Domain decomposition multiple scattering solvers by semi-infinite and infinite arrays of discrete identical scatterers in two dimensions

We present domain decomposition (DD) numerical approaches for the simulation of scattering involving semi-infinite, infinite but not necessarily periodic, as well as large finite arrays of identical obstacles in free space in two dimensions. The DD approach relies on dividing the array computational domain into infinite/finite non-overlapping copies of a unit cell structure consisting of obstacles enclosed by fictitious infinite vertical walls and subsequently solving the array scattering problem via connecting Robin data on vertical walls through Robin-to-Robin (RtR) maps. The unit cell RtR maps, in turn, are computed in the Fourier domain using layer potentials and their associated Boundary Integral Operators. The unit cell RtR maps are the key ingredient in the computation of certain transmission maps associated with semi-infinite arrays of scatterers via operator Riccatti equations. These semi-infinite transmission operators are subsequently used to obtain the solution of scattering problems involving infinite periodic arrays of scatterers, as well as infinite structures that present defects such as gaps in the periodic arrangement of scatterers. Furthermore, the same unit cell RtR maps are the building blocks of DD solutions of scattering problems by very large but finite arrays of scatterers. The very large ensuing DD linear systems are solved via direct methods that employ hierarchical Schur complements. A variety of numerical results are presented to illustrate the effectiveness of the DD approach for solving scattering problems by semi-infinite and infinite arrays of identical scatterers.This article is part of the theme issue 'Analytically grounded full-wave methods for advances in computational electromagnetics'.