Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs

In the present paper we are interested in the Finite Difference (FD) discretization of elliptic second order PDEs of the form \[ -\sum_{i,j=1}^d {\frac{\partial }{\partial x_i}} \left(a_{i,j}(x) {\frac{\partial }{\partial x_j}} u(x)\right)=b(x) \] over $(0,1)^d$ with Dirichlet boundary conditions. The resulting sequence of algebraic systems is characterized by matrices expressible in terms of weighted sums of dyads. The considered representation formulas are then used in order to find deep relationships with Generalized Locally Toeplitz sequences \cite{tilliloc2} and matrix-valued Linear Positive Operators \cite{Sergo}. As a direct consequence we obtain a complete understanding of the distributional spectral properties of these FD matrix sequences, that are used in order to devise an efficient numerical (iterative) solution of the associated linear systems.