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

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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 
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     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/}
}
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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/