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We consider Finite Difference discretizations of an elliptic second order PDE as — £\ ^7 laij(x)-rp—u(x) j = b(x) over (0, l) with Dirichlet boundary conditions, where the dxd matrix A{x) = (aij(x)) is symmetric, uniformly positive definite and whose entries are Riemann integrable. We choose the discretization so that the resulting matrices {An(A)}n form a sequence of Hermitian positive definite matrices. The eigenvalue distribution has been studied and characterized [23] in terms of weighted multidimensional Szego formulas. Here by using some tools introduced in a preceding paper [17] we analyze the spectral behaviour of the preconditioned matrix sequences {An{B)An{A)}n so that B(x) is symmetric positive definite and with Riemann integrable entries. Some issues on efficient preconditioning strategies are discussed as well.

A note on the asymptotic spectra of finite difference discretizations of second order elliptic partial differential equations