Technical Report 2019-002

Block Generalized Locally Toeplitz Sequences: Theory and Applications

C. Garoni and S. Serra-Capizzano

April 2019

When dealing with a large linear system arising from the numerical discretization of a differential equation (DE), the knowledge of the spectral distribution of the associated matrix has proved to be a useful information for designing/analyzing appropriate solvers|especially, preconditioned Krylov and multigrid solvers for the considered system. 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 }, 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.

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