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

摘要

The integral equation method is widely used in numerical simulations of 2D/3Dacoustic and electromagnetic scattering problems, that needs a large number ofvalues of the Green's functions. A significant topic is the scattering problemsin periodic domains, that 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 introduce a new FFT-based fastalgorithm to compute the 2D/3D quasi-periodic Green's functions for bothHelmholtz equations and Maxwell's equations. The convergence results and errorestimates are also investigated in this paper. At the end of this paper, thenumerical examples will be given to show that when large number of values areneeded, the new algorithm is very competitive.