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.
Highlights
The theory of generalized locally Toeplitz (GLT) sequences stems from Tilli’s work on locallyToeplitz (LT) sequences [1] and from the spectral theory of Toeplitz matrices [2,3,4,5,6,7,8,9,10,11,12]
It was carried forward in [13,14,15,16], and was recently extended by Barbarino [17]. This theory is a powerful apparatus for computing the asymptotic spectral distribution of matrices arising from the numerical discretization of continuous problems, such as integral equations (IEs) and, especially, differential equations (DEs)
The experience reveals that virtually any kind of numerical methods for the discretization of DEs gives rise to structured matrices An whose asymptotic spectral distribution, as the mesh fineness parameter n tends to infinity, can be computed through the theory of GLT sequences
Summary
The theory of generalized locally Toeplitz (GLT) sequences stems from Tilli’s work on locally. In the context of Galerkin and collocation IgA discretizations of elliptic DEs, the spectral distribution computed through the theory of GLT sequences in a series of recent papers [21,22,23,24,25] was exploited in [31,32,33] to devise and analyze optimal and robust multigrid solvers for IgA linear systems.
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