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 Trudy Inst. Mat. i Mekh. UrO RAN, 2014, Volume 20, Number 1, Pages 32–42 (Mi timm1027)

Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties

N. V. Baidakovaab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Ural Federal University named after the First President of Russia B. N. Yeltsin

Abstract: We consider a natural class of composite finite elements that provides the $m$th-order smoothness of the resulting piecewise polynomial function on the triangulated domain and does not require information on neighboring elements. It is known that, to provide the required convergence rate, the “smallest angle condition” must be often imposed on the triangulation in the finite element method; i.e., the smallest possible values of the smallest angles of the triangles must be lower bounded. On the other hand, the negative role of the smallest angle can be weakened (but not excluded completely) by choosing appropriate interpolation conditions. As shown earlier, for a large number of methods of choosing interpolation conditions in the construction of simple (noncomposite) finite elements, 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 present paper, a similar result is proved for some class of composite finite elements.

Keywords: multidimensional interpolation, finite element method, smallest angle condition, spline functions on triangulations.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2015, 288, suppl. 1, 29–39

Bibliographic databases:

UDC: 517.51

Citation: N. V. Baidakova, “Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties”, Trudy Inst. Mat. i Mekh. UrO RAN, 20, no. 1, 2014, 32–42; Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 29–39

Citation in format AMSBIB
\Bibitem{Bai14} \by N.~V.~Baidakova \paper Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2014 \vol 20 \issue 1 \pages 32--42 \mathnet{http://mi.mathnet.ru/timm1027} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3364189} \elib{http://elibrary.ru/item.asp?id=21258480} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2015 \vol 288 \issue , suppl. 1 \pages 29--39 \crossref{https://doi.org/10.1134/S0081543815020042} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000352991400003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84958249679}