A Spectral Element Reduced Basis Method in Parametric CFD

  • Martin W. Hess
  • Gianluigi Rozza
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14,259 degrees of freedom. The steady-state snapshot solutions define a reduced order space, which allows to accurately evaluate the steady-state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation (Karniadakis and Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford, 2005) in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.



This work was supported by European Union Funding for Research and Innovation through the European Research Council (project H2020 ERC CoG 2015 AROMA-CFD project 681447, P.I. Prof. G. Rozza).


  1. 1.
    G. Karniadakis, S. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. (Oxford University Press, Oxford, 2005)CrossRefGoogle Scholar
  2. 2.
    C. Canuto, M.Y. Hussaini, A. Quarteroni, Th.A. Zhang, Spectral Methods Fundamentals in Single Domains (Springer – Scientific Computation, New York, 2006)zbMATHGoogle Scholar
  3. 3.
    C. Canuto, M.Y. Hussaini, A. Quarteroni, Th.A. Zhang, Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics (Springer – Scientific Computation, New York, 2007)zbMATHGoogle Scholar
  4. 4.
    A.T. Patera, A spectral element method for fluid dynamics; laminar flow in a channel expansion. J. Comput. Phys. 54(3), 468–488 (1984)CrossRefGoogle Scholar
  5. 5.
    H. Herrero, Y. Maday, F. Pla, RB (Reduced Basis) for RB (Rayleigh–Bénard). Comput. Methods Appl. Mech. Eng. 261–262, 132–141 (2013)CrossRefGoogle Scholar
  6. 6.
    L. Fick, Y. Maday, A. Patera, T. Taddei, A reduced basis technique for long-time unsteady turbulent flows. J. Comput. Phys. (submitted). arxiv:
  7. 7.
    J.S. Hesthaven, G. Rozza, B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics (Springer, Berlin, 2016)Google Scholar
  8. 8.
    G. Pitton, A. Quaini, G. Rozza, Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: applications to Coanda effect in cardiology. J. Comput. Phys. 344, 534–557 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Burger, Numerical Methods for Incompressible Flow. Lecture Notes (UCLA, Los Angeles, 2010)Google Scholar
  10. 10.
    C.D. Cantwell, D. Moxey, A. Comerford, A. Bolis, G. Rocco, G. Mengaldo, D. de Grazia, S. Yakovlev, J.-E. Lombard, D. Ekelschot, B. Jordi, H. Xu, Y. Mohamied, C. Eskilsson, B. Nelson, P. Vos, C. Biotto, R.M. Kirby, S.J. Sherwin, Nektar++: an open-source spectral/hp element framework. Comput. Phys. Commun. 192, 205–219 (2015)CrossRefGoogle Scholar
  11. 11.
    T. Lassila, A. Manzoni, A. Quarteroni, G. Rozza, Model Order Reduction in Fluid Dynamics: Challenges and Perspectives, vol. 9, ed. by A. Quarteroni, G. Rozza. Reduced Order Methods for Modelling and Computational Reduction (Springer International Publishing, MS&A, Cham, 2014), pp. 235–273Google Scholar
  12. 12.
    G. Pitton, G. Rozza, On the application of reduced basis methods to bifurcation problems in incompressible fluid dynamics. J. Sci. Comput. 73, 157 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics (Springer, Berlin, 2013)Google Scholar
  14. 14.
    A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations (Springer, Berlin, 1994)zbMATHGoogle Scholar
  15. 15.
    R. Wille, H. Fernholz, Report on the first European Mechanics Colloquium, on the Coanda effect. J. Fluid Mech. 23(4), 801–819 (1965)CrossRefGoogle Scholar
  16. 16.
    A. Quaini, R. Glowinski, S. Čanić, A computational study on the generation of the Coanda effect in a mock heart chamber. RIMS Kôkyûroku series, No. 2009-4 (2016)Google Scholar
  17. 17.
    Y. Maday, E.M. Ronquist, A reduced-basis element method. C. R. Math. 335(2), 195–200 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A.E. Lovgren, Y. Maday, E.M. Ronquist, A reduced basis element method for the steady stokes problem. ESAIM: Math. Model. Numer. Anal. 40(3), 529–552 (2006)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.SISSA mathLabInternational School for Advanced StudiesTriesteItaly

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