# On the Scalability of the Schwarz Method

## TL;DR

This paper investigates the convergence and scalability of the one-level Parallel Schwarz method for complex domain decomposition problems, especially those with interior subdomains and multiple overlaps, extending existing theoretical results.

## Contribution

It develops a systematic framework to analyze the Schwarz method's convergence for interior subdomains with arbitrary overlaps, broadening previous limited scenarios.

## Key findings

- Bounded the norm of the Schwarz iteration operator for interior subdomains.
- Extended scalability analysis to multiple overlapping subdomains.
- Provided theoretical insights applicable to solvation models in chemistry.

## Abstract

In this article, we analyse the convergence behaviour and scalability properties of the one-level Parallel Schwarz method (PSM) for domain decomposition problems in which the boundaries of many subdomains lie in the interior of the global domain. Such problems arise, for instance, in solvation models in computational chemistry. Existing results on the scalability of the one-level PSM are limited to situations where each subdomain has access to the external boundary, and at most only two subdomains have a common overlap. We develop a systematic framework that allows us to bound the norm of the Schwarz iteration operator for domain decomposition problems in which subdomains may be completely embedded in the interior of the global domain and an arbitrary number of subdomains may have a common overlap.