Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

Abstract

Access from the University of Nottingham repository:http://eprints.nottingham.ac.uk/1437/1/cliffe_et_al_o2_article.pdf Copyright and reuse:The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions. This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdf A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Abstract. In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses rotational and reflectional or O(2) symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual Weighted Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented. Here, particular attention is devoted to the problem of flow through a cylindrical pipe with a sudden expansion, which represents a notoriously difficult computational problem.Key words. Incompressible flows, bifurcation problems, a posteriori error estimation, adaptivity, discontinuous Galerkin methods, O(2) symmetry 1. Introduction. In this article, we study the stability of the three-dimensional incompressible Navier-Stokes equations in the case when the underlying system possesses both rotational and reflectional symmetry, or more precisely, O(2) symmetry. To this end, we are interested in numerically estimating the critical Reynolds number Re, at which a (pitchfork) bifurcation point first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al. [13], for example. The work in this article expands upon our recent work in [12] and [10] to include problems whose geometries exhibit both rotational and reflectional symmetry. The detection of bifurcation points in this setting is now well understood, for example, see Golubitsky and Schaeffer [19]. For the purposes of this article, we assume that a symmetric steady state solution to the incompressible Navier-Stokes equations undergoes a steady pitchfork bifurcation at a critical value of the Reynolds number. Estimation of the critical...

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