TY - JOUR

T1 - Block generalized locally toeplitz sequences

T2 - Theory and applications in the multidimensional 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 partial differential equation (PDE), 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 multilevel 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 PDEs. 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 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 [Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I., Springer, Cham, 2017], multilevel GLT sequences [Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II., Springer, Cham, 2018], and block GLT sequences [Barbarino, Garoni, and Serra-Capizzano, Electron. Trans. Numer. Anal., 53 (2020), pp. 28-112]. We also present several emblematic applications of this theory in the context of PDE 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 partial differential equation (PDE), 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 multilevel 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 PDEs. 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 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 [Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I., Springer, Cham, 2017], multilevel GLT sequences [Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II., Springer, Cham, 2018], and block GLT sequences [Barbarino, Garoni, and Serra-Capizzano, Electron. Trans. Numer. Anal., 53 (2020), pp. 28-112]. We also present several emblematic applications of this theory in the context of PDE discretizations.

KW - Asymptotic distribution of singular values and eigenvalues

KW - B-splines

KW - Discontinuous Galerkin methods

KW - Finite differences

KW - Finite elements

KW - Isogeometric analysis

KW - Multilevel block generalized locally Toeplitz matrices

KW - Multilevel block Toeplitz matrices

KW - Numerical discretization of partial differential equations

KW - Tensor products

UR - http://www.scopus.com/inward/record.url?scp=85080042448&partnerID=8YFLogxK

U2 - 10.1553/etna_vol53s113

DO - 10.1553/etna_vol53s113

M3 - Article

AN - SCOPUS:85080042448

VL - 53

SP - 113

EP - 216

JO - Electronic Transactions on Numerical Analysis

JF - Electronic Transactions on Numerical Analysis

SN - 1068-9613

ER -