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.
Highlights
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
We are interested in numerically estimating the critical Reynolds number Re, at which a bifurcation point first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al [13], for example
In [11] we considered the application of the Dual Weighted Residual (DWR) a posteriori error estimation technique to compute the eigenvalues μ for a series of parameter values λ0, while in [10] the error estimation was directed at computing the critical parameter value, i.e. the value where the steady solution first loses stability
Summary
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 derivation of a computable error estimator for the critical parameter of interest, namely Re, based on exploiting the Dual Weighted Residual (DWR) a posteriori error estimation technique is undertaken and implemented within an adaptive finite element algorithm The application of this approach to study steady pitchfork bifurcations highlights the numerical performance of the error estimation techniques developed in this article. To the best of our knowledge, our article represents the first attempt to derive a posteriori error bounds on critical parameter values for the hydrodynamic stability problem in the O(2) setting
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