Skip to main content
SpringerLink
Log in

On the circumradius condition for piecewise linear triangular elements

  • Original Paper
  • Area 2
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We discuss the error analysis of linear interpolation on triangular elements. We claim that the circumradius condition is more essential than the well-known maximum angle condition for convergence of the finite element method, especially for the linear Lagrange finite element. Numerical experiments show that this condition is the best possible. We also point out that the circumradius condition is closely related to the definition of surface area.

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
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. The subscript ‘\(E\)’ of \(A_E\) stands for ‘Elementary’.

  2. By the statement (i) given after Corollary 2, we realize that if the circumradius condition does not hold then the maximum angle condition does not hold either.

  3. In [16, p.12], Radó wrote wrongly that the second result was by Fréchet.

References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003)

    MATH  Google Scholar 

  2. Babuška, I., Aziz, A.K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214–226 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  3. Besicovitch, A.S.: On the definition of the area of a surface by means of inscribed polyhedra. J. London Math. Soc. 19, 139–141 (1944)

    Google Scholar 

  4. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  5. Cesari, K.: Surface Area. Princeton University Press, USA (1956)

    MATH  Google Scholar 

  6. Ciarlet, P.G.: The Finite Element Methods for Elliptic Problems, North Holland, 1978, reprint by SIAM 2008

  7. Fréchet, M.: Note on the area of a surface. In: Records of proceedings at meetings, Proc. London Math. Soc., 24, page xlviii (1926). doi:10.1112/plms/s2-24.1.1-s

  8. Hannukainen, A., Korotov, S., Křížek, M.: The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120, 79–88 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jamet, P.: Estimations d’erreur pour des elements finis droits presque degeneres, R.A.I.R.O. Anal. Numer. 10, 43–61 (1976)

    MATH  MathSciNet  Google Scholar 

  10. Kobayashi, K.: On the interpolation constants over triangular elements (in Japanese). RIMS Kokyuroku 1733, 58–77 (2011)

    Google Scholar 

  11. Kobayashi, K., Tsuchiya, T.: A Babuška-Aziz type proof of the circumradius condition. Jpn. J. Indus. Appl. Math. 31, 193–210 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. Liu, X., Kikuchi, F.: Analysis and estimation of error constants for \(P_0\) and \(P_1\) interpolations over triangular finite elements. J. Math. Sci. Univ. Tokyo 17, 27–78 (2010)

    MATH  MathSciNet  Google Scholar 

  13. Rademacher, H.: Über partielle und totale differenzierbarkeit von funktionen mehrerer vaiabeln und über die transformation der doppelintegrale. Math. Ann. 79, 340–359 (1919). doi:10.1007/BF01498415

    Article  MathSciNet  Google Scholar 

  14. Rademacher, H.: Über partielle und totale differenzierbarkeit von funktionen mehrerer vaiabeln II. Math. Ann. 81, 52–63 (1920). doi:10.1007/BF01563620

    Article  MathSciNet  Google Scholar 

  15. Radò, T.: Length and Area. American Mathematical Society, USA (1948)

    MATH  Google Scholar 

  16. Radò, T.: On the Problem of Plateau. Springer (1933), reprint by Chelsea (1951)

  17. Rand, A.: Delaunay refinement algorithms for numerical methods. Ph.D. dissertation, Carnegie Mellon University, 2009

  18. Saks, S.: Theory of the Integral. Warszawa-Lwów, 1937, reprint by Dover 2005

  19. Young, W.H.: On the triangulation method of defining the area of a surface. Proc. London Math. Soc. 19, 117–152 (1921). doi:10.1112/plms/s2-19.1.117

    Article  Google Scholar 

  20. Zames, F.: Surface area and the cylinder area paradox. Coll. Math. J. 8, 207–211 (1977). This article is downloadable from http://mathdl.maa.org/mathDL/22

  21. Zlámal, M.: On the finite element method. Numer. Math. 12, 394–409 (1968)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The first author is supported by Inamori Foundation and JSPS Grant-in-Aid for Young Scientists (B) 22740059. The second author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 22440139 and Grant-in-Aid for Scientific Research (B) 23340023. The authors thank the anonymous referee for valuable comments and for drawing the authors’ attention to Besicovitch’s paper [3].

Author information

Authors and Affiliations

  1. Graduate School of Commerce and Management, Hitotsubashi University, Kunitachi, Japan

    Kenta Kobayashi

  2. Graduate School of Science and Engineering, Ehime University, Matsuyama, Japan

    Takuya Tsuchiya

Authors
  1. Kenta Kobayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takuya Tsuchiya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takuya Tsuchiya.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kobayashi, K., Tsuchiya, T. On the circumradius condition for piecewise linear triangular elements. Japan J. Indust. Appl. Math. 32, 65–76 (2015). https://doi.org/10.1007/s13160-014-0161-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-014-0161-5

Keywords

Mathematics Subject Classification

Search

Navigation