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

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