×

On the generalization of the Synge-Křížek maximum angle condition for \(d\)-simplices. (English) Zbl 1426.65177

This paper is concerned with the generalization of the maximum angle condition, proposed by J. L. Synge [The hypercircle in mathematical physics. A method for the approximate solution of boundary value problems. Cambridge: At the University Press (1957; Zbl 0079.13802)] and M. Křížek [SIAM J. Numer. Anal. 29, No. 2, 513–520 (1992; Zbl 0755.41003)] for triangular and tetrahedral elements, respectively, for the case of higher-dimensional simplicial finite elements. The relations to other angle-type conditions commonly used in interpolation theory and in finite element analysis are also discussed.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
41A05 Interpolation in approximation theory

References:

[1] Brandts, J.; Hannukainen, A.; Korotov, S.; Křížek, M., On angle conditions in the finite element method, SeMA J., 56, 81-95 (2011) · Zbl 1242.65235
[2] Synge, J. L., The Hypercircle in Mathematical Physics (1957), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0079.13802
[3] Babuška, I.; Aziz, A. K., On the angle condition in the finite element method, SIAM J. Numer. Anal., 13, 214-226 (1976) · Zbl 0324.65046
[4] 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) · Zbl 0304.65076
[5] Křížek, M., On the maximum angle condition for linear tetrahedral elements, SIAM J. Numer. Anal., 29, 513-520 (1992) · Zbl 0755.41003
[6] S.W. Cheng, T.K. Dey, H. Edelsbrunner, M.A. Facello, S.H. Teng, Sliver exudation, in: Proc. 15-th ACM Symp. Comp. Geometry, 1999, pp. 1-13.; S.W. Cheng, T.K. Dey, H. Edelsbrunner, M.A. Facello, S.H. Teng, Sliver exudation, in: Proc. 15-th ACM Symp. Comp. Geometry, 1999, pp. 1-13.
[7] Edelsbrunner, H., Triangulations and meshes in computational geometry, Acta Numer., 9, 133-213 (2000) · Zbl 1004.65024
[8] 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) · Zbl 1255.65196
[9] Khademi, A.; Korotov, S.; Vatne, J. E., On interpolation error on degenerating prismatic elements, Appl. Math., 63, 237-258 (2018) · Zbl 1488.65632
[10] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland, Amsterdam · Zbl 0383.65058
[11] Brandts, J.; Korotov, S.; Křížek, M., Generalization of the Zlámal condition for simplicial finite elements in \(R^d\), Appl. Math., 56, 417-424 (2011) · Zbl 1240.65327
[12] Jamet, P., Estimation d’erreur pour des éléments finis droits presque dégénérés, RAIRO Anal. Numér., 10, 43-61 (1976) · Zbl 0346.65052
[13] Subbotin, Yu. N., Dependence of estimates of a multidimensional piecewise polynomial approximation on the geometric characteristics of the triangulation, Tr. Mat. Inst. Steklova, 189, Article 117 pp. (1989) · Zbl 0716.41008
[14] Rand, A., Average interpolation under the maximum angle condition, SIAM J. Numer. Anal., 50, 2538-2559 (2012) · Zbl 1267.65166
[15] Hannukainen, A.; Korotov, S.; Křížek, M., Generalizations of the Synge-type condition in the finite element method, Appl. Math., 62, 1-13 (2017) · Zbl 1424.65220
[16] Baidakova, N. V., On Jamet’s esimates for the finite element method with interpolation at uniform nodes of a simplex, Sib. Adv. Math., 28, 1-22 (2018)
[17] Eriksson, F., The law of sines for tetrahedra and \(n\)-simplices, Geom. Dedicata, 7, 71-80 (1978) · Zbl 0375.50008
[18] 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) · Zbl 0188.22604
[19] Zlámal, M., On the finite element method, Numer. Math., 12, 394-409 (1968) · Zbl 0176.16001
[20] Brandts, J.; Korotov, S.; Křížek, M.; Šolc, J., On nonobtuse simplicial partitions, SIAM Rev., 51, 317-335 (2009) · Zbl 1172.51012
[21] Maehara, H., On dihedral angles of a simplex, J. Math. Res., 5, 79-83 (2013)
[22] Hannukainen, A.; Korotov, S.; Křížek, M., Maximum angle condition for \(n\)-dimensional simplicial elements, (Proc. of ENUMATH-2017, Voss, Norway (2019), Springer), 769-778
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.