Skip to main content
SpringerLink
Log in

On error indicators for optimizing parameters in stabilized methods

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

Abstract

Numerical solution of convection-dominated problems requires special techniques to suppress spurious oscillations in approximate solutions. Often, stabilized methods are applied which involve user-chosen parameters. These parameters significantly influence the quality of the solution but their optimal choice is usually not known. One possibility is to define them in an adaptive way by minimizing an error indicator characterizing the quality of the approximate solution. A non-trivial requirement on the error indicator is that its minimization with respect to the stabilization parameters should suppress spurious oscillations without smearing layers. In this paper, a new error indicator is introduced and its suitability is tested on two newly proposed benchmark problems for which previously proposed indicators do not provide satisfactory results.

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. Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion–Reaction and Flow Problems, 2nd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  2. John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part II – Analysis for P 1 and Q 1 finite elements. Comput. Methods Appl. Mech. Eng. 197, 1997–2014 (2008)

    Article  MATH  Google Scholar 

  3. John, V., Knobloch, P., Savescu, S.B.: A posteriori optimization of parameters in stabilized methods for convection–diffusion problems - Part I. Comput. Methods Appl. Mech. Eng. 200, 2916–2929 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. John, V., Knobloch, P.: Adaptive computation of parameters in stabilized methods for convection–diffusion problems. In: Cangiani, A., Davidchack, R.L., Georgoulis, E.H., Gorban, A., Levesley, J., Tretyakov, M.V. (eds.) Numerical Mathematics and Advanced Applications 2011, Proceedings of ENUMATH 2011, pp 275–283. Springer, Berlin (2013)

  5. Lukáš, P.: Optimization of parameters in SDFEM for different spaces of parameters. Appl. Math. Comput. 267, 711–715 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Lukáš, P.: A posteriori optimization of parameters in the SUPG method for higher degree FE spaces. In: Knobloch, P. (ed.) Boundary and Interior Layers, Computational and Asymptotic Methods – BAIL 2014, Lect. Notes Comput. Sci. Eng., vol. 108, pp 171–182. Springer (2015)

  7. Lukáš, P., Knobloch, P.: Adaptive techniques in SOLD methods. Appl. Math. Comput. 319, 24–30 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Bernardi, C., Girault, V., Hecht, F.: A posteriori analysis of a penalty method and application to the Stokes problem. Math. Models Methods Appl. Sci. 13, 1599–1628 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ainsworth, M., Allendes, A., Barrenechea, G.R., Rankin, R.: On the adaptive selection of the parameter in stabilized finite element approximations. SIAM J. Numer. Anal. 51, 1585–1609 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Silvester, D.: Optimal low order finite element methods for incompressible flow. Comput. Methods Appl. Mech. Eng. 111, 357–368 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, New York (1978)

    MATH  Google Scholar 

  12. Šolín, P.: Partial Differential Equations and the Finite Element Method. Wiley, Hoboken (2005)

    MATH  Google Scholar 

  13. Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review. Comput. Methods Appl. Mech. Eng. 196, 2197–2215 (2007)

    Article  MATH  Google Scholar 

  15. Giles, M.B., Pierce, N.A.: An introduction to the adjoint approach to design. Flow Turbul Combust 65, 393–415 (2000)

    Article  MATH  Google Scholar 

  16. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006)

    MATH  Google Scholar 

  17. Alnæs, M.S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, Ch., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS project version 1.5. Archive of Numerical Software 3(100), 9–23 (2015). https://doi.org/10.11588/ans.2015.100.20553

    Google Scholar 

Download references

Funding

The work of P. Knobloch and P. Lukáš was supported through the grant No. 16-03230S of the Czech Science Foundation.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, Sokolovská 49/83, 186 75, Praha 8, Prague, Czech Republic

    Petr Knobloch & Petr Lukáš

  2. University of Nevada, 1664 N Virginia St, Reno, NV, 89557, USA

    Pavel Solin

Authors
  1. Petr Knobloch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr Lukáš
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pavel Solin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petr Lukáš.

Additional information

Communicated by: Pavel Solin

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Knobloch, P., Lukáš, P. & Solin, P. On error indicators for optimizing parameters in stabilized methods. Adv Comput Math 45, 1853–1862 (2019). https://doi.org/10.1007/s10444-019-09662-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-019-09662-4

Mathematics Subject Classification (2010)

Search

Navigation