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

• Published in: Axioms Vol. 7; no. 3; p. 49
• Main Authors: Garoni, Carlo, Mazza, Mariarosa, Serra-Capizzano, Stefano
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-09-2018
• Subjects: B-splines; curl–curl operator; Differential equations; discontinuous Galerkin methods; Discretization; discretization of systems of differential equations; Eigenvalues; Fineness; finite difference methods; Finite element method; Galerkin method; generalized locally Toeplitz sequences; higher-order finite element methods; Information technology; isogeometric analysis; Linear algebra; Numerical analysis; Partial differential equations; spectral (eigenvalue) and singular value distributions; Theory; time harmonic Maxwell’s equations and magnetostatic problems; New York; United States > US; Sweden
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms7030049

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.