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

In this paper we study the structural properties of matrices coming from high-precision Finite Difference (FD) formulae, when discretizing elliptic (or semielliptic) differential operators L(a,u) of the formimageStrong relationships with Toeplitz structures and Linear Positive Operators (LPO) are highlighted. These results allow one to give a detailed analysis of the eigenvalues localisation/distribution of the arising matrices. The obtained spectral analysis is then used to define optimal Toeplitz preconditioners in a very compact and natural way and, in addition, to prove Szegö-like and Widom-like ergodic theorems for the spectra of the related preconditioned matrices. A wide numerical experimentation, confirming the theoretical results, is also reported.