We consider the Finite Difference discretization of an
elliptic second order PDE as 
$-\sum_{i,j=1}^d {\partial \over \partial x_i} \left(a_{i,j}(x)
{\partial \over \partial x_j} u(x)\right)=b(x)$ over a bounded domain and we
study the singular value  distribution (the eigenvalue distribution in the
Hermitian case) in terms of weighted multidimensional Szeg\H o formulas
involving the data of the problem (the matrix $A(x)=\left(a_{i,j}(x)\right)$
and the domain) and the used formulas for the derivatives
${\partial \over \partial x_j}$, $j=1,\ldots,d$.
Some applications of this theoretic analysis and of the extremal
spectral behaviour to the numerical solution of the corresponding PDEs
linear systems are briefly discussed.