On the Scalability of the Parallel Schwarz Method in One-Dimension

TL;DR

This paper rigorously analyzes the scalability limitations of the one-level parallel Schwarz method for 1D Laplace problems, providing insights that could extend to higher-dimensional cases.

Contribution

It offers the first systematic, rigorous quantification of the lack of scalability of the classical Schwarz method in one dimension.

Findings

Quantifies the non-scalability of the method in 1D

Provides a framework for extending analysis to higher dimensions

Highlights conditions under which scalability is achieved or lost

Abstract

In contrast with classical Schwarz theory, recent results in computational chemistry have shown that for special domain geometries, the one-level parallel Schwarz method can be scalable. This property is not true in general, and the issue of quantifying the lack of scalability remains an open problem. Even though heuristic explanations are given in the literature, a rigorous and systematic analysis is still missing. In this short manuscript, we provide a first rigorous result that precisely quantifies the lack of scalability of the classical one-level parallel Schwarz method for the solution to the one-dimensional Laplace equation. Our analysis technique provides a possible roadmap for a systematic extension to more realistic problems in higher dimensions.