  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.
