ETNA  Electronic Transactions on Numerical Analysis

Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A1011 Wien, Dr. Ignaz SeipelPlatz 2
Tel. +431515 81/DW 3420, Fax +431515 81/DW 3400 https://verlag.oeaw.ac.at, email: verlag@oeaw.ac.at 

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BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)

ETNA  Electronic Transactions on Numerical Analysis ISBN 9783700182580 Online Edition Research Article
Giovanni Barbarino,
Carlo Garoni,
Stefano SerraCapizzano
Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case ()
S. 113  216doi:10.1553/etna_vol53s113 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol53s113
Abstract: In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from thenumerical discretization of a partial differential equation (PDE), knowledge of the spectral distribution of the associated matrix has proved to beuseful 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 meshfineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to asequence $\\{A_n\\}_n$, 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 GLTsequence. In particular, multilevel block GLT sequences are encountered in the discretization of systems of PDEs as well as in the higherorder finite element or discontinuousGalerkin 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 SerraCapizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I., Springer, Cham, 2017],multilevel GLT sequences[Garoni and SerraCapizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II., Springer, Cham, 2018],and block GLT sequences[Barbarino, Garoni, and SerraCapizzano, Electron. Trans. Numer. Anal., 53 (2020), pp. 28–112].We also present several emblematic applications of this theory in the context of PDE discretizations. Keywords: asymptotic distribution of singular values and eigenvalues, multilevel block Toeplitz matrices, multilevel block generalized locally Toeplitz matrices, numerical discretization of partial differential equations, finite differences, finite elements, isogeometric analysis, discontinuous Galerkin methods, tensor products, Bsplines Published Online: 2020/01/30 09:36:23 Object Identifier: 0xc1aa5572 0x003b41cb Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 10689613. …

Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A1011 Wien, Dr. Ignaz SeipelPlatz 2
Tel. +431515 81/DW 3420, Fax +431515 81/DW 3400 https://verlag.oeaw.ac.at, email: verlag@oeaw.ac.at 