TY - JOUR

T1 - Block generalized locally toeplitz sequences

T2 - Theory and applications in the unidimensional case

AU - Barbarino, Giovanni

AU - Garoni, Carlo

AU - Serra-Capizzano, Stefano

N1 - Funding Information:
∗Received August 12, 2019. Accepted November 6, 2019. Published online on January 30, 2020. Recommended by L. Reichel. Carlo Garoni acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome “Tor Vergata”, CUP E83C18000100006, and the support obtained by the Beyond Borders Programme of the University of Rome “Tor Vergata” through the project ASTRID, CUP E84I19002250005. †Faculty of Sciences, Scuola Normale Superiore, Italy (giovanni.barbarino@sns.it). ‡Department of Mathematics, University of Rome “Tor Vergata”, Italy; Department of Science and High Technology, University of Insubria, Italy (garoni@mat.uniroma2.it; carlo.garoni@uninsubria.it). §Department of Humanities and Innovation, University of Insubria, Italy; Department of Information Technology, Uppsala University, Sweden (stefano.serrac@uninsubria.it; stefano.serra@it.uu.se).
Publisher Copyright:
Copyright © 2020, Kent State University.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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 An arising from virtually any kind of numerical discretization of DEs. Indeed, when the mesh-fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}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.

AB - 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 An arising from virtually any kind of numerical discretization of DEs. Indeed, when the mesh-fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}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.

KW - Asymptotic distribution of singular values and eigenvalues

KW - B-splines

KW - Block generalized locally Toeplitz matrices

KW - Block Toeplitz matrices

KW - Discontinuous Galerkin methods

KW - Finite differences

KW - Finite elements

KW - Isogeometric analysis

KW - Numerical discretization of differential equations

KW - Tensor products

UR - http://www.scopus.com/inward/record.url?scp=85080048427&partnerID=8YFLogxK

U2 - 10.1553/etna_vol53s28

DO - 10.1553/etna_vol53s28

M3 - Article

AN - SCOPUS:85080048427

SN - 1068-9613

VL - 53

SP - 28

EP - 112

JO - Electronic Transactions on Numerical Analysis

JF - Electronic Transactions on Numerical Analysis

ER -