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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.

Keywords

Dual Weighted Residual (DWR) Discontinuous Galerkin finite Element Approximation Posteriori Error Estimates Interior Penalty Discontinuous Galerkin Method Symmetry Breaking Bifurcation 
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References

  1. 1.
    Aston, P.: Analysis and computation of symmetry-breaking bifurcation and scaling laws using group theoretic methods.SIAM J.Math.Anal.22, 139–152 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blackburn, H., Sherwin, S., Barkley, D.: Convective instability and transient growth in steady and pulsatile stenotic flows.J.Fluid Mech.607, 267–277 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of non-linear problems.3.Simple bifurcation points.Numer.Math.38(1), 1–30 (1981)Google Scholar
  4. 4.
    Cliffe, K., Hall, E., Houston, P.: Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows.SIAM J.Sci.Comput.31, 4607–4632 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cliffe, K., Hall, E., Houston, P., Phipps, E., Salinger, A.: Adaptivity and a posteriori error control for bifurcation problems I: The Bratu problem.Commun.Comput.Phys.8, 845–865 (2010)MathSciNetGoogle Scholar
  6. 6.
    Cliffe, K., Hall, E., Houston, P., Phipps, E., Salinger, A.: Adaptivity and a posteriori error control for bifurcation problems II: Incompressible fluid flow in open systems with Z 2symmetry.J.Sci.Comput.47(3), 389–418 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cliffe, K., Hall, E., Houston, P., Phipps, E., Salinger, A.: Adaptivity and a posteriori error control for bifurcation problems III: Incompressible fluid flow in open systems with O(2) symmetry.J.of Sci.Comput.52(1), 153–179 (2012).In pressGoogle Scholar
  8. 8.
    Cliffe, K., Spence, A., Tavener, S.: O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe.Internat.J.Numer.Methods Fluids 32, 175–200 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cockburn, B., Kanschat, G., Schötzau, D.: The local discontinuous Galerkin method for the Oseen equations.Math.Comp.73, 569–593 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local discontinuous Galerkin methods for the Stokes system.SIAM J.Numer.Anal.40, 319–343 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Golubitsky, M., Schaeffer, D.: Singularities and Groups in Bifurcation Theory, Vol I.Springer, New York (1985)zbMATHGoogle Scholar
  12. 12.
    Houston, P., Süli, E.: A note on the design of hp–adaptive finite element methods for elliptic partial differential equations.Comput.Methods Appl.Mech.Engrg.194(2–5), 229–243 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Sherwin, S., Blackburn, H.: Three–dimensional instabilities and transition of steady and pulsatile axisymmetric stenotics flows.J.Fluid Mech.533, 297–327 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Werner, B., Spence, A.: The computation of symmetry-breaking bifurcation points.SIAM J.Numer.Anal.21, 388–399 (1984)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Nottingham, University ParkNottinghamUK

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