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The integral equation method is widely used in numerical simulations of 2D/3Dacoustic and electromagnetic scattering problems, which needs a large number ofvalues of the Green's functions. A significant topic is the scattering problemsin periodic domains, where the corresponding Green's functions arequasi-periodic. The quasi-periodic Green's functions are defined by series thatconverge too slowly to be used for calculations. Many mathematicians havedeveloped several efficient numerical methods to calculate quasi-periodicGreen's functions. In this paper, we will propose a new FFT-based fastalgorithm to compute the 2D/3D quasi-periodic Green's functions for both theHelmholtz equations and Maxwell's equations. The convergence results and errorestimates are also investigated in this paper. Further, the numerical examplesare given to show that, when a large number of values are needed, the newalgorithm is very competitive.

An FFT-based algorithm for efficient computation of Green's functions for the Helmholtz and Maxwell's equations in periodic domains