Abstract
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 bifurcation occurs when the underlying physical system possesses rotational and reflectional or 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 adaptively refined computational meshes are presented. Here, particular attention is devoted to the problem of flow through a cylindrical pipe with a sudden expansion, which represents a notoriously difficult computational problem.
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Amestoy, P.R., Duff, I.S., L’Excellent, J.-Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184, 501–520 (2000)
Amestoy, P.R., Duff, I.S., Koster, J., L’Excellent, J.-Y.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)
Amestoy, P.R., Guermouche, A., L’Excellent, J.-Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001)
Aston, P.J.: Analysis and computation of symmetry-breaking bifurcation and scaling laws using group theoretic methods. SIAM J. Math. Anal. 22, 139–152 (1991)
Becker, R., Rannacher, R.: An optimal control approach to a-posteriori error estimation in finite element methods. In: Iserles, A. (ed.) Acta Numerica, pp. 1–102. Cambridge University Press, Cambridge (2001)
Blackburn, H.M., Sherwin, S.J., Barkley, D.: Convective instability and transient growth in steady and pulsatile stenotic flows. J. Fluid Mech. 607, 267–277 (2008)
Brezzi, F., Rappaz, J., Raviart, P.A.: Finite dimensional approximation of non-linear problems 3. Simple bifurcation points. Numer. Math. 38(1), 1–30 (1981)
Cliffe, K.A., Garratt, T.J., Spence, A.: Eigenvalues of the discretized Navier-Stokes equations with application to the detection of Hopf bifurcations. Adv. Comput. Math. 1, 337–356 (1993)
Cliffe, K.A., Spence, A., Tavener, S.J.: O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe. Int. J. Numer. Methods Fluids 32, 175–200 (2000)
Cliffe, K.A., Hall, E., Houston, P.: Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows. SIAM J. Sci. Comput. 31, 4607–4632 (2010)
Cliffe, K.A., Hall, E., Houston, P., Phipps, E.T., Salinger, A.G.: Adaptivity and a posteriori error control for bifurcation problems I: The Bratu problem. Commun. Comput. Phys. 8, 845–865 (2010)
Cliffe, K.A., Hall, E., Houston, P.: Adaptivity and a posteriori error control for bifurcation problems II: Incompressible fluid flow in open systems with Z 2 symmetry. J. Sci. Comput. 47(3), 389–418 (2011)
Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40, 319–343 (2002)
Cockburn, B., Kanschat, G., Schötzau, D.: The local discontinuous Galerkin method for the Oseen equations. Math. Comput. 73, 569–593 (2004)
Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74, 1067–1095 (2005)
Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. In: Iserles, A. (ed.) Acta Numerica, pp. 105–158. Cambridge University Press, Cambridge (1995)
Fearn, R.M., Mullin, T., Cliffe, K.A.: Nonlinear flow phenomena in a symmetric sudden expansion. J. Fluid Mech. 211, 595–608 (1990)
Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. I. Springer, New York (1985)
Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Springer, New York (1988)
Houston, P., Süli, E.: Adaptive finite element approximation of hyperbolic problems. In: Barth, T., Deconinck, H. (eds.) Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics. Lect. Notes Comput. Sci. Eng., vol. 25, pp. 269–344. Springer, Berlin (2002)
Keller, H.B.: Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems. Academic Press, New York (1977)
Larson, M.G., Barth, T.J.: A posteriori error estimation for discontinuous Galerkin approximations of hyperbolic systems. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)
Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia (1998)
Mullin, T., Seddon, J.R.T., Mantle, M.D., Sederman, A.J.: Bifurcation phenomena in the flow through a sudden expansion in a circular pipe. Phys. Fluids 21 (2009)
Pironneau, O., Hecht, F., Le Hyaric, A., Morice, J.: Freefem++. Technical report (2010). www.freefem.org/ff++/
Sherwin, S.J., Blackburn, H.M.: Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotics flows. J. Fluid Mech. 533, 297–327 (2005)
Vanderbauwhede, A.: Local Bifurcation and Symmetry. Pitman, London (1982)
Werner, B., Spence, A.: The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399 (1984)
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Cliffe, K.A., Hall, E.J.C., Houston, P. et al. Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry. J Sci Comput 52, 153–179 (2012). https://doi.org/10.1007/s10915-011-9545-8
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DOI: https://doi.org/10.1007/s10915-011-9545-8