RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy Inst. Mat. i Mekh. UrO RAN: Year: Volume: Issue: Page: Find

 Trudy Inst. Mat. i Mekh. UrO RAN, 2011, Volume 17, Number 3, Pages 83–97 (Mi timm723)

Influence of smoothness on the error of approximation of derivatives under local interpolation on triangulations

N. V. Baidakovaab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Ural Federal University

Abstract: The paper is concerned with one problem of function interpolation on a triangle. We consider a large class of interpolation conditions guaranteeing the smoothness of order $m$ of the resulting piecewise polynomial function on the triangulated domain. It is known that, for smoothness $m\ge1$, the known upper estimates for the error of approximation of derivatives of order $2$ and above by derivatives of interpolation polynomials defined on a triangulation element contain the sine of the smallest angle in the denominator. As a result, the “smallest angle condition” must be imposed on the triangulation. It was shown earlier that the influence of the smallest angle could be weakened (which does not mean that it can be eliminated in all cases). The principal aim of this paper is to show that, for a large number of methods of choosing interpolation conditions, including traditional conditions, the influence of the smallest angle of the triangle on the error of approximation of derivatives of a function by derivatives of the interpolation polynomial is essential for a number of derivatives of order $2$ and above for $m\ge1$. In the case $m=0$, the influence of the middle (largest) angle is important. As a consequence, the results on the unimprovability of the upper estimates obtained earlier are strengthened.

Keywords: multidimensional interpolation, finite element method, approximation.

Full text: PDF file (232 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2012, 277, suppl. 1, 33–47

Bibliographic databases:

UDC: 517.51

Citation: N. V. Baidakova, “Influence of smoothness on the error of approximation of derivatives under local interpolation on triangulations”, Trudy Inst. Mat. i Mekh. UrO RAN, 17, no. 3, 2011, 83–97; Proc. Steklov Inst. Math. (Suppl.), 277, suppl. 1 (2012), 33–47

Citation in format AMSBIB
\Bibitem{Bai11} \by N.~V.~Baidakova \paper Influence of smoothness on the error of approximation of derivatives under local interpolation on triangulations \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2011 \vol 17 \issue 3 \pages 83--97 \mathnet{http://mi.mathnet.ru/timm723} \elib{http://elibrary.ru/item.asp?id=17870123} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2012 \vol 277 \issue , suppl. 1 \pages 33--47 \crossref{https://doi.org/10.1134/S0081543812050057} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000305909000005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84863604863} 

• http://mi.mathnet.ru/eng/timm723
• http://mi.mathnet.ru/eng/timm/v17/i3/p83

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. V. Baidakova, “Otsenki sverkhu velichiny pogreshnosti approksimatsii proizvodnykh v konechnom elemente Sie–Klafa–Tochera”, Tr. IMM UrO RAN, 18, no. 4, 2012, 80–89
2. N. V. Baidakova, “Novye otsenki velichin pogreshnosti approksimatsii proizvodnykh pri interpolyatsii funktsii mnogochlenami tretei stepeni na treugolnike”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:1(2) (2013), 15–19
3. N. V. Baidakova, “Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 29–39
4. N. V. Baidakova, “A triangular finite element with new approximation properties”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 74–84
5. N. V. Baǐdakova, “On Jamet's estimates for the finite element method with interpolation at uniform nodes of a simplex”, Siberian Adv. Math., 28:1 (2018), 1–22
6. A. A. Klyachin, “Postroenie triangulyatsii ploskikh oblastei metodom izmelcheniya”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 2(39), 18–28
7. N. V. Baidakova, “Lineinaya interpolyatsiya na tetraedre”, Tr. IMM UrO RAN, 24, no. 4, 2018, 80–84
8. A. A. Klyachin, “Postroenie treugolnoi setki dlya oblastei, ogranichennykh zamknutymi prostymi krivymi”, Matematicheskaya fizika i kompyuternoe modelirovanie, 21:3 (2018), 31–38
•  Number of views: This page: 262 Full text: 78 References: 23 First page: 1