In this paper we are concerned with the spectral analysis of the sequence of preconditioned matrices {Pn-1(a, m, k)An(a, m, k)}n, where An (a, m, k) is the n × n symmetric matrix coming from a high-order Finite Difference discretization of the problem {(-)k(dk/dxk(a(x)dk/dx ku(x))) = f(x) on Ω = (0, 1), (ds/dx su(x))|∂Ω = 0 s = 0, . . . , k - 1. The coefficient function a(x) is assumed to be positive or with a finite number of zeros. The matrix Pn (a, m, k) is a Toeplitz based preconditioner constructed as Dn1/2 (a, m, k)An(1, m, k)D n1/2 (a, m, k), where Dn (a, m, k) is the suitably scaled diagonal part of An (a, m, k). The main result is the proof of the asymptotic clustering around unity of the eigenvalues of the preconditioned matrices. In addition, the "strength" of the cluster shows some interesting dependencies on the order k, on the regularity features of a(x) and on the presence of the zeros of a(x). The multidimensional case is analyzed in depth in a twin paper [38]. Copyright © 2000, Kent State University.

Serra Capizzano, S., TABLINO POSSIO, C. (2000). High-order finite difference schemes and Toeplitz based preconditioners for elliptic problems. ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS, 11, 55-84.

High-order finite difference schemes and Toeplitz based preconditioners for elliptic problems

TABLINO POSSIO, CRISTINA
2000

Abstract

In this paper we are concerned with the spectral analysis of the sequence of preconditioned matrices {Pn-1(a, m, k)An(a, m, k)}n, where An (a, m, k) is the n × n symmetric matrix coming from a high-order Finite Difference discretization of the problem {(-)k(dk/dxk(a(x)dk/dx ku(x))) = f(x) on Ω = (0, 1), (ds/dx su(x))|∂Ω = 0 s = 0, . . . , k - 1. The coefficient function a(x) is assumed to be positive or with a finite number of zeros. The matrix Pn (a, m, k) is a Toeplitz based preconditioner constructed as Dn1/2 (a, m, k)An(1, m, k)D n1/2 (a, m, k), where Dn (a, m, k) is the suitably scaled diagonal part of An (a, m, k). The main result is the proof of the asymptotic clustering around unity of the eigenvalues of the preconditioned matrices. In addition, the "strength" of the cluster shows some interesting dependencies on the order k, on the regularity features of a(x) and on the presence of the zeros of a(x). The multidimensional case is analyzed in depth in a twin paper [38]. Copyright © 2000, Kent State University.
Articolo in rivista - Articolo scientifico
finite differences, Toeplitz and Vandermonde matrices, clustering and preconditioning, ergodic theorems, spectral distribution
English
2000
11
55
84
none
Serra Capizzano, S., TABLINO POSSIO, C. (2000). High-order finite difference schemes and Toeplitz based preconditioners for elliptic problems. ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS, 11, 55-84.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/384
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