@TechReport{ it:2019-005,
author = {Giovanni Barbarino and Carlo Garoni and Stefano
Serra-Capizzano},
title = {Block Generalized Locally {T}oeplitz Sequences: Theory and
Applications in the Multidimensional Case},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2019},
number = {2019-005},
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 partial differential
equation (PDE), 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 multilevel 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
PDEs. 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 multilevel GLT
sequence or one of its ``relatives'', i.e., a multilevel
block GLT sequence or a (multilevel) reduced GLT sequence.
In particular, multilevel block GLT sequences are
encountered in the discretization of systems of PDEs as
well as in the higher-order finite element or discontinuous
Galerkin approximation of scalar/vectorial PDEs.
In this work, we systematically develop the theory of
multilevel block GLT sequences as an extension of the
theories of (unilevel) GLT sequences \cite{GLT-bookI},
multilevel GLT sequences \cite{GLT-bookII}, and block GLT
sequences \cite{bg}. We also present several emblematic
applications of this theory in the context of PDE
discretizations.}
}