On non-local exchange and scattering operators in domain decomposition methods

Abstract. We study non-local exchange and scattering operators arising in domain decomposition algorithms for solving elliptic problems on domains in R2. Motivated by recent formulations of the Optimized Schwarz Method introduced by Claeys, we rigorously analyze the behavior of a family of non-local exchange operators Πγ, defined in terms of boundary integral operators associated to the fundamental solution for −∆ + γ−2, with γ > 0. Our first main result establishes precise estimates comparing Πγ to its local counterpart Π0 as γ → 0, providing a quantitative bridge between the classical and non-local formulations of the Optimized Schwarz Method. In addition, we investigate the corresponding scattering operators, proving norm estimates that relate them to their classical analogues through a detailed analysis of the associated Dirichlet-to-Neumann operators. Our results clarify the relationship between classical and non-local formulations of domain decomposition methods and yield new insights that are essential for the analysis of these algorithms, particularly in the presence of cross points and for domains with curvilinear polygonal boundaries.