Geodesic
Modified Delaunay empty sphere condition in the problem of approximation of the gradient
Izvestiya. Mathematics , Tome 80 (2016) no. 3, pp. 549-556.

Voir la notice de l'article provenant de la source Math-Net.Ru

The classical Schwarz example shows that piecewise-linear approximation of smooth functions does not necessary yield convergence of the derivatives. However, in the planar case, the required convergence holds if the triangulation of the grid satisfies the empty sphere condition (that is, it is a Delaunay triangulation). These results do not extend to the multidimensional case, as is shown by our published examples. We give a modified empty sphere condition that also guarantees the necessary approximation in the multidimensional case.
Keywords: empty sphere condition, piecewise-linear approximation.
Mots-clés : Delaunay triangulation 
@article{IM2_2016_80_3_a5,
     author = {V. A. Klyachin},
     title = {Modified {Delaunay} empty sphere condition in the problem of approximation of the gradient},
     journal = {Izvestiya. Mathematics },
     pages = {549--556},
     publisher = {mathdoc},
     volume = {80},
     number = {3},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a5/}
}
TY  - JOUR
AU  - V. A. Klyachin
TI  - Modified Delaunay empty sphere condition in the problem of approximation of the gradient
JO  - Izvestiya. Mathematics 
PY  - 2016
SP  - 549
EP  - 556
VL  - 80
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a5/
LA  - en
ID  - IM2_2016_80_3_a5
ER  - 
%0 Journal Article
%A V. A. Klyachin
%T Modified Delaunay empty sphere condition in the problem of approximation of the gradient
%J Izvestiya. Mathematics 
%D 2016
%P 549-556
%V 80
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a5/
%G en
%F IM2_2016_80_3_a5
V. A. Klyachin. Modified Delaunay empty sphere condition in the problem of approximation of the gradient. Izvestiya. Mathematics , Tome 80 (2016) no. 3, pp. 549-556. http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a5/

[1] A. V. Skvortsov, N. S. Mirza, Algoritmy postroeniya i analiza triangulyatsii, Izd. Tomsk. un-ta, Tomsk, 2006, 168 pp.

[2] V. A. Klyachin, A. A. Shirokii, “Triangulyatsiya Delone mnogomernykh poverkhnostei”, Vestn. SamGU. Estestvennonauchn. ser., 2010, no. 4(78), 51–55

[3] V. A. Klyachin, A. A. Shirokii, “The Delaunay triangulation for multidimensional surfaces and its approximative properties”, Russian Math. (Iz. VUZ), 56:1 (2012), 27–34 | DOI | MR | Zbl

[4] V. A. Klyachin, “On a multidimensional analogue of the Schwarz example”, Izv. Math., 76:4 (2012), 681–687 | DOI | DOI | MR | Zbl

[5] B. R. Gelbaum, J. M. H. Olmsted, Counterexamples in analysis, The Mathesis Series, Holden-Day, Inc., San Francisco–London–Amsterdam, 1964, xxiv+194 pp. | MR | Zbl | Zbl

[6] S. N. Borovikov, I. E. Ivanov, I. A. Kryukov, “Modelirovanie prostranstvennykh techenii idealnogo gaza s ispolzovaniem tetraedralnykh setok”, Matem. modelirovanie, 18:8 (2006), 37–48 | Zbl

[7] S. N. Borovikov, I. A. Kryukov, I. E. Ivanov, “Postroenie neregulyarnykh treugolnykh setok na krivolineinykh granyakh na osnove triangulyatsii Delone”, Matem. modelirovanie, 17:8 (2005), 31–45 | MR | Zbl

[8] P. J. Sun, B. Huang, X. M. Wang, “The design of weldment model based on improved Delaunay triangulation algorithm”, Applied Mechanics and Materials, 411–414 (2013), 1414–1418 | DOI

[9] J. L. Synge, The hypercircle in mathematical physics: a method for the approximate solution of boundary value problems, Cambridge Univ. Press, Cambridge, 1957, xii+424 pp. | MR | Zbl

[10] P. Jamet, “Estimations d'erreur pour des elements finis droits presque dégénérées”, Rev. Française Automat. Informat. Recherche Opérationnelle. Analyse Numérique, 10:R-1 (1976), 43–60 | MR | Zbl

[11] Yu. N. Subbotin, “Mnogomernaya kusochno polinomialnaya interpolyatsiya”, Metody approksimatsii i interpolyatsii, VTsN, Novosibirsk, 1981, 148–153 | MR | Zbl

[12] Yu. N. Subbotin, “The dependence of estimates of a multidimensional piecewise polynomial approximation on the geometric characteristics of the triangulation”, Proc. Steklov Inst. Math., 189 (1990), 135–159 | MR | Zbl

[13] J. R. Shewchuk, What is a good linear element? Interpolation, conditioning, anisotropy, and quality measures, Department of electrical engineering and computer sciences University of California, Berkeley, CA, 2002, 66 pp. www.cs.berkeley.edu/~jrs/papers/elemj.pdf