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Transverse magnetic (TM) scattering of an electromagnetic wave from aperiodic dielectric diffraction grating can mathematically be described by avolume integral equation. This volume integral equation, however, in generalfails to feature a weakly singular integral operator. Nevertheless, after asuitable periodization, the involved integral operator can be efficientlyevaluated on trigonometric polynomials using the fast Fourier transform (FFT)and iterative methods can be used to solve the integral equation. UsingFredholm theory, we prove that a trigonometric Galerkin discretization appliedto the periodized integral equation converges with optimal order to thesolution of the scattering problem. The main advantage of this FFT-baseddiscretization scheme is that the resulting numerical method is particularlyeasy to implement, avoiding for instance the need to evaluate quasiperiodicGreen's functions.

A Trigonometric Galerkin Method for Volume Integral Equations Arising in TM Grating Scattering