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 [Image] 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
Serra Capizzano, S., TABLINO POSSIO, C. (1999). Spectral and structural analysis of high precision finite difference matrices for elliptic operators. LINEAR ALGEBRA AND ITS APPLICATIONS, 293(1-3), 85-131.
Spectral and structural analysis of high precision finite difference matrices for elliptic operators
TABLINO POSSIO, CRISTINA
1999
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 [Image] 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 reportedI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.