FFT-based integral equation method introduced recently [IEEE Trans. Antennas and Propagation, 48 (2000), pp. 1862–1864] for the evaluation of transverse electric–polarized electromagnetic scattering from a bounded, penetrable inhomogeneity in two-dimensional space. Roughly speaking, this method is based on Fourier smoothing of the integral operator and the refractive index n(x). Here we prove that the solution of the resulting integral equation approximates the solution of the exact integral equation with higher-order accuracy, even when n(x) is a discontinuous function—as suggested by the numerical experiments contained in the paper mentioned above. In detail, we relate the convergence rates of the computed interior and exterior fields to the regularity of the scatterer, and we demonstrate, with a few numerical examples, that the predicted convergence rates are achieved in practice.