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

TL;DR

This work rigorously links Claeys' non-local exchange and scattering operators in domain decomposition to the classical local formulations as the non-local parameter $\gamma$ tends to zero. It develops boundary-integral representations and a detailed DtN spectral theory to quantify how the non-local operators $Π_\gamma$ and $S_\gamma$ approximate their local counterparts $Π_0$ and $S_0$, with precise rates that depend on boundary geometry and energy constraints encoded by the sets $X_{\gamma}(M)$. The results establish sharp bounds for the DtN operator $T_γ$ and reveal how domain features such as cross-points and convex vertices affect convergence, including optimality statements via explicit counterexamples. The findings provide a rigorous foundation for using non-local exchange in Helmholtz-type domain decomposition and inform numerical implementations, especially in complex geometries where cross-points arise.

Abstract

We study non-local exchange and scattering operators arising in domain decomposition algorithms for solving elliptic problems on domains in $\mathbb{R}^2$. 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 $γ\to 0$, 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.

Figures (1)

• Figure 1: Examples from Cl of a sub-domain partition without cross-points (left) and with cross-points denoted with blue dots (right).

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Proposition 2.1
• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 2.2
• Lemma 3.1