Superalgebraically Convergent Smoothly-Windowed Lattice Sums for Doubly Periodic Green Functions in Three-Dimensional Space
For researchers in computational electromagnetics and acoustics, this provides a practical solution to the long-standing problem of slow convergence in doubly periodic Green function evaluations, with rigorous proof of superalgebraic convergence.
This paper introduces a highly efficient algorithm for evaluating quasi-periodic Green functions and a boundary-integral equation method for wave scattering by doubly periodic arrays, achieving superalgebraically fast convergence of lattice sums through smooth windowing, in contrast to the slow convergence without windowing.
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.