Summary

Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications

2012

Session Number:D1L-D

Session:

Number:809

On the Interpolation Constants over Triangular Elements

Kenta Kobayashi,  

pp.809-811

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.809

PDF download (388.8KB)

Summary:
We present remarkable formulas which give sharp upper bounds for the interpolation constants on the triangles. These constants play an important role in the interpolation theory, in an a priori error estimation in the finite element analysis and in many other areas. The proposed formulas provide better upper bounds than the former ones. Moreover, they are convenient in practical calculations. In the proof of the formulas, we employ numerical verification method.

References:

[1] Arbenz, P. (1982) Computable finite element error bounds for Poisson's equation. IMA Journal of Numerical Analysis, 2, 475-479.

[2] Arcangeli, R. & Gout, J. L. (1976) Sur l'évaluation de I'erreur d'interpolation de Lagrange dans un ouvert de Rn. R.A.I.R.O. Analyse Numérique, 10, 5-27.

[3] Babuška, I. & Aziz, A. K. (1976) On the angle condition in the finite element method. SIAM Journal on Numerical Analysis, 13, 214-226 .

[4] Kobayashi, K. (2011) On the interpolation constants over triangular elements (in Japanese). RIMS Kokyuroku, 1733, 58-77.

[5] Kikuchi, F. & Liu, X. (2007) Estimation of interpolation error constants for the P0 and P1 triangular finite elements. Comput. Methods Appl. Mech. Engrg., 196, 3750-3758.

[6] Laugesen, R. S. & Siudeja, B. A. (2010) Minimizing Neumann fundamental tones of triangles: An optimal Poincaré inequality. J. Differential Equations, 249, 118-135.

[7] Liu, X. & Kikuchi, F. (2010) Analysis and estimation of error constants for P0 and P1 interpolations over triangular finite elements. J. Math. Sci. Univ. Tokyo, 17, 27-78.

[8] Meinguet, J. & Descloux, J. (1977) An operatortheoretical approach to error estimation. Numer. Math., 27, 307-326.

[9] Nakao, M. T. & Yamamoto, N. (2001) A guaranteed bound of the optimal constant in the error estimates for linear triangular element. Computing Supplementum, 15, 163-173.

[10] Natterer, F. (1975) Berechenbare Fehlerschranken für die Methode der finite Elemente. International Series of Numerical Mathematics, 28, 109-121.

[11] Payne, L. E. & Weinberger, H. F. (1960) An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal., 5, 286-292.

[12] Zlámal, M. (1968) On the Finite Element Method. Numerische Mathematik, 12, 394-409.