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Axioms 2018, 7(3), 49; https://doi.org/10.3390/axioms7030049

Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

1
Institute of Computational Science, University of Italian Switzerland, 6900 Lugano, Switzerland
2
Department of Science and High Technology, University of Insubria, 22100 Como, Italy
3
Division of Numerical Methods in Plasma Physics, Max Planck Institute for Plasma Physics, 85748 Garching bei München, Germany
4
Department of Information Technology, Uppsala University, P.O. Box 337, SE-751 05 Uppsala, Sweden
*
Author to whom correspondence should be addressed.
Received: 9 May 2018 / Revised: 6 July 2018 / Accepted: 16 July 2018 / Published: 19 July 2018
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Abstract

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 differential equations (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 DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations. View Full-Text
Keywords: spectral (eigenvalue) and singular value distributions; generalized locally Toeplitz sequences; discretization of systems of differential equations; higher-order finite element methods; discontinuous Galerkin methods; finite difference methods; isogeometric analysis; B-splines; curl–curl operator; time harmonic Maxwell’s equations and magnetostatic problems spectral (eigenvalue) and singular value distributions; generalized locally Toeplitz sequences; discretization of systems of differential equations; higher-order finite element methods; discontinuous Galerkin methods; finite difference methods; isogeometric analysis; B-splines; curl–curl operator; time harmonic Maxwell’s equations and magnetostatic problems
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Garoni, C.; Mazza, M.; Serra-Capizzano, S. Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications. Axioms 2018, 7, 49.

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