Error estimates over infinite intervals of some discretizations of evolution equations
type: Item
instance of: scholarly article
title: Error estimates over infinite intervals of some discretizations of evolution equations
zbMATH Open document ID: 0573.65038
DOI: 10.1007/BF01934901
published in: BIT
publication date: 1984-01-01T00:00:00Z
review text: The author considers the initial value problem in a reflexive B-space \(V: du/dt+F(t,u)=0\), \(u(0)=u_ 0\in V\), \(t>0\). The operator \(F: V\to V'\) is not necessarily bounded. Four cases are investigated: F monotone, F strongly monotone, F conservative, F non-monotone (the case (F(t,u)- F(t,v), u-v)\(\leq -\rho_ 0\| u-v\|^ 2\), \(\forall u,v\in V\) with \(\rho_ 0>0)\). The (implicit) \(\vartheta\)-method is applied: \(v(t-k)- v(t)+kF(t+(1-\vartheta)k,\vartheta v(t)+(1-\vartheta)v(t+k))=0\) with \(0\leq \vartheta \leq 1\). It is assumed that this nonlinear equation has a unique solution in V. In each of the discussed cases estimates for the error \(e(t)=u(t)-v(t)\) are given in terms of e(0) and of the truncation error \[ R_{\vartheta}(t,u)=F(t+(1-\vartheta)k,\vartheta u(t)+(1- \vartheta)u(t+k))-k^{-1}\int^{t+k}_{t}F(s,u(s))ds. \] Many remarks on the behaviour of solutions are joint, also the so called 'forced solution' (slowly increasing) in the non-monotone case is discussed. One of the sections of the paper is devoted to estimates of the truncation error. The last section discusses the so called boundary value techniques, in the context of the forced solution for the non-monotone case. The method presented here is composed of the repeated midpoint formula and the backward Euler on the last step only.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 3915525
zbMATH Keywords: theta-method
zbMATH Keywords: stability
zbMATH Keywords: forced solution
zbMATH Keywords: stiffness
zbMATH Keywords: initial value problem
zbMATH Keywords: reflexive B-space
zbMATH Keywords: monotone
zbMATH Keywords: strongly monotone
zbMATH Keywords: conservative
zbMATH Keywords: non-monotone
zbMATH Keywords: truncation error
author: Owe Axelsson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01934901
OpenAlex ID: W2093367230
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description: scientific article; zbMATH DE number 3915525