Previous |  Up |  Next


MSC: 65N12, 65N15, 65N30
Full entry | Fulltext not available (moving wall 24 months)      Feedback
finite elements; error bounds; divergence; maximum angle condition; triangulation
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.
[1] Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics Teubner, Leipzig; Technische Univ., Chemnitz (1999). MR 1716824 | Zbl 0934.65121
[2] Babuška, I., Aziz, A. K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976), 214-226. DOI 10.1137/0713021 | MR 0455462 | Zbl 0324.65046
[3] Bank, R. E., Yserentant, H.: A note on interpolation, best approximation, and the saturation property. Numer. Math. (2014), doi:10.1007/s00211-014-0687-0. DOI 10.1007/s00211-014-0687-0 | MR 3383332
[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.: The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120 (2012), 79-88. DOI 10.1007/s00211-011-0403-2 | MR 2885598 | Zbl 1255.65196
[6] Jamet, P.: Estimations d'erreur pour des éléments finis droits presque dégénérés. Rev. Franc. Automat. Inform. Rech. Operat. {\it 10}, Analyse numer., R-1 (1976), 43-60. MR 0455282
[7] Kobayashi, K., Tsuchiya, T.: A Babuška-Aziz type proof of the circumradius condition. Japan J. Ind. Appl. Math. 31 (2014), 193-210. DOI 10.1007/s13160-013-0128-y | MR 3167084 | Zbl 1295.65011
[8] Křížek, M.: On semiregular families of triangulations and linear interpolation. Appl. Math., Praha 36 (1991), 223-232. MR 1109126 | Zbl 0728.41003
[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.
Partner of
EuDML logo