An analysis of some element-by-element techniques
type: Item
instance of: scholarly article
title: An analysis of some element-by-element techniques
zbMATH Open document ID: 0694.65013
DOI: 10.1016/0045-7825(89)90052-2
published in: Computer Methods in Applied Mechanics and Engineering
publication date: 1989-01-01T00:00:00Z
review text: The paper provides an analysis of some EBE preconditioners \(P_{EBE}\) originally introduced by \textit{T. J. R. Hughes}, \textit{I. Levit} and \textit{J. Winger} [ibid. 36, 241-354 (1983; Zbl 0487.73083)]. The EBE preconditioning effects are compared with those obtained by diagonal scaling. This comparison is made for finite element approximations \(Ax=b\) to the simple model problem \(-\Delta u=f\) in \(\Omega =(0,1)^ d\) and \(u=0\) on \(\partial \Omega\) for \(d=1,2,3\). On the basis of the numerical computation of the corresponding spectral condition numbers \(\kappa\) (.) for finer and finer grids, the author observes that \(\kappa (P^{- 1}_{diag}A)\simeq \alpha \kappa (P^{-1}_{EBE}A)\) with some \(\alpha\) between 3 and 9, where \(P_{diag}=diag(A).\) This relation would imply that the EBE preconditioners \(P_{EBE}\) cannot improve the asymptotic behaviour of \(\kappa (P^{-1}_{EBE}A)\) in comparison with \(\kappa\) (A). For instance, for second-order elliptic boundary value problems, we would have \(\kappa (P^{- 1}_{EBE}A)=O(\kappa (A))=O(h^{-2})\) if the discretization parameter h \((h=O(n^{-1/d}_{nodes}))\) tends to zero. Thus, it would be desirable to analyze the EBE ``preconditioners'' for more complicated problems (strongly discontinuous coefficients, irregular domains, systems of PDEs etc.) instead of the model problems considered in this paper.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 4137241
zbMATH Keywords: iterative methods
zbMATH Keywords: EBE-methods
zbMATH Keywords: EBE preconditioning
zbMATH Keywords: scaling
zbMATH Keywords: finite element
zbMATH Keywords: spectral condition numbers
author: A. J. Wathen
reviewed by: Ulrich Langer
describes a project that uses: nag
describes a project that uses: NAG
MaRDI profile type: Publication
full work available at URL: 0045-7825(89)90052-2
OpenAlex ID: W1965268863
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description: scientific article; zbMATH DE number 4137241