Geodesic
Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 1, pp. 32-42 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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.
Mots-clés : multidimensional interpolation
Keywords: finite element method, smallest angle condition, spline functions on triangulations. 
@article{TIMM_2014_20_1_a3,
     author = {N. V. Baidakova},
     title = {Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {32--42},
     year = {2014},
     volume = {20},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a3/}
}
TY  - JOUR
AU  - N. V. Baidakova
TI  - Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2014
SP  - 32
EP  - 42
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a3/
LA  - ru
ID  - TIMM_2014_20_1_a3
ER  - 
%0 Journal Article
%A N. V. Baidakova
%T Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties
%J Trudy Instituta matematiki i mehaniki
%D 2014
%P 32-42
%V 20
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a3/
%G ru
%F TIMM_2014_20_1_a3
N. V. Baidakova. Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 1, pp. 32-42. http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a3/

[1] Ciarlet P. G., Raviart P. A., “General Lagrange and Hermite interpolation in $R^n$ with applications to finite element methods”, Arch. Rational Mech. Anal., 46:3 (1972), 177–199 | DOI | MR | Zbl

[2] Ženišek A., “Interpolation polynomials on the triangle”, Numer. Math., 15 (1970), 283–296 | DOI | MR | Zbl

[3] Bramble J. H., Zlamal M., “Triangular elements in the finite element method”, Math. Comp., 24:112 (1970), 809–820 | DOI | MR

[4] Zlamal M., Ženišek A., “Mathematical aspect of the finite element method”, Technical, physical and mathematical principles of the finite element method, eds. V. Kolar et al., Acad. VED, Praha, 1971, 15–39

[5] Synge J. L., The hypercircle in mathematical physics, A method for the approximate solution of boundary value problems, Cambridge University Press, Cambridge, 1957, 440 pp. | MR | Zbl

[6] Babuška I., Aziz A. K., “On the angle condition in the finite element method”, SIAM J. Numer. Anal., 13:2 (1976), 214–226 | DOI | MR | Zbl

[7] Subbotin Yu. N., “Zavisimost otsenok mnogomernoi kusochno-polinomialnoi approksimatsii ot geometricheskikh kharakteristik triangulyatsii”, Tr. MIAN, 189, 1989, 117–137 | MR | Zbl

[8] Subbotin Yu. N., “Zavisimost otsenok approksimatsii interpolyatsionnymi polinomami pyatoi stepeni ot geometricheskikh kharakteristik treugolnika”, Tr. In-ta matematiki i mekhaniki UrO RAN, 2, 1992, 110–119 | MR | Zbl

[9] Latypova N. V., “Error estimates for approximation by polynomials of degree $4k+3$ on the triangle”, Proc. Steklov Inst. Math., Suppl. 1, 2002, S190–S213 | MR | Zbl

[10] Baidakova N. V., “On some interpolation process by polynomials of degree $4m+1$ on the triangle”, Russian J. Numer. Anal. Math. Modelling, 14:2 (1999), 87–107 | DOI | MR | Zbl

[11] Subbotin Yu. N., “A new cubic element in the FEM”, Proc. Steklov Inst. Math., Suppl. 2, 2005, S176–S187 | MR | Zbl

[12] Baidakova N. V., “A Method of Hermite interpolation by polynomials of the third degree on a triangle”, Proc. Steklov Inst. Math., Suppl. 2, 2005, S49–S55 | MR | Zbl

[13] Ženišek A., “Maximum-angle condition and triangular finite elements of Hermite type”, Math. Comp., 64:211 (1995), 929–941 | DOI | MR | Zbl

[14] Latypova N. V., “Pogreshnost kusochno-kubicheskoi interpolyatsii na treugolnike”, Vestn. Udmurt. un-ta. Ser. Matematika, 2003, 3–10

[15] Matveeva Yu. V., “Ob ermitovoi interpolyatsii mnogochlenami tretei stepeni na treugolnike s ispolzovaniem smeshannykh proizvodnykh”, Izv. Sarat. un-ta, (Nov. ser. Matematika. Mekhanika. Informatika), 7:1 (2007), 23–27

[16] Baidakova N. V., “Otsenki sverkhu velichiny pogreshnosti approksimatsii proizvodnykh v konechnom elemente Sie–Klafa–Tochera”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18, no. 4, 2012, 80–89

[17] Baidakova N. V., “Vliyanie gladkosti na pogreshnost approksimatsii proizvodnykh pri lokalnoi interpolyatsii na triangulyatsiyakh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17, no. 3, 2011, 83–97

[18] Korneichuk N. P., Tochnye konstanty v teorii priblizheniya, Nauka, M., 1987, 422 pp. | MR