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On Equivalence of Maximum Angle Conditions for Tetrahedral Finite Element Meshes

  • Ali Khademi
  • Sergey KorotovEmail author
  • Jon Eivind Vatne
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 131)

Abstract

In this paper we prove that two versions of the maximum angle condition used for various convergence results in the finite element analysis are equivalent in the case of tetrahedral meshes.

References

  1. 1.
    Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1999)Google Scholar
  2. 2.
    Apel, T., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47, 277–293 (1992)Google Scholar
  3. 3.
    Babuška, I., Aziz, A.K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214–226 (1976)Google Scholar
  4. 4.
    Baidakova, N.V.: On Jamet’s estimates for the finite element method with interpolation at uniform nodes of a simplex. Sib. Adv. Math. 28, 1–22 (2018)Google Scholar
  5. 5.
    Barnhill, R.E., Gregory, J.A.: Sard kernel theorems on triangular domains with applications to finite element error bounds. Numer. Math. 25, 215–229 (1976)Google Scholar
  6. 6.
    Brandts, J., Hannukainen, A., Korotov, S., Křížek, M.: On angle conditions in the finite element method. SeMA J. 56, 81–95 (2011)Google Scholar
  7. 7.
    Brandts, J., Korotov, S., Křížek, M.: Generalization of the Zlámal condition for simplicial finite elements in R d. Appl. Math. 56, 417–424 (2011)Google Scholar
  8. 8.
    Cheng, S.W., Dey, T.K., Edelsbrunner, H., Facello, M.A., Teng, S.H.: Sliver exudation. In: Proceedings of the 15th ACM Symposium on Computational Geometry, pp. 1–13. ACM, New York (1999)Google Scholar
  9. 9.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)Google Scholar
  10. 10.
    Edelsbrunner, H.: Triangulations and meshes in computational geometry. Acta Numer. 9, 133–213 (2000)Google Scholar
  11. 11.
    Eriksson, F.: The law of sines for tetrahedra and n-simplices. Geom. Dedicata 7, 71–80 (1978)Google Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Hannukainen, A., Korotov, S., Křížek, M.: Generalizations of the Synge-type condition in the finite element method. Appl. Math. 62, 1–13 (2017)Google Scholar
  14. 14.
    Hannukainen, A., Korotov, S., Křížek, M.: Maximum angle condition for n-dimensional simplicial elements. In: Radu, F., et al. (eds.) Proceedings of the Twelfth European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2017), Voss, Norway. Lecture Notes in Computational Science and Engineering, vol. 126, pp. 769–775. Springer, Cham (2019)Google Scholar
  15. 15.
    Jamet, P.: Estimation de l’erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér. 10, 43–60 (1976)Google Scholar
  16. 16.
    Khademi, A., Korotov, S., Vatne, J.E.: On interpolation error on degenerating prismatic elements. Appl. Math. 63, 237–258 (2018)Google Scholar
  17. 17.
    Kobayashi, K., Tsuchiya, T.: On the circumradius condition for piecewise linear triangular elements. Jpn. J. Ind. Appl. Math. 32, 65–76 (2015)Google Scholar
  18. 18.
    Kobayashi, K., Tsuchiya, T.: A priori error estimates for Lagrange interpolation on triangles. Appl. Math. 60, 485–499 (2015)Google Scholar
  19. 19.
    Kobayashi, K., Tsuchiya, T.: Extending Babuška-Aziz theorem to higher-order Lagrange interpolation. Appl. Math. 61, 121–133 (2016)Google Scholar
  20. 20.
    Křížek, M.: On semiregular families of triangulations and linear interpolation. Appl. Math. 36, 223–232 (1991)Google Scholar
  21. 21.
    Křížek, M.: On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29, 513–520 (1992)Google Scholar
  22. 22.
    Křížek, M., Neittaanmäki, P.: Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer, Dordrecht (1996)Google Scholar
  23. 23.
    Kučera, V.: Several notes on the circumradius condition. Appl. Math. 61, 287–298 (2016)Google Scholar
  24. 24.
    Mao, S., Shi, Z.: Error estimates of triangular finite elements under a weak angle condition. J. Comput. Appl. Math. 230, 329–331 (2009)Google Scholar
  25. 25.
    Oswald, P.: Divergence of FEM: Babuška-Aziz triangulations revisited. Appl. Math. 60, 473–484 (2015)Google Scholar
  26. 26.
    Rand, A.: Average interpolation under the maximum angle condition. SIAM J. Numer. Anal. 50, 2538–2559 (2012)Google Scholar
  27. 27.
    Subbotin, Yu. N.: Dependence of estimates of a multidimensional piecewise polynomial approximation on the geometric characteristics of the triangulation. Tr. Mat. Inst. Steklova 189, 117 (1989)Google Scholar
  28. 28.
    Synge, J.L.: The Hypercircle in Mathematical Physics. Cambridge University Press, Cambridge (1957)Google Scholar
  29. 29.
    Ženíšek, A.: The convergence of the finite element method for boundary value problems of a system of elliptic equations (in Czech). Apl. Mat. 14, 355–377 (1969)Google Scholar
  30. 30.
    Zlámal, M.: On the finite element method. Numer. Math. 12, 394–409 (1968)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computing, Mathematics and PhysicsWestern Norway University of Applied SciencesBergenNorway

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