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 Trudy Inst. Mat. i Mekh. UrO RAN, 2008, Volume 14, Number 3, Pages 43–57 (Mi timm39)

On some interpolation third-degree polynomials on a three-dimensional simplex

N. V. Baidakova

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: The interpolation problem under consideration is connected with the finite element method in $\mathbb R^3$. In most cases, when finite elements are constructed by means of the partition of a given domain in $\mathbb R^2$ into triangles and interpolation of the Hermite or Birkhoff type, the sine of the smallest angle of the triangle appears in the denominators of the error estimates for the derivatives. In the case of $\mathbb R^m$ ($m\ge3$), the ratio of the radius of the inscribed sphere to the diameter of the simplex is used as an analog of this characteristic. This makes it necessary to impose constraints on the triangulation of the domain. The recent investigations by a number of authors reveal that, in the case of triangles, the smallest angle in the error estimates for some interpolation processes can be replaced by the middle or the greatest one, which makes it possible to weaken the triangulation requirements. There are fewer works of this kind for $m\ge3$, and the error estimates are given there in terms of other characteristics of the simplex. In this paper, methods are suggested for constructing an interpolation third-degree polynomial on a simplex in $\mathbb R^3$. These methods allow one to obtain estimates in terms of a new characteristic of a rather simple form and weaken the triangulation requirements.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2009, 264, suppl. 1, S44–S59

Bibliographic databases:

UDC: 517.51

Citation: N. V. Baidakova, “On some interpolation third-degree polynomials on a three-dimensional simplex”, Trudy Inst. Mat. i Mekh. UrO RAN, 14, no. 3, 2008, 43–57; Proc. Steklov Inst. Math. (Suppl.), 264, suppl. 1 (2009), S44–S59

Citation in format AMSBIB
\Bibitem{Bai08} \by N.~V.~Baidakova \paper On some interpolation third-degree polynomials on a~three-dimensional simplex \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2008 \vol 14 \issue 3 \pages 43--57 \mathnet{http://mi.mathnet.ru/timm39} \elib{https://elibrary.ru/item.asp?id=11929744} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2009 \vol 264 \issue , suppl. 1 \pages S44--S59 \crossref{https://doi.org/10.1134/S0081543809050046} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000265511100004} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-65349101861} 

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This publication is cited in the following articles:
1. N. V. Baidakova, “Algoritm postroeniya ermitovykh konechnykh elementov tretei stepeni”, Sib. elektron. matem. izv., 13 (2016), 799–814
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