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The numerical analysis of bifurcation problems with application to fluid mechanics

Published online by Cambridge University Press:  21 March 2001

K. A. Cliffe,
A. Spence and
S. J. Tavener
K. A. Cliffe
Affiliation:
AEA Technology, Harwell Laboratory, Didcot, Oxfordshire OX11 0RA, England. E-mail: andrew.cliffe@aeat.co.uk
A. Spence
Affiliation:
School of Mathematics, University of Bath, Claverton Down, Bath BA2 7AY, England. E-mail: A.Spence@bath.ac.uk
S. J. Tavener
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. E-mail: tavener@math.psu.edu
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Abstract

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In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

Information

Type
Research Article
Information
Acta Numerica , Volume 9 , January 2000, pp. 39 - 131
Copyright
© Cambridge University Press 2000