Abstract
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 form (−) k d k dx k a(x) d k dx k u(x) . Strong 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.
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