Optimal convergence results for finite elements on extremely deformed meshes
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Sep 21, 2024
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Abstract
This short notes provides an optimal analysis of finite element convergence on meshes containing a so-called band of caps. These structures consist of a zig-zag arrangement of `degenerating' triangles which violate the classical maximum angle condition. Previously a necessary condition on the geometry of such a structure was given by Ku\v{c}era to ensure convergence of finite elements. Here we prove that the condition is also sufficient, providing an optimal analysis of this special case of meshes. This result provides only the second known optimal analysis of finite element convergence on special meshes containing elements violating the maximum angle condition.
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Kučera, V.
(2024).
Optimal convergence results for finite elements on extremely deformed meshes.
Proceedings Of The Conference Algoritmy, , 199 - 203.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2173/1041
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References
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[3] A. Hannukainen, S. Korotov and M. Křı́žek, The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120, 1 (2012), pp. 78–88.
[4] K. Kobayashi and T. Tsuchiya, On the circumradius condition for piecewise linear triangular elements. Japan J. Ind. Appl. Math. 32 (2015), 65-76.
[5] V. Kučera: Several notes on the circumradius condition, Appl. Math. 61, 3 (2016), pp. 287-
298.
[6] V. Kučera: On necessary and sufficient conditions for finite element convergence, http://arxiv.org/abs/1601.02942 (preprint), Numer. Math. (submitted).
[7] P. Oswald, Divergence of the FEM: Babuška-Aziz triangulations revisited. Appl. Math. 60, 5 (2015), pp. 473–484.