Computation of Asymptotic Spectral Distributions for Sequences of Grid Operators
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L NUMERICAL METHODS
Computation of Asymptotic Spectral Distributions for Sequences of Grid Operators S. V. Morozova,b,*, S. Serra-Capizzanoc,d,**, and E. E. Tyrtyshnikova,b,e,f,*** a
Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991 Russia b Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, 119333 Russia c University of Insubria, Como, 22100 Italy d Uppsala University, Uppsala, SE-751 05 Sweden e Siedlce University, Siedlce, 08-110 Poland f Moscow Center for Fundamental and Applied Mathematics, Moscow, 119234 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received March 23, 2020; revised May 11, 2020; accepted July 7, 2020
Abstract—The asymptotic spectral properties of matrices of grid operators on polygonal domains in the plane are studied in the case of refining triangular grids. Methods available for analyzing spectral distributions are largely based on tool of the theory of generalized locally Toeplitz sequences (GLT theory). In this paper, we show that the matrices of grid operators on nonrectangular domains do not form GLT sequences. A method for calculating spectral distributions in such cases is proposed. Generalizations of GLT sequences are introduced, and preconditioner based on them are proposed. Keywords: Toeplitz matrices, locally Toeplitz sequences, GLT sequences, discretization of partial differential equations, eigenvalues, singular values, preconditioning DOI: 10.1134/S0965542520110093
1. INTRODUCTION The theory related to the study of spectral properties of matrix sequences has rapidly developed over the last decades. Most publications on this subject fall into two categories: works studying the general properties of eigenvalue and singular value distributions of matrix sequences [1–6], for example, matrix sequences produced by discretizing partial differential equations, and works studying the asymptotic behavior of individual eigenvalues of large Toeplitz matrices [7–10]. In recent years, much attention has been focused on the theory of generalized locally Toeplitz (GLT) sequences of matrices [4]. There are numerous examples of using GLT theory in the study of spectral properties of matrix sequences arising in solving partial differential equations, but the expansion of the applicability range of the theory of GLT sequences remains an open question. In this work, we discuss an example showing that the GLT theory can be insufficient for describing spectral distributions of discretization matrices even for elementary operators defined on nonrectangular domains. At the same time, we find that matrix sequence can frequently be reduced to GLT sequences by applying similarity transformations. Based on this observation, a generalization is constructed that makes it possible to work with operators defined on nonrectangular domains. To avoid restricting the results to particular differential operators and discretization methods, all arguments are given in
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