TY - JOUR

T1 - Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space

AU - Bruno, Oscar P.

AU - Shipman, Stephen P.

AU - Turc, Catalin

AU - Venakides, Stephanos

N1 - Funding Information:
The authors gratefully acknowledge support from AFOSR and NSF under contracts FA9550-15-1-0043 and DMS-1411876 (O.P.B.); NSF DMS-0807325 (S.P.S.); NSF DMS-1008076 (C.T.) and NSF DMS-0707488 and NSF DMS-1211638 (S.V.).
Publisher Copyright:
© 2016 The Author(s) Published by the Royal Society.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - This work, 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 smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function-that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) 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.

AB - This work, 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 smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function-that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) 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.

KW - Scattering,Periodic Green function,Lattice sum,Smooth truncation,Super-algebraic convergence,Boundary-integral equations

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U2 - 10.1098/rspa.2016.0255

DO - 10.1098/rspa.2016.0255

M3 - Article

AN - SCOPUS:84980047914

SN - 1364-5021

VL - 472

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2191

M1 - 20160255

ER -