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hp-Adaptive discontinuous Galerkin methods for bifurcation phenomena in open flows

Cliffe, Andrew; Hall, Edward; Houston, Paul

Authors

Andrew Cliffe Andrew.Cliffe@nottingham.ac.uk

Edward Hall Edward.Hall@nottingham.ac.uk

Paul Houston Paul.Houston@nottingham.ac.uk

Abstract

In this article we consider the a posteriori error estimation and adaptive mesh refinement of hp-version 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 either reflectional Z_2 symmetry, or rotational and reflectional 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 hp-adaptively refined computational meshes are presented for both two- and three-dimensional problems. In the latter case, particular attention is devoted to the problem of flow through a cylindrical pipe with a sudden expansion, which represents a notoriously difficult computational problem.

Journal Article Type Article
Publication Date Mar 30, 2014
Journal Computers and Mathematics with Applications
Print ISSN 0898-1221
Electronic ISSN 0898-1221
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 67
Issue 4
Institution Citation Cliffe, A., Hall, E., & Houston, P. (2014). hp-Adaptive discontinuous Galerkin methods for bifurcation phenomena in open flows. Computers and Mathematics with Applications, 67(4), doi:10.1016/j.camwa.2013.09.024
DOI https://doi.org/10.1016/j.camwa.2013.09.024
Publisher URL https://www.sciencedirect.com/science/article/pii/S0898122113005919
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0




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