Adaptivity and a Posteriori Error Control for Bifurcation Problems II: Incompressible Fluid Flow in Open Systems with Z 2 Symmetry

Abstract

Access from the University of Nottingham repository:http://eprints.nottingham.ac.uk/1257/1/cliffe_et_al_z2_bifurcation.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 or Hopf bifurcation occurs when the underlying physical system possesses reflectional or Z 2 symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard DualWeighted-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.Key words. Incompressible flows, bifurcation problems, a posteriori error estimation, adaptivity, discontinuous Galerkin methods, Z 2 symmetry 1. Introduction. Due to the highly nonlinear governing equations and complex geometries involved, understanding fluid flow remains one of the fundamental engineering challenges today. Often it is impossible to obtain analytical solutions to problems and numerical methods must instead be exploited. Broadly speaking, there are two numerical approaches to understanding the Navier-Stokes equations: the first involves direct 'simulation' of the time dependent problem, while the second, less commonly used approach focuses on applying nonlinear analysis to compute paths of steady solutions using numerical continuation methods and to determine their stability based on eigenvalue information. In this article we focus on the latter technique, specifically where we seek to understand how the solution structure changes as one parameter of interest is varied; in the case of the incompressible Navier-Stokes equations this parameter is typically the Reynolds number. We are particularly interested in the location of critical parameters at which a bifurcation first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al. [16], for example.Over the past few decades, ...

High-order/ $$hp$$ -adaptive discontinuous Galerkin finite element methods for acoustic problems

“…For this reason DWR has been already used in many fields. For example in [22] DWR has been used to accurately compute the lift and drag of wing profile, moreover in [23,24] DWR has been used to accurately compute eigenvalues of eigenvalue problems from different areas of physics.…”

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with 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].…”

“…also [10]. In contrast, flow through a cylindrical pipe with a sudden expansion remains poorly understood.…”

“…If a single real-valued eigenvalue crosses the imaginary axis, then a steady bifurcation occurs; on the other hand, if a complex conjugate pair cross the imaginary axis then a Hopf bifurcation occurs, in which case a time dependent solution will exist. Throughout this paper we will be concerned only with steady bifurcations, however, for the application of the methodology employed in this article to Hopf bifurcations, see [11,10], where problems exhibiting Z 2 -symmetry are considered. In [11] we considered the application of the DWR a posteriori error estimation technique to compute the eigenvalues µ for a series of parameter values λ 0 , while in [10] the error estimation was directed specifically at computing the critical parameter value, i.e.…”

Adaptivity and a Posteriori Error Control for Bifurcation Problems I: the Bratu Problem

“…Numerical experiments presented in both one-and two-dimensions clearly highlight the practical performance of the proposed a posteriori error indicator within an automatic adaptive mesh refinement strategy. The extension of these ideas to more complex problems involving incompressible fluid flows in open systems will be considered in the companion articles [11,12].…”