Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case

Abstract

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

Fast Parallel Solver for the Space-time IgA-DG Discretization of the Diffusion Equation

“…In the context of (nodal) DG methods [33] The solution of system (2.1) yields the approximate solution of problem (1.1); see [10] for details. The main result of [10] is reported in Theorem 2.1 below; see also [8,Sect. 6.2] for a more recent and lucid proof.…”

“…Suppose the hypotheses of Theorem 2.1 are satisfied. Then, on the basis of Theorem 2.2 and the theory of (block) generalized locally Toeplitz (GLT) sequences [7,8,30,31,52,53], we expect that the sequence of preconditioned matrices…”

A systematic approach to reduced GLT

“…A similar scheme can be found in [20], where all the points are the same, except for GLT1, GLT2 and GLT6, that can be deduced from the results in [6,8]. Moreover, GLT8 has been substituted with its more powerful version from [10] and GLT9 has been expanded to include the fact that every measurable function is a GLT symbol for some sequence.…”

“…Following the lead of the classical GLT sequences, the next step is to generalize the space of reduced GLT to the case of block sequences, studied in [9,10], in order to tackle also systems of PDEs and high-order approximations on generic domains.…”

“…It stems from Tilli's work on locally Toeplitz (LT) sequences [32] and from the spectral theory of Toeplitz matrices. Nowadays, the main and most comprehensive sources in the literature for theory and applications of GLT spaces are the books [9,10,19,20], in which we can find a careful and complete description of GLT sequences, block GLT sequences, and their respective multivariate versions. For more information, check also the articles [4][5][6][7][8]27,28].…”