Reduced Basis Approximation of Parametrized Advection-Diffusion PDEs with High Péclet Number

  • Paolo Pacciarini
  • Gianluigi RozzaEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)


In this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.


Bilinear Form Reference Domain Galerkin Projection Reduce Basis Method Online Stage 
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  1. 1.
    A. Brooks, T. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    L. Dedè, Reduced basis method for parametrized elliptic advection-reaction problems. J. Comput. Math. 28(1), 122–148 (2010)zbMATHMathSciNetGoogle Scholar
  3. 3.
    F. Gelsomino, G. Rozza, Comparison and combination of reduced-order modelling techniques in 3D parametrized heat transfer problems. Math. Comput. Model. Dyn. Syst. 17(4), 371–394 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Huynh, G. Rozza, S. Sen, A. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. Acad. Sci. Paris 345(8), 473–478 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    F. Negri, G. Rozza, A. Manzoni, A. Quarteroni, Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35(5), A2316–A2340 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    P. Pacciarini, G. Rozza, Stabilized reduced basis method for parametrized advection-diffusion PDEs. Comput. Methods Appl. Mech. Eng. 18, 1–18 (2014)CrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Quarteroni, Numerical Models for Differential Problems. MS&A Modeling, Simulation and Applications, vol. 8 (Springer, Milan, 2014)Google Scholar
  8. 8.
    A. Quarteroni, G. Rozza, A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1, 1(3) (2011)MathSciNetGoogle Scholar
  9. 9.
    A. Quarteroni, G. Rozza, A. Quaini, Reduced basis methods for optimal control of advection-diffusion problem, in Advances in Numerical Mathematics, Moscow/Houston, 2007, ed. by W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, Y. Vassilevski, pp. 193–216Google Scholar
  10. 10.
    A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23 (Springer, Berlin, 1994)Google Scholar
  11. 11.
    G. Rozza, D. Huynh, N. Nguyen, A. Patera, Real-time reliable simulation of heat transfer phenomena, in ASME – American Society of Mechanical Engineers – Heat Transfer Summer Conference Proceedings, HT2009 3, San Francisco, pp. 851–860, 2009Google Scholar
  12. 12.
    G. Rozza, D. Huynh, A. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    G. Rozza, T. Lassila, A. Manzoni, Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map, in Spectral and High Order Methods for Partial Differential Equations ed. by J. Hesthaven, E.M. Rønquist. Lecture Notes in Computational Science and Engineering, vol. 76 (Springer, Berlin/Heidelberg, 2011), pp. 307–315Google Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.MOX – Modellistica e Calcolo Scientifico, Dipartimento di Matematica F. BrioschiPolitecnico di MilanoMilanoItaly
  2. 2.SISSA mathLabInternational School for Advanced StudiesTriesteItaly

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