title: Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space.
abstract: 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.
authors: Applied and Computational Mathematics , Caltech, Pasadena, CA 91125, USA.
authors: Oscar P
authors: OP
authors: Bruno
authors: Department of Mathematics , Louisiana State University , Baton Rouge, LA 70803, USA.
authors: Stephen P
authors: 0000-0001-6620-6528
authors: SP
authors: Shipman
authors: Department of Mathematical Sciences , New Jersey Institute of Technology , Newark, NJ 07102, USA.
authors: Catalin
authors: C
authors: Turc
authors: Department of Mathematics , Duke University , Durham, NC 27708, USA.
authors: Stephanos
authors: S
authors: Venakides
keywords: boundary-integral equations; lattice sum; periodic Green function; scattering; smooth truncation; super-algebraic convergence