#### Volume 54, pp. 31-50, 2021.

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

Carlos Echeverría, Jörg Liesen, and Petr Tichý

### 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 two-dimensional 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 of Echeverría et al. given in [Electron. Trans. Numer. Anal., 48 (2018), pp. 40–62].

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### Key words

multiplicative Schwarz method, iterative methods, convergence analysis, singularly perturbed problems, Shishkin mesh discretization, block diagonal dominance

### AMS subject classifications

15A60, 65F10, 65F35

### Links to the cited ETNA articles

[4] | Vol. 48 (2018), pp. 40-62 Carlos Echeverría, Jörg Liesen, Daniel B. Szyld, and Petr Tichý: Convergence of the multiplicative Schwarz method for singularly perturbed convection-diffusion problems discretized on a Shishkin mesh |

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