On the effect of boundary conditions on the scalability of Schwarz methods

TL;DR

This paper investigates how different boundary conditions affect the convergence and scalability of one-level Schwarz methods, extending understanding beyond classical cases to mixed boundary conditions.

Contribution

It provides new analysis of convergence and scalability for Schwarz methods under mixed boundary conditions, which was previously less understood.

Findings

Scalability depends on boundary condition configurations

Convergence behavior varies with mixed boundary conditions

Extended theoretical results for different boundary setups

Abstract

In contrast with classical Schwarz theory, recent results have shown that for special domain geometries, one-level Schwarz methods can be scalable. This property has been proved for the Laplace equation and external Dirichlet boundary conditions. Much less is known if mixed boundary conditions are considered. This short manuscript focuses on the convergence and scalability analysis of one-level parallel Schwarz method and optimized Schwarz method for several different external configurations of boundary conditions, i.e., mixed Dirichlet, Neumann and Robin conditions.