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

✍ Scribed by Stefano Serra Capizzano; Cristina Tablino Possio

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 330 KB
• Volume: 293
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00022-1

✦ Synopsis

In this paper we study the structural properties of matrices coming from high-precision Finite Dierence (FD) formulae, when discretizing elliptic (or semielliptic) differential operators vY u of the form

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 de®ne optimal Toeplitz preconditioners in a very compact and natural way and, in addition, to prove Szeg o-like and Widom-like ergodic theorems for the spectra of the related preconditioned matrices. A wide numerical experimentation, con®rming the theoretical results, is also reported.