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 Trudy Inst. Mat. i Mekh. UrO RAN, 2012, Volume 18, Number 4, Pages 80–89 (Mi timm868)

Upper estimates for the error of approximation of derivatives in a finite element of Hsieh–Clough–Tocher type

N. V. Baidakovaab

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

Abstract: For a triangle $T$, we obtain upper estimates for the error of approximation of derivatives of a function $f\in W^4M$ by derivatives of a piecewise polynomial function $P_3$ that defines a composite Hsieh–Clough–Tocher element. In the obtained error estimates, the negative influence of the smallest angle $\alpha$ of the triangle $T$ on the error of approximation of derivatives is decreased as compared to most often used classical estimates for noncomposite elements. Contrary to expectations, the behavior of the obtained upper estimates with respect to the angle $\alpha$ turned out to be similar to the estimates for the fifth-order polynomial $\widetilde P_5$ defining a “purely polynomial” (noncomposite) finite element that were found by Yu. N. Subbotin. However, the Hsieh–Clough–Tocher element may have an advantage over the polynomial $\widetilde P_5$, which provides the same smoothness, because the implementation of the finite element method for finding $P_3$ requires 12 free parameters, whereas the implementation of this method for finding $\widetilde P_5$ requires 21 parameters.

Keywords: multidimensional interpolation, finite element method, approximation.

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Citation: N. V. Baidakova, “Upper estimates for the error of approximation of derivatives in a finite element of Hsieh–Clough–Tocher type”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 4, 2012, 80–89

Citation in format AMSBIB
\Bibitem{Bai12} \by N.~V.~Baidakova \paper Upper estimates for the error of approximation of derivatives in a~finite element of Hsieh--Clough--Tocher type \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2012 \vol 18 \issue 4 \pages 80--89 \mathnet{http://mi.mathnet.ru/timm868} \elib{http://elibrary.ru/item.asp?id=18126470} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. 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
2. A. A. Klyachin, “Otsenka pogreshnosti vychisleniya funktsionala, soderzhaschego proizvodnye vtorogo poryadka, na treugolnoi setke”, Sib. elektron. matem. izv., 16 (2019), 1856–1867
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