On Non-Local Exchange and Scattering Operators in Domain Decomposition Methods

We study non-local exchange and scattering operators arising in domain decomposition algorithms for solving elliptic problems on domains in ℝ 2 . 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 Π γ , defined in terms of boundary integral operators associated to the fundamental solution for −Δ + γ − 2 , with γ > 0. Our first main result establishes precise estimates comparing Π γ to its local counterpart Π 0 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 analogs 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.