From Partial Differential Equations to generalized Locally Toepliz sequences

Starting from the Finite Difference discretization of a general second order partial differential equation (PDE), we introduce the notion of generalized Locally Toeplitz sequence of matrices. The singular value distribution (the eigenvalue distribution in the Hermitian case)is studied and characterized for generalized Locally Toeplitz sequences in terms of weighted multidimensional Szego" formulas which extend previous results due to the second author and concerning the unilevel case. The application of this theoretic analysis to the numerical solution of elliptic PDEs is finally discussed.