A bound on powers of linear operators, with relevance to numerical stability
type: Item
instance of: scholarly article
title: A bound on powers of linear operators, with relevance to numerical stability
zbMATH Open document ID: 1028.47008
DOI: 10.1016/S0893-9659(01)00091-X
author: Yanyan Li
published in: Applied Mathematics Letters
publication date: 2003-03-10T00:00:00Z
review text: The authors consider a bounded linear Banach space operator \(T\) with spectrum in the closed unit disk. It is assumed that there are two integers \(k,m\) such that \((I-zT)^{-k-1}=H_1(z)+\cdots+H_m(z)\) in the open unit disk, where \(H_j\) are operator-valued holomorphic functions with certain growth restrictions. The main result is an upper bound for the norms of the consecutive powers: \(\|T^n\|\leq C_m/{n+k \choose k}\), where the constant \(C_m\) can be specified in terms of \(H_j\). The authors then give an application of this result to the operator \(T\) of the Crank-Nicolson finite difference method for the numerical solution of the heat equation. The application yields a new stability estimate for \(\|T^n\|\).
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 1878830
zbMATH Keywords: power bounded operators
zbMATH Keywords: resolvent condition
zbMATH Keywords: numerical analysis
zbMATH Keywords: stability analysis
zbMATH Keywords: heat equation
zbMATH Keywords: finite-difference scheme
zbMATH Keywords: Crank-Nicolson method
reviewed by: Khristo N. Boyadzhiev
MaRDI profile type: Publication
cites work: Q5593264
cites work: A discrete Hille-Yosida-Phillips theorem
cites work: On resolvent conditions and stability estimates
cites work: Q3139557
cites work: Q4352376
cites work: A note about ritt's condition, related resolvent conditions and power bounded operators
cites work: Q3126808
cites work: Maximum norm contractivity of discretization schemes for the heat equation
cites work: Maximum norm contractivity in the numerical solution of the one-dimensional heat equation
cites work: Q4855462
cites work: Resolvent conditions for discretizations of diffusion-convection-reaction equations in several space dimensions
cites work: Q3254057
Wikidata QID: Q127211887
description: scientific article; zbMATH DE number 1878830