Generalized Locally Toeplitz sequences: spectral analysis and applications to discretized Partial Differential equations

Starting from the finite difference discretization of an elliptic second order PDE as-∑ (i,j=1) (d) ∂/∂x i(a i,j(x) ∂/∂x ju(x)=b(x)over a bounded domain, 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. This extends preceding results attributed to Tilli which concern the unilevel case. The application of this theoretic analysis to the numerical solution of PDEs is finally discussed.