#### Volume 53, pp. 28-112, 2020.

## Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case

Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano

### Abstract

In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numerical discretization of a differential equation (DE), knowledge of the spectral distribution of the associated matrix has proved to be useful information for designing/analyzing appropriate solvers–-especially, preconditioned Krylov and multigrid solvers–-for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material). The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from virtually any kind of numerical discretization of DEs. Indeed, when the mesh-fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence or one of its “relatives”, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar/vectorial DEs. This work is a review, refinement, extension, and systematic exposition of the theory of block GLT sequences. It also includes several emblematic applications of this theory in the context of DE discretizations.

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### Key words

asymptotic distribution of singular values and eigenvalues, block Toeplitz matrices, block generalized locally Toeplitz matrices, numerical discretization of differential equations, finite differences, finite elements, isogeometric analysis, discontinuous Galerkin methods, tensor products, B-splines.

### AMS subject classifications

15A18, 15B05, 47B06, 65N06, 65N30, 65N25, 15A60, 15A69, 65D07

### ETNA articles which cite this article

Vol. 53 (2020), pp. 113-216 Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano: Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case |

Vol. 55 (2022), pp. 76-91 Manuel Bogoya, Stefano Serra-Capizzano, and Ken Trotti: Upper Hessenberg and Toeplitz Bohemian matrix sequences: a note on their asymptotical eigenvalues and singular values |

Vol. 55 (2022), pp. 585-617 Giovanni Barbarino, Carlo Garoni, Mariarosa Mazza, and Stefano Serra-Capizzano: Rectangular GLT sequences |

Vol. 58 (2023), pp. 136-163 Matthias Bolten, Sven-Erik Ekström, Isabella Furci, and Stefano Serra-Capizzano: A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices |

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