Skip to main content
SpringerLink
Log in

Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almroth, B.O., Stern, P., Brogan, F.A.: Automatic choice of global shape functions in structural analysis. AIAA J. 16, 525–528 (1978)

    Article  Google Scholar 

  2. Barrault, M., Maday, Y., Nguyen, N., Patera, A.: An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus de l’Académie des Sciences Series I 339, 667–672 (2004)

    MATH  MathSciNet  Google Scholar 

  3. Brezzi, F.: On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. R.A.I.R.O. 8, 129–151 (1974)

    MATH  MathSciNet  Google Scholar 

  4. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Verlag, Berlin (1991)

    Book  MATH  Google Scholar 

  5. Deparis, S., Løvgren, E.: Stabilized reduced basis approximation of incompressible three-dimensional Navier-Stokes equations in parametrized deformed domains. J. Sci. Comput. 50(1), 198–212 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dihlmann, M.A., Haasdonk, B.: Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems. Tech. rep., University of Stuttgart (2013). Submitted to Computational Optimization and Applications

  7. Discacciati, M.: Domain decomposition methods for the coupling of surface and groundwater flows. Ph.D. thesis, EPFL, Lausanne (2004)

  8. Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Eftang, J.L., Patera, A.T.: Port reduction in parametrized component static condensation: approximation and a posteriori error estimation. Int. J. Numer. Methods Eng. (2013). doi:10.1002/nme.4543

  10. Gerner, A.L., Veroy, K.: Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput. 34(5), A2812–A2836 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Grepl, M.A., Patera, A.T.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. Math. Model. Numer. Anal. 39, 157–181 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. Math. Model. Numer. Anal. 42, 277–302 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Huynh, D.B.P., Knezevic, D.J., Patera, A.T.: A static condensation reduced basis element method: Approximation and a posteriori error estimation. Math. Model. Numer. Anal. 47(1), 213–251 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Huynh, D.B.P., Rozza, G., Sen, S., Patera, A.T.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. Comptes Rendus de l’Académie des Sciences Series I 345, 473–478 (2007)

    MATH  MathSciNet  Google Scholar 

  15. Iapichino, L.: Reduced basis method for the solution of parametrized PDEs in repetitive and complex networks with application to CFD. Ph.D. thesis, EPFL, Lausanne (2012)

  16. Iapichino, L., Quarteroni, A., Rozza, G.: A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Computer Methods in Applied Mechanics and Engineering 221–222, 62–82 (2012)

  17. Layton, W.J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for the steady Stokes problem. Math. Model. Numer. Anal. 40, 529–552 (2006)

    Article  Google Scholar 

  19. Maday, Y., Rønquist, E.M.: A reduced-basis element method. J. Sci. Comput. 17, 447–459 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Maday, Y., Rønquist, E.M.: The reduced basis element method: application to a thermal fin problem. J. Sci. Comput. 26, 240–258 (2004)

    MATH  Google Scholar 

  21. Maier, I., Haasdonk, B.: A Dirichlet-Neumann reduced basis method for homogeneous domain decomposition problems. Appl. Numer. Math. 78, 31–48 (2013)

    Article  MathSciNet  Google Scholar 

  22. Manzoni, A.: Reduced models for optimal control, shape optimization and inverse problems in haemodynamics. Ph.D. thesis, EPFL, Lausanne (2012)

  23. Oliveira, I.B., Patera, A.T.: Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim. Eng. 8, 43–65 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Patera, A.T., Rozza, G.: Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Grad. Monogr. Mech. Eng. (2007) [ http://augustine.mit.edu]

  25. Prud’homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J. Fluids Eng. 124, 70–80 (2002)

    Article  Google Scholar 

  26. Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1(3) (2011). doi:10.1186/2190-5983-1-3

  27. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford University Press (1999)

  28. Rozza, G.: Shape design by optimal flow control and reduced basis techniques: applications to bypass configurations in haemodynamics. Ph.D. thesis, EPFL, Lausanne (2005)

  29. Rozza, G., Huynh, D.B.P., Manzoni, A.: Reduced basis approximation and a posteriori error estimation for Stokes flow in parametrized geometries: roles of the inf-sup stability constants. Numer. Math. 125, 115–152 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15, 229–275 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196, 1244–1260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sen, S., Veroy, K., Huynh, D.B.P., Deparis, S., Nguyen, N.C., Patera, A.T.: “Natural norm” a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217, 37–62 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Smetana, K.: A new certification framework for the port reduced static condensation reduced basis element method. Computer Methods in Applied Mechanics and Engineering, Submitted (2014)

  34. Toselli, A., Widlund, O.: Domain decomposition methods - algorithms and theory. Springer-Verlag, Berlin (2005)

    MATH  Google Scholar 

  35. Vallaghé, S., Patera, A.T.: The static condensation reduced basis element method for a mixed-mean conjugate heat exchanger model. SIAM J. Sci. Comput. 36(3), B294–B320 (2013)

    Article  Google Scholar 

  36. Veroy, K., Prud’homme, C., Rovas, D.V., Patera, A.T.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of 16th AIAA computational fluid dynamics conference (2003). Paper 2003-3847

  37. Xu, J., Zikatanov, L.: Some observations on Babuška and Brezzi theories. Numer. Math. 94, 195–202 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Immanuel Martini & Bernard Haasdonk

  2. SISSA MathLab, International School for Advanced Studies, Office A-435, Via Bonomea 265, 34136, Trieste, Italy

    Gianluigi Rozza

Authors
  1. Immanuel Martini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianluigi Rozza
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernard Haasdonk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Immanuel Martini.

Additional information

Communicated by: Editors of Special Issue on MoRePas

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martini, I., Rozza, G. & Haasdonk, B. Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system. Adv Comput Math 41, 1131–1157 (2015). https://doi.org/10.1007/s10444-014-9396-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-014-9396-6

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation