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Numerical Mathematics and Advanced Applications 2011 pp 333–340Cite as

Application of hp–Adaptive Discontinuous Galerkin Methods to Bifurcation Phenomena in Pipe Flows

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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 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 hp–adaptively refined computational meshes are presented.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    K. A. Cliffe, E. J. C. Hall & P. Houston

Authors
  1. K. A. Cliffe
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  2. E. J. C. Hall
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  3. P. Houston
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Corresponding author

Correspondence to P. Houston .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Leicester, Leicester, LE!1 7RH, United Kingdom

    Andrea Cangiani

  2. , Department of Mathematics, University of Leicester, Leicester, LE1 7RL, United Kingdom

    Ruslan L. Davidchack

  3. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Emmanuil Georgoulis

  4. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  5. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

  6. , School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

    Michael V. Tretyakov

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Cliffe, K.A., Hall, E.J.C., Houston, P. (2013). Application of hp–Adaptive Discontinuous Galerkin Methods to Bifurcation Phenomena in Pipe Flows. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_36

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