Divergence of FEM: Babuška-Aziz triangulations revisited

Peter Oswald

Applications of Mathematics (2015)

  • Volume: 60, Issue: 5, page 473-484
  • ISSN: 0862-7940

Abstract

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By re-examining the arguments and counterexamples in I. Babuška, A. K. Aziz (1976) concerning the well-known maximum angle condition, we study the convergence behavior of the linear finite element method (FEM) on a family of distorted triangulations of the unit square originally introduced by H. Schwarz in 1880. For a Poisson problem with polynomial solution, we demonstrate arbitrarily slow convergence as well as failure of convergence if the distortion of the triangulations grows sufficiently fast. This seems to be the first formal proof of divergence of the FEM for a standard elliptic problem with smooth solution.

How to cite

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Oswald, Peter. "Divergence of FEM: Babuška-Aziz triangulations revisited." Applications of Mathematics 60.5 (2015): 473-484. <http://eudml.org/doc/271633>.

@article{Oswald2015,
abstract = {By re-examining the arguments and counterexamples in I. Babuška, A. K. Aziz (1976) concerning the well-known maximum angle condition, we study the convergence behavior of the linear finite element method (FEM) on a family of distorted triangulations of the unit square originally introduced by H. Schwarz in 1880. For a Poisson problem with polynomial solution, we demonstrate arbitrarily slow convergence as well as failure of convergence if the distortion of the triangulations grows sufficiently fast. This seems to be the first formal proof of divergence of the FEM for a standard elliptic problem with smooth solution.},
author = {Oswald, Peter},
journal = {Applications of Mathematics},
keywords = {finite elements; error bounds; divergence; maximum angle condition; triangulation; finite elements; error bounds; divergence; maximum angle condition; triangulation},
language = {eng},
number = {5},
pages = {473-484},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Divergence of FEM: Babuška-Aziz triangulations revisited},
url = {http://eudml.org/doc/271633},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Oswald, Peter
TI - Divergence of FEM: Babuška-Aziz triangulations revisited
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 5
SP - 473
EP - 484
AB - By re-examining the arguments and counterexamples in I. Babuška, A. K. Aziz (1976) concerning the well-known maximum angle condition, we study the convergence behavior of the linear finite element method (FEM) on a family of distorted triangulations of the unit square originally introduced by H. Schwarz in 1880. For a Poisson problem with polynomial solution, we demonstrate arbitrarily slow convergence as well as failure of convergence if the distortion of the triangulations grows sufficiently fast. This seems to be the first formal proof of divergence of the FEM for a standard elliptic problem with smooth solution.
LA - eng
KW - finite elements; error bounds; divergence; maximum angle condition; triangulation; finite elements; error bounds; divergence; maximum angle condition; triangulation
UR - http://eudml.org/doc/271633
ER -

References

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  1. Apel, T., Anisotropic Finite Elements: Local Estimates and Applications, Advances in Numerical Mathematics Teubner, Leipzig; Technische Univ., Chemnitz (1999). (1999) Zbl0934.65121MR1716824
  2. Babuška, I., Aziz, A. K., 10.1137/0713021, SIAM J. Numer. Anal. 13 (1976), 214-226. (1976) Zbl0324.65046MR0455462DOI10.1137/0713021
  3. Bank, R. E., Yserentant, H., 10.1007/s00211-014-0687-0, Numer. Math. (2014), doi:10.1007/s00211-014-0687-0. (2014) MR3383332DOI10.1007/s00211-014-0687-0
  4. Hannukainen, A., Juntunen, M., Huhtala, A., Finite Element Methods I, course notes A.Mat-1.3650, Univ. Helsinki, 2015, . 
  5. Hannukainen, A., Korotov, S., Křížek, M., 10.1007/s00211-011-0403-2, Numer. Math. 120 (2012), 79-88. (2012) Zbl1255.65196MR2885598DOI10.1007/s00211-011-0403-2
  6. Jamet, P., Estimations d'erreur pour des éléments finis droits presque dégénérés, Rev. Franc. Automat. Inform. Rech. Operat. , Analyse numer., R-1 (1976), 43-60. (1976) MR0455282
  7. Kobayashi, K., Tsuchiya, T., 10.1007/s13160-013-0128-y, Japan J. Ind. Appl. Math. 31 (2014), 193-210. (2014) Zbl1295.65011MR3167084DOI10.1007/s13160-013-0128-y
  8. Křížek, M., On semiregular families of triangulations and linear interpolation, Appl. Math., Praha 36 (1991), 223-232. (1991) Zbl0728.41003MR1109126
  9. Ludwig, L., A discussion on the maximum angle condition/counterexample for the convergence of the FEM, Manuscript, TU Dresden, 2011. 
  10. Schwarz, H. A., Sur une définition erroneé de l'aire d'une surface courbe, Gesammelte Mathematische Abhandlungen, vol. 2 Springer, Berlin (1890), 309-311, 369-370. (1890) 

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