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 thenumerical discretization of a partial differential equation (PDE), knowledge of the spectral distribution of the associated matrix has proved to beuseful 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 $A_n$arising from virtually any kind of numerical discretization of PDEs. Indeed, when the mesh-fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to asequence $\\{A_n\\}_n$, 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 GLTsequence. 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 discontinuousGalerkin 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 and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I., Springer, Cham, 2017],multilevel GLT sequences[Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II., Springer, Cham, 2018],and block GLT sequences[Barbarino, Garoni, and Serra-Capizzano, Electron. Trans. Numer. Anal., 53 (2020), pp. 28–112].We also present several emblematic applications of this theory in the context of PDE discretizations.