Advertisement

Maximum Angle Condition for n-Dimensional Simplicial Elements

  • Antti Hannukainen
  • Sergey Korotov
  • Michal KřížekEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

In this paper the Synge maximum angle condition for planar triangulations is generalized for higher-dimensional simplicial partitions. In addition, optimal interpolation properties are presented for linear simplicial elements which can degenerate in certain ways.

Notes

Acknowledgements

The authors are indebted to Prof. Jan Brandts, Prof. Takuya Tsuchiya, and Prof. Jon Eivind Vatne for valuable suggestions. The third author was supported by RVO 67985840 of the Czech Republic and Grant no. 18-09628S of the Grant Agency of the Czech Republic.

References

  1. 1.
    T. Apel, Anisotropic Finite Elements: Local Estimates and Applications. Advances in Applied Mathematics (B.G. Teubner, Stuttgart, 1999)Google Scholar
  2. 2.
    T. Apel, M. Dobrowolski, Anisotropic interpolation with applications to the finite element method. Computing 47, 277–293 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    I. Babuška, A.K. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214–226 (1976)MathSciNetCrossRefGoogle Scholar
  4. 4.
    R.E. Barnhill, J.A. Gregory, Sard kernel theorems on triangular domains with applications to finite element error bounds. Numer. Math. 25, 215–229 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Brandts, S. Korotov, M. Křížek, On the equivalence of ball conditions for simplicial finite elements in R d. Appl. Math. Lett. 22, 1210–1212 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. Brandts, S. Korotov, M. Křížek, Generalization of the Zlámal condition for simplicial finite elements in R d. Appl. Math. 56, 417–424 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    S.W. Cheng, T.K. Dey, H. Edelsbrunner, M.A. Facello, S.H. Teng, Sliver exudation, in Proceedings of 15-th ACM Symposium on Computational Geometry (1999), pp. 1–13Google Scholar
  8. 8.
    P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978)zbMATHGoogle Scholar
  9. 9.
    H. Edelsbrunner, Triangulations and meshes in computational geometry. Acta Numer. 9, 133–213 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    F. Eriksson, The law of sines for tetrahedra and n-simplices. Geom. Dedicata 7, 71–80 (1978)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.W. Gaddum, The sums of dihedral and trihedral angles in a tetrahedron. Am. Math. Mon. 59, 370–371 (1952)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Hannukainen, S. Korotov, M. Křížek, The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120, 79–88 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Hannukainen, S. Korotov, M. Křížek, Generalizations of the Synge-type condition in the finite element method. Appl. Math. 62, 1–13 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    P. Jamet, Estimation de l’erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér. 10, 43–60 (1976)zbMATHGoogle Scholar
  15. 15.
    K. Kobayashi, T. Tsuchiya, On the circumradius condition for piecewise linear trian-gular elements. Jpn. J. Ind. Appl. Math. 32, 65–76 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    K. Kobayashi, T. Tsuchiya, A priori error estimates for Lagrange intrepolation on triangles. Appl. Math. 60, 485–499 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    K. Kobayashi, T. Tsuchiya, Extending Babuška-Aziz theorem to higher-order Largange interpolation. Appl. Math. 61, 121–133 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Křížek, On semiregular families of triangulations and linear interpolation. Appl. Math. 36, 223–232 (1991)MathSciNetzbMATHGoogle Scholar
  19. 19.
    M. Křížek, On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29, 513–520 (1992)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Křížek, P. Neittaanmäki, Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications (Kluwer Academic Publishers, Dordrecht, 1996)CrossRefGoogle Scholar
  21. 21.
    V. Kučera, A note on necessary and sufficient condition for convergence of the finite element method, in Proceedings of Conference on Applied Mathematics 2015, ed. by J. Brandts et al. (Institute of Mathematical, Prague, 2015), pp. 132–139Google Scholar
  22. 22.
    V. Kučera, Several notes on the circumradius condition. Appl. Math. 61, 287–298 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    V. Kučera, On necessary and sufficient conditions for finite element convergence. Numer. Math (submitted). Arxiv 1601.02942Google Scholar
  24. 24.
    S. Mao, Z. Shi, Error estimates of triangular finite elements under a weak angle condition. J. Comput. Appl. Math. 230, 329–331 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    P. Oswald, Divergence of FEM: Babuška-Aziz triangulations revisited. Appl. Math. 60, 473–484 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    K. Rektorys, Survey of Applicable Mathematics I (Kluwer Academic Publishers, Dordrecht, 1994)CrossRefGoogle Scholar
  27. 27.
    J.L. Synge, The Hypercircle in Mathematical Physics (Cambridge University Press, Cambridge, 1957)CrossRefGoogle Scholar
  28. 28.
    A. Ženíšek, The convergence of the finite element method for boundary value problems of a system of elliptic equations (in Czech). Appl. Math. 14, 355–377 (1969)zbMATHGoogle Scholar
  29. 29.
    M. Zlámal, On the finite element method. Numer. Math. 12, 394–409 (1968)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Antti Hannukainen
    • 1
  • Sergey Korotov
    • 2
  • Michal Křížek
    • 3
    Email author
  1. 1.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland
  2. 2.Department of Computing, Mathematics and PhysicsWestern Norway University of Applied SciencesBergenNorway
  3. 3.Institute of Mathematics, Czech Academy of SciencesPrague 1Czech Republic

Personalised recommendations