Adaptivity and a posteriori error control for bifurcation problems. II: Incompressible fluid flow in open systems with \(Z_{2}\) symmetry. (English) Zbl 1311.76052
Summary: 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.
For part I, see [Commun. Comput. Phys. 8, 845–865 (2010)].
For part I, see [Commun. Comput. Phys. 8, 845–865 (2010)].
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
incompressible flows; bifurcation problems; \(a\) \(posteriori\) error estimation; adaptivity; discontinuous Galerkin methods; \(Z _{2}\) symmetryReferences:
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