Domain decomposition methods and preconditioning strategies using generalized locally Toepltiz tools: proposals, analysis, and numerical validation

TL;DR

This paper uses generalized locally Toeplitz (GLT) sequences to analyze the spectral properties and convergence of Schwarz domain decomposition methods, providing explicit spectral expressions and insights into their efficiency and scalability.

Contribution

It introduces a GLT-based spectral analysis framework for Schwarz methods, offering explicit spectral characterizations and a unified understanding of their convergence behavior.

Findings

GLT symbols describe spectral evolution with mesh refinement

Restricted Schwarz variants improve parallel efficiency

Numerical experiments confirm theoretical predictions

Abstract

In the current work we present a spectral analysis of the additive and multiplicative Schwarz methods within the framework of domain decomposition techniques, by investigating the spectral properties of these classical Schwarz preconditioning matrix-sequences, with emphasis on their convergence behavior and on the effect of transmission operators. In particular, after a general presentation of various options, we focus on restricted variants of the Schwarz methods aimed at improving parallel efficiency, while preserving their convergence features. In order to rigorously describe and analyze the convergence behavior, we employ the theory of generalized locally Toeplitz (GLT) sequences, which provides a robust framework for studying the asymptotic spectral distribution of the discretized operators arising from Schwarz iterations. By associating each operator sequence with the appropriate GLT symbol, we derive explicit expressions for the GLT symbols of the convergence factors, for both additive and multiplicative Schwarz methods. The GLT-based spectral approach offers a unified and systematic understanding of how the spectrum evolves with mesh refinement and overlap size (in the algebraic case). Our analysis not only deepens the theoretical understanding of classical Schwarz methods, but also establishes a foundation for examining future restricted or hybrid Schwarz variants using symbolic spectral tools. These results enable the prediction of the remarkable efficiency of block Jacobi/Gauss--Seidel and block additive/multiplicative Schwarz preconditioners for GLT sequences, as further illustrated through a wide choice of numerical experiments.