Adaptivity and a posteriori error control for bifurcation problems II: Incompressible fluid flow in open systems with Z_2 symmetry

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 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.

High-order hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows

This article is concerned with the construction of general isotropic and anisotropic adaptive strategies, as well as hp-mesh refinement techniques, in combination with dual-weighted-residual a posteriori error indicators for the discontinuous Galerkin finite element discretization of compressible fluid flow problems.

Chaotic mixing of a competitive-consecutive reaction

The evolution of a competitive-consecutive chemical reaction is computed numerically in a two-dimensional chaotic fluid flow with initially segregated reactants. Results from numerical simulations are used to evaluate a variety of reduced models commonly adopted to model the full advection-reaction-diffusion problem. Particular emphasis is placed upon fast reactions, where the yield varies most significantly with Peclet number (the ratio of diffusive to advective time scales). When effects of the fluid mechanical mixing are strongest, we find that the yield of the reaction is underestimated by a one-dimensional lamellar model that ignores the effects of fluid mixing, but overestimated by two other lamellar models that include fluid mixing.