Skip to main content
SpringerLink
Log in

Anisotropic metrics for finite element meshes using a posteriori error estimates: Poisson and Stokes equations

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this paper, metrics derived from a posteriori error estimates for the Poisson problem and for the Stokes system solved by some finite element methods are presented. Numerical examples of mesh adaptation in two dimensions of the space are given and show that these metrics detect the singular behavior of the solution, in particular its anisotropy.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Alauzet F, Frey P (2005) Anisotropic mesh adaptation for CFD computations. Comput Methods Appl Mech Eng 194(48–49):5068–5082

    MathSciNet  MATH  Google Scholar 

  2. Bernardi C, Girault V, Hecht F (2003) A posteriori analysis of a penalty method and application to the Stokes problem. Math Models Methods Appl Sci 13(11):1599–1628

    Article  MathSciNet  MATH  Google Scholar 

  3. Cao W (2007) An interpolation error estimates on anisotropic meshes in \(\mathcal{R}^n\) and optimal metrics for mesh refinement. SIAM J Numer Anal 45(6):2368–2391

    Article  MathSciNet  MATH  Google Scholar 

  4. Carstensen C, Funken SA (2001) Averaging technique for FE - a posteriori error control in elasticity.part ii: λ-independent estimates. Comput Methods Appl Mech Eng 190:4663–4675

    Article  MathSciNet  MATH  Google Scholar 

  5. Carstensen C, Verfürth R (1999) Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J Numer Anal 36(5):1571–1587

    Article  MathSciNet  MATH  Google Scholar 

  6. Creusé E, Kunert G, Nicaise S (2004) A posteriori error estimation for the Stokes problem: Anisotropic and isotropic discretizations. Math Models Methods Appl Sci 14(9):1297–1341

    Article  MathSciNet  MATH  Google Scholar 

  7. Dolešjí V (1998) Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput Vis Sci 1:165–178

    Article  Google Scholar 

  8. Formaggia L, Micheletti S, Perotto S (2004) Anisotropic mesh adaptation in computational fluid dynamics: application to the advection–diffusion–reaction and the stokes problems. Appl Numer Math 51(4):511–533. http://dx.doi.org/10.1016/j.apnum.2004.06.007

    Google Scholar 

  9. Girault V, Raviart P-A (1986) Finite Element methods for Navier–Stokes equations theory and algorithms, vol. 5. Springer Series in Computational Mathematics, Springer, New York

  10. Hecht F (1998) The mesh adapting software: bamg. INRIA report. http://www-c.inria.fr/gamma/cdrom/www/bamg/eng.htm

  11. Hecht F, Georges PL (1999) Nonisotropic grids. CRC, Boca Raton, FL

    Google Scholar 

  12. Hecht F, Kuate R (2008) Metric generation for a given error estimation. In: Proceedings of 17th international meshing roundtable. Springer, Berlin, pp 569–584

  13. Hecht F, Pironneau O Freefem++, language for finite element method and PDEs., Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, http://www.freefem.org/ff++/

  14. Houston P, Süli E (2000) Adaptive Lagrange-Galerkin methods for unsteady convection-diffusion problems. Math Comput 70:77–106

    Article  Google Scholar 

  15. Kröner D, Ohlberger M (1999) A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. Math Comput 69(229):25–39

    Article  Google Scholar 

  16. Kunert G, Verfürth R (2000) Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numerishe Mathematik 86:283–303

    Article  MATH  Google Scholar 

  17. Picasso M (2003) An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: application to elliptic and parabolic problems. SIAM J Sci Comput 24(4):1328–1355 (electronic)

    Google Scholar 

  18. Sonar T, Süli E (1998) A dual graph-norm refinement indicator for finite volume approximations of the Euler equations. Numerishe Mathematik 78:619–658

    Article  MATH  Google Scholar 

  19. Verfürth R (1993) A posteriori error estimators and adaptative mesh-refinement techniques. J Comput Appl Math 50:67–83

    Article  Google Scholar 

  20. Yan N, Zhou Z (2009) A priori and a posteriori error analysis of edge stabilization galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J Comput Appl Math 223(1):198–217

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was carried out in the EGSISTES project supported by the French National Research Agency, ANR, under contract ANR-06-SECU-004-02. The author also thanks Serge Nicaise for fruitful discussions.

Author information

Authors and Affiliations

  1. UVHC, LAMAV, 59313, Valenciennes, France

    Raphaël Kuate

  2. Université Paris-Est, IFSTTAR, MACS, 75732, Paris, France

    Raphaël Kuate

Authors
  1. Raphaël Kuate
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raphaël Kuate.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuate, R. Anisotropic metrics for finite element meshes using a posteriori error estimates: Poisson and Stokes equations. Engineering with Computers 29, 497–505 (2013). https://doi.org/10.1007/s00366-012-0276-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-012-0276-8

Keywords

Search

Navigation