Adaptive Discontinuous Galerkin Methods for Eigenvalue Problems Arising in Incompressible Fluid Flows

Abstract

Abstract. In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. 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 eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem, as well as the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

“…Here, the elements are marked for refinement/derefinement on the basis of the size of the elemental error indicators |η κ |, using the fixed fraction refinement algorithm with refinement and derefinement fractions set to 25% and 10%, respectively. Then, in the third example we present results for the cylindrical pipe with a sudden expansion, where we use both the error estimate derived in Section 4 and the error estimate for the eigenvalues, see [11]. In this case we compare the numerical results with recent experimental data, see Mullin et al [25].…”

“…This problem has been considered recently by Sherwin & Blackburn [26] et al [7] and also as a test problem in Cliffe et al [11]. In this setting, with a Poiseuille flow profile at the inlet, a steady O(2) symmetry breaking occurs with azimuthal wave number m = 1 when Re 0 = 721.05272346 to 8 decimal places.…”

“…For a detailed discussion, we refer to the series of articles [6,21,17,23]. We remark that, in the discussion that follows, we concentrate only on error estimation for the critical parameter value found using (3.9); error estimation of the eigenvalue can be performed in an analogous manner, see Cliffe et al [11].…”

“…In particular, the quality of the (approximate) error representation and (approximate) a posteriori bound are studied through these numerical examples. We then apply the techniques developed in this article alongside those developed in [11] to the problem of a sudden expansion in a cylindrical pipe. Finally, we summarize the work presented in this article and draw some conclusions in Section 8.…”

“…If a single real-valued eigenvalue crosses the imaginary axis, then a steady bifurcation occurs; on the other hand, if a complex conjugate pair cross the imaginary axis then a Hopf bifurcation occurs, in which case a time dependent solution will exist. Throughout this paper we will be concerned only with steady bifurcations, however, for the application of the methodology employed in this article to Hopf bifurcations, see [11,10], where problems exhibiting Z 2 -symmetry are considered. In [11] we considered the application of the DWR a posteriori error estimation technique to compute the eigenvalues µ for a series of parameter values λ 0 , while in [10] the error estimation was directed specifically at computing the critical parameter value, i.e.…”