Analysis of the multiplicative Schwarz method for matrices with a special block structure

TL;DR

This paper investigates the convergence properties of the algebraic multiplicative Schwarz method for systems with matrices exhibiting a specific block structure, relevant in domain decomposition for PDEs, without requiring symmetry or positive definiteness.

Contribution

The paper provides a generalized convergence analysis of the multiplicative Schwarz method for block-structured matrices, extending previous models to more general, non-symmetric cases.

Findings

Derived error bounds based on block diagonal dominance.

The analysis applies to non-symmetric and non-M/H-matrix systems.

Generalizes previous results from one-dimensional models.

Abstract

We analyze the convergence of the (algebraic) multiplicative Schwarz method applied to linear algebraic systems with matrices having a special block structure that arises, for example, when a (partial) differential equation is posed and discretized on a domain that consists of two subdomains with an overlap. This is a basic situation in the context of domain decomposition methods. Our analysis is based on the algebraic structure of the Schwarz iteration matrices, and we derive error bounds that are based on the block diagonal dominance of the given system matrix. Our analysis does not assume that the system matrix is symmetric (positive definite), or has the $M$- or $H$-matrix property. Our approach is motivated by and significantly generalizes an analysis for a special one-dimensional model problem given in [4].