Precise finite element error analysis by Yamamoto's explicit inversion formula for tridiagonal matrices -- an extension of Babuška-Osborn's theorems
type: Item
instance of: scholarly article
title: Precise finite element error analysis by Yamamoto's explicit inversion formula for tridiagonal matrices -- an extension of Babuška-Osborn's theorems
zbMATH Open document ID: 1044.65068
author: Takuya Tsuchiya
published in: Numerische Mathematik
publication date: 2003-08-19T00:00:00Z
review text: The paper is concerned with error estimates for linear spline approximations \(u_h\) to boundary value problems of the form \(-(pu')'= f\) on \((a, b)\), \(u(a)= u(b)= 0\), with \(p\in L^\infty(a, b)\) bounded away from zero. The error analysis is based on an explicit formula for the inverse of the (tridiagonal) discretized matrix, and a splitting of the difference between this inverse and the Green's function into an approximation and an interpolation error term, resulting in a corresponding splitting of the error \(u -u_h\). Particular emphasis is put on the question if, for a sufficiently ``rough'' coefficient function \(p\), the convergence \(u- u_h\to 0\) can be arbitrarily slow, which has been answered affirmatively by \textit{I. Babuska} and \textit{J. Osborn} [Math. Comput. 69, No. 230, 443--462 (2000; Zbl 0940.65086)] for a more specific boundary value problem. Using their construction of \(p\), the author proves that the same holds true for the more general boundary value problem under consideration here, by detailed estimates of approximation and interpolation error.
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 65L20
zbMATH DE Number: 1967711
zbMATH Keywords: linear finite elements
zbMATH Keywords: error estimates
zbMATH Keywords: convergence
reviewed by: Michael Plum
MaRDI profile type: Publication
full work available at URL: s00211-002-0423-z
OpenAlex ID: W2052388901
DOI: 10.1007/S00211-002-0423-Z
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description: scientific article; zbMATH DE number 1967711