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

Abstract

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 $\kappa$ called symbol. We prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $\kappa$, then the sequences of the BJ, BGS, and MS preconditioners are GLT sequences with symbol $\kappa$. For the AS preconditioner, we prove that $\{P_n^{AS}(A_n)\}_n$ is a GLT sequence with symbol $\kappa^{AS}\approx\kappa$, and $\kappa^{AS}=\kappa$ 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.