arXiv subjects

Abdessadek Rifqui

Publications and source records attributed to Abdessadek Rifqui.

2 recordsLinked to original sources

arXiv2026

Block Schwarz methods and preconditioning strategies using Generalized locally Toeplitz tools - part I: analysis of the preconditioners and numerical validation

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 GLT symbolic spectral tools. Numerical experiments are presented, while, based on the study in the current work, the analysis of preconditioned matrix-sequences and proposals of new Schwarz preconditioners are given in a twin paper, ideally part II of the present work.

math.NA↗

arXiv2026

Analysis of Block Jacobi/Gauss-Seidel and additive/multiplicative Schwarz preconditioning through the theory of GLT sequences, with applications to domain decomposition discretizations

When a linear differential problem is discretized by a linear numerical method characterized by a mesh fineness parameter $n$, the computation of the numerical solution reduces to solving a linear discrete problem identified by a matrix $A_n$ whose size grows with $n$. The sequence of discretization matrices $\{A_n\}_n$ often falls within the class of generalized locally Toeplitz (GLT) sequences, even when the numerical method belongs to the family of domain decomposition methods (DDMs), as illustrated herein through examples. Four widely used preconditioners for DDM discretization matrices are the block Jacobi (BJ), block Gauss--Seidel (BGS), additive Schwarz (AS), and multiplicative Schwarz (MS) preconditioners. In this paper, we provide formal definitions of the BJ/BGS/AS/MS preconditioners for arbitrary multilevel block matrices. These definitions and the associated notations are inspired by the theory of GLT sequences and are proposed as alternatives to those commonly used by the DDM community. We analyze the structure of the BJ/BGS/AS/MS preconditioners when applied to multilevel block matrices $A_n$ belonging to a GLT sequence $\{A_n\}_n$. Every GLT sequence $\{A_n\}_n$ is uniquely associated with a special function $κ$ called symbol. We prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $κ$, then the sequences of the BJ, BGS, and MS preconditioners are GLT sequences with symbol $κ$. For the AS preconditioner, we prove that $\{P_n^{AS}(A_n)\}_n$ is a GLT sequence with symbol $κ^{AS}\approxκ$, and $κ^{AS}=κ$ whenever the overlaps in the subdomains used for the construction of $P_n^{AS}(A_n)$ vanish as $n\to\infty$. A numerical validation of these results in the context of isogeometric DDMs is presented.

math.NA↗