The approximation of generalized turning points by projection methods with superconvergence to the critical parameter
type: Item
instance of: scholarly article
title: The approximation of generalized turning points by projection methods with superconvergence to the critical parameter
zbMATH Open document ID: 0596.65040
DOI: 10.1007/BF01389452
published in: Numerische Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133088
review text: The authors consider mildly nonlinear equations (1) \(F(z,\lambda)=x+f(x,\beta,\lambda)=0\) where \(z=(x,\beta)\) and F is a \(C^ r\)-map from an open set in \(Z=X\times R^{p-1}\) into the Banach space X. The range of \(F_ z\) is assumed to have co-dimension at most one. For the computation of singular solutions, where this co-dimension is exactly one, augmented systems of the form (2) \(F(z,\lambda)=0\), \(g_ i(z,\lambda)=0\), \(i=1,...,p\), are constructed as in the authors' earlier paper in [SIAM J. Numer. Anal. 21, 176-185 (1984; Zbl 0536.65031)]. Then the error is analyzed when the characterizing equations (2) are discretized by some general projection scheme. These results are extensions of those of \textit{F. Brezzi}, \textit{J. Rappaz}, \textit{P. Raviart} [Numer. Math. 36, 1-25 (1980; Zbl 0488.65021), 37, 1-28 (1981; Zbl 0525.65036) and 38, 1-30 (1981; Zbl 0525.65037)] and are closely related to those of \textit{J. P. Fink} and the reviewer [ibid. 45, 323-343 (1984; Zbl 0555.65034)]. A procedure is developed for the computation of the singular points. Moreover, for certain two point boundary value problems collocation at Gauss points is shown to achieve superconvergence.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65L15
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 34L99
zbMATH DE Number: 3960892
zbMATH Keywords: mildly nonlinear
zbMATH Keywords: Banach space
zbMATH Keywords: singular solutions
zbMATH Keywords: projection scheme
zbMATH Keywords: collocation
zbMATH Keywords: superconvergence
author: Andreas Griewank
author: George W. Reddien
reviewed by: Werner C. Rheinboldt
describes a project that uses: AUTO
MaRDI profile type: Publication
Wikidata QID: Q57389736
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description: scientific article; zbMATH DE number 3960892