Block generalized locally Toeplitz sequences: theory and applications in the unidimensional 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 differential equation (DE), 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 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 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/vectorial DEs. This work is a review, refinement, extension, and systematic exposition of the theory of block GLT sequences. It also includes several emblematic applications of this theory in the context of DE discretizations. Key words. asymptotic distribution of singular values and eigenvalues, block Toeplitz matrices, block generalized locally Toeplitz matrices, numerical discretization of differential equations, finite differences, finite elements, isogeometric analysis, discontinuous Galerkin methods, tensor products, B-splines.

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

“…It was then carried forward in [62,63], and it was finally developed in a systematic way in [40,Chapter 7] and [41,Chapter 4]. The theory of block LT sequences was originally suggested in [63,Section 3.3], carried forward in [45], and developed in a systematic way in [8,Chapter 3]. In this chapter, we address the multidimensional version of the theory of block LT sequences, also known as the theory of multilevel block LT sequences.…”

“…In [8], starting from the original intuition in [63,Section 3.3] and based on the recent contributions [3,6,7,9,39,42,45,46], the theory of block GLT sequences has been developed in a systematic way as an extension of the theory of GLT sequences. The focus of [8], however, is only on the unidimensional (or unilevel) version of the theory, which allows one to face only unidimensional PDEs (i.e., ordinary differential equations). In this work, we complete [8] by covering the multidimensional (or multilevel) version of the theory, also known as the theory of multilevel block GLT sequences.…”

“…The focus of [8], however, is only on the unidimensional (or unilevel) version of the theory, which allows one to face only unidimensional PDEs (i.e., ordinary differential equations). In this work, we complete [8] by covering the multidimensional (or multilevel) version of the theory, also known as the theory of multilevel block GLT sequences. Such a completion is of the utmost importance in practical applications; in particular, it provides the necessary tools for computing the spectral distribution of multilevel block matrices arising from the discretization of systems of PDEs [63,Section 3.3] and from the higher-order FE or discontinuous Galerkin (DG) approximation of scalar/vectorial PDEs [12,34,44,47].…”

“…Chapter 6 is devoted to applications. The exposition in this work is conducted on an abstract level; for motivations and insights we recommend that the reader takes a look at the extended introduction of [8]. Needless to say, the reader who knows [8] will be certainly facilitated in reading this work.…”

“…It can be shown that ER(f ) is closed and Λ(f ) ⊆ ER(f ) a.e. [8,Lemma 2.2]. In the case where the eigenvalue functions λ j (f ) : D → C, j = 1, .…”