The GLT class as a Generalized Fourier Analysis and applications

Recently, the class of Generalized Locally Toeplitz (GLT) sequences has been introduced [17, 18] as a generalization both of classical Toeplitz sequences and of variable coefficient differential operators and, for every sequence of the class, it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum [3, 21] in terms of a function (the symbol) that can be easily identified; see also [20]. This generalizes the notion of a symbol for differential operators (discrete and continuous) or for Toeplitz sequences for which it is identified through the Fourier coefficients and is related to the classical Fourier Analysis. The GLT class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations, products, and inversion when the sequence which is inverted shows a sparsely vanishing symbol (sparsely vanishing symbol = a symbol which vanishes at most in a set of zero Lebesgue measure). Furthermore, the GLT class virtually includes any approximation by local methods (Finite Difference, Finite Element, Isogeometric Analysis [4, 5, 19] etc) of partial differential equations (PDEs) and, based on this, we demonstrate that our results on GLT sequences can be used in a PDE setting in various directions: 1. as a generalized Fourier Analysis for the design and for the study of preconditioned iterative and semi-iterative methods, when dealing with variable coefficients, non rectangular domains, non uniform gridding or triangulations, 2. for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case, ∗Department of Science and High Technology, Insubria University, Via Valleggio 11, 22100 Como (ITALY); Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden. Email: [email protected]; [email protected]