Superalgebraically Convergent Smoothly-Windowed Lattice Sums for Doubly Periodic Green Functions in Three-Dimensional Space
Abstract
This paper, Part I in a two-part series, presents (i) A simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) An associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain "Wood frequencies" at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on use of smooth windowing functions, gives rise to lattice sums which converge superalgebraically fast--that is, faster than any power of the number of terms used--in sharp contrast with the extremely slow convergence exhibited by the corresponding sums in absence of smooth windowing. (The Wood-frequency problem is treated in Part II.) A proof presented in this paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.
Keywords
Cite
@article{arxiv.1307.1176,
title = {Superalgebraically Convergent Smoothly-Windowed Lattice Sums for Doubly Periodic Green Functions in Three-Dimensional Space},
author = {Oscar P. Bruno and Stephen P. Shipman and Catalin Turc and Stephanos Venakides},
journal= {arXiv preprint arXiv:1307.1176},
year = {2016}
}