A priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems
type: Item
instance of: scholarly article
title: A priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems
zbMATH Open document ID: 0883.65051
DOI: 10.1016/S0377-0427(96)00149-5
author: Takuya Tsuchiya
author: Ivo Babuška
published in: Journal of Computational and Applied Mathematics
publication date: 1998-03-23T00:00:00Z
review text: The authors study the parametrized nonlinear equation \(F:\Lambda\times H^1_0(J)\to H^{-1}(J)\) with one parameter \(\lambda\in\Lambda\subset \mathbb{R}\) defined as a nonlinear boundary value problem \[ F(\lambda,u)= 0,\quad (\lambda, u)\in\Lambda\times H^1_0(J),\tag{1} \] \[ (F(\lambda,u),v)= \int_J[a(\lambda, x,u'(x)) v'+ f(\lambda, x,u(x))v]dx,\quad \forall v\in H^1_0(J),\tag{2} \] where \(J:=(b,c)\subset\mathbb{R}\) is a bounded interval, and \(a,b:\Lambda\times J\times\mathbb{R}\to \mathbb{R}\) are sufficiently smooth functions, i.e. \(F\) is a second-order differential operator in divergence form. After redefining (1) and (2) using the Sobolev space \(W^{1, \infty}(J)\), the operator \(F\) becomes as smooth as the functions \(a\) and \(f\), and it is a Fredholm operator in a certain open set whose finite element solutions form smooth manifolds under suitable conditions. Making use of the extended implicit function theorem due to \textit{F. Brezzi}, \textit{J. Rappaz} and \textit{P. A. Raviart} [Numer. Math. 36, 1-25 (1980; Zbl 0488.65021); ibid. 37, 1-28 (1981; Zbl 0525.65036); ibid. 38, 1-30 (1981; Zbl 0525.65037)], the authors develop a thorough theory of a priori error estimates of finite element solutions of (1) on regular branches and on branches around turning points in the case of strongly nonlinear problems, i.e. if \(a(\lambda,x,y)\) in (2) is nonlinear with respect to \(y\). Including strongly nonlinear problems extends essentially the theory of the prior works.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 1004377
zbMATH Keywords: nonlinear boundary value problem
zbMATH Keywords: Sobolev space
zbMATH Keywords: Fredholm operator
zbMATH Keywords: finite element
zbMATH Keywords: error estimates
zbMATH Keywords: turning points
zbMATH Keywords: strongly nonlinear problems
reviewed by: Otu Vaarmann
describes a project that uses: PITCON
MaRDI profile type: Publication
cites work: Q4146134
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Wikidata QID: Q126552125
description: scientific article; zbMATH DE number 1004377