Geodesic
The $C^1$-approximation of the level surfaces of functions defined on irregular meshes
Sibirskij žurnal industrialʹnoj matematiki, Tome 13 (2010) no. 2, pp. 69-78.

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

We consider the problem of interpolating the level surfaces of functions in some classes (Lipschitz functions, continuously differentiable functions, functions whose gradient satisfies the Hölder condition, and twice continuously differentiable functions) given their values at the nodes of irregular meshes. We derive geometric conditions on the triangulations of a sequence of finite collections of points which guarantee that the gradients of piecewise linear approximations converge. We illustrate the sharpness of these conditions with Schwartz's example. We propose a method for approximating level surfaces which guarantees $C^1$-convergence without any restrictions on the location of nodes.
Mots-clés : triangulation
Keywords: approximation of the gradient, level surface, Voronoi diagram. 
@article{SJIM_2010_13_2_a6,
     author = {V. A. Klyachin and E. A. Pabat},
     title = {The $C^1$-approximation of the level surfaces of functions defined on irregular meshes},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {69--78},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2010_13_2_a6/}
}
TY  - JOUR
AU  - V. A. Klyachin
AU  - E. A. Pabat
TI  - The $C^1$-approximation of the level surfaces of functions defined on irregular meshes
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2010
SP  - 69
EP  - 78
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2010_13_2_a6/
LA  - ru
ID  - SJIM_2010_13_2_a6
ER  - 
%0 Journal Article
%A V. A. Klyachin
%A E. A. Pabat
%T The $C^1$-approximation of the level surfaces of functions defined on irregular meshes
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2010
%P 69-78
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2010_13_2_a6/
%G ru
%F SJIM_2010_13_2_a6
V. A. Klyachin; E. A. Pabat. The $C^1$-approximation of the level surfaces of functions defined on irregular meshes. Sibirskij žurnal industrialʹnoj matematiki, Tome 13 (2010) no. 2, pp. 69-78. http://geodesic.mathdoc.fr/item/SJIM_2010_13_2_a6/

[1] Miklyukov V. M., Geometricheskii analiz. Differentsialnye formy, pochti resheniya, pochti kvazikonformnye otobrazheniya, Izd-vo VolGU, Volgograd, 2007

[2] Zhuravlev I. V., Igumnov A. Yu., Miklyukov V. M., “An implicit function theorem”, Rocky Mountain J. Math., 36:1 (2006), 357–365 | DOI | MR | Zbl

[3] Gelbaum B., Olmsted Dzh., Kontrprimery v analize, PLATON, Volgograd, 1997

[4] Preparata F., Sheimos M., Vychislitelnaya geometriya, Mir, M., 1989 | MR | Zbl