@TechReport{ it:2019-004,
author = {Giovanni Barbarino and Carlo Garoni and Stefano
Serra-Capizzano},
title = {Block Generalized Locally {T}oeplitz Sequences: Theory and
Applications in the Unidimensional Case},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2019},
number = {2019-004},
month = jul,
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), the knowledge of the spectral distribution
of the associated matrix has proved to be a 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.}
}