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

Cliffe, Andrew; Hall, Edward; Houston, Paul

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.

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), https://doi.org/10.1016/j.camwa.2013.09.024

• Journal Article Type: Article
• Acceptance Date: Sep 13, 2013
• Online Publication Date: Oct 28, 2013
• Publication Date: Mar 30, 2014
• Deposit Date: Apr 25, 2012
• Publicly Available Date: Oct 28, 2013
• Journal: Computers and Mathematics with Applications
• Print ISSN: 0898-1221
• Electronic ISSN: 0898-1221
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 67
• Issue: 4
• DOI: https://doi.org/10.1016/j.camwa.2013.09.024
• Public URL: https://nottingham-repository.worktribe.com/output/724103
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0898122113005919