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

TL;DR

This paper introduces a new FFT-based algorithm for efficiently computing quasi-periodic Green's functions in 2D and 3D for Helmholtz and Maxwell's equations, improving computational speed for large-scale scattering problems.

Contribution

A novel FFT-based method for fast calculation of quasi-periodic Green's functions, with convergence analysis and demonstrated efficiency in large-scale computations.

Findings

The algorithm significantly accelerates Green's function computations.

Convergence and error estimates are rigorously established.

Numerical examples confirm the method's competitiveness for large data sets.

Abstract

The integral equation method is widely used in numerical simulations of 2D/3D acoustic and electromagnetic scattering problems, which needs a large number of values of the Green's functions. A significant topic is the scattering problems in periodic domains, where the corresponding Green's functions are quasi-periodic. The quasi-periodic Green's functions are defined by series that converge too slowly to be used for calculations. Many mathematicians have developed several efficient numerical methods to calculate quasi-periodic Green's functions. In this paper, we will propose a new FFT-based fast algorithm to compute the 2D/3D quasi-periodic Green's functions for both the Helmholtz equations and Maxwell's equations. The convergence results and error estimates are also investigated in this paper. Further, the numerical examples are given to show that, when a large number of values are needed, the new algorithm is very competitive.