The convergence rate of the minimal residual method for the Stokes problem
type: Item
instance of: scholarly article
title: The convergence rate of the minimal residual method for the Stokes problem
zbMATH Open document ID: 0837.65026
author: A. J. Wathen
author: B. Fischer
author: David Silvester
published in: Numerische Mathematik
publication date: 1996-05-21T00:00:00Z
review text: The solution of the classical Stokes equation in fluid dynamics leads to the solution of a system \(Ax= b\) with \(A\) symmetric but indefinite. The iterative solution of this system by the minimal residual method converges like \(|r_k|\leq e_k|r_0|\), where \(r_k\) is the residual of step \(k\). The value \(e_k= \min_p \max_{x\in E} |p(x)|\), where \(p\) is a polynomial of degree at most \(k\), with constant term equal to 1. Thus means that it is the equioscillation value of the minimax polynomial for \(E\). This set \(E\) is an inclusion set for the eigenvalues of the iteration matrix, but which does not contain 0. This in this case \(E\) consists of at least one positive and one negative interval. There exist no explicit expressions for \(e_k\), but in this paper, the asymptotics for \(e= \lim_{k\to \infty} e^{1/k}_k\) are given in the form \(1- \alpha^{3/2}(bc/ad)^{1/2}+ O(\alpha^{5/2})\) when \(E= [- a, - b\alpha]\cup [c\alpha^2, d]\). Typically, \(\alpha\) is related to the discretization parameter \(k\) for the Stokes equation. There is no theoretical evidence that the estimate is optimal, but numerical experiments show that it is quite close.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 65N06
zbMATH DE Number: 802469
zbMATH Keywords: symmetric indefinite matrix
zbMATH Keywords: convergence
zbMATH Keywords: Stokes equation
zbMATH Keywords: minimal residual method
zbMATH Keywords: iteration matrix
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: s002110050138
OpenAlex ID: W2082001547
DOI: 10.1007/S002110050138
description: scientific article; zbMATH DE number 802469