Spectral and structural analysis of high precision finite difference matrices for elliptic operators

In this paper we study the structural properties of matrices coming from hi gh-precision Finite Difference (FD) formulae, when discretizing elliptic (o r semielliptic) differential operators L(a, u) of the form (-)(k)(dx(k)/d(k)(a(x)d(k)/dx(k)u(x))). Strong relationships with Toeplitz structures and Linear Positive Operators (LIPO) are highlighted. These results allow one to give a detailed analysi s of the eigenvalues localisation/distribution of the arising matrices. The obtained spectral analysis is then used to define optimal Toeplitz precond itioners in a very compact and natural way and, in addition, to prove Szego -like and Widom-like ergodic theorems for the spectra of the related precon ditioned matrices. A wide numerical experimentation, confirming the theoret ical results, is also reported. (C) 1999 Elsevier Science Inc. All rights r eserved.