Geodesic
A method of Hermite interpolation by polynomials of the third degree on a triangle
Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 47-52 Cet article a éte moissonné depuis la source Math-Net.Ru

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As a rule, in constructing triangular finite elements of Hermite or Birkhoff type, the denominators of interpolation error bounds contain the sine of the minimum angle in the triangle. This leads to the necessity to impose some restrictions on the triangulation of the domain. Excluding the paper by Yu. N. Subbotin published in the present issue, the author does not know any description of the cases where the minimum angle is absent in the estimates of all derivatives up to order $n$ inclusive when a function is interpolated by Hermite or Birkhoff's polynomial of degree $n$. In this paper, a new method of Hermite interpolation of a function in two variables on a triangle by polynomials of degree 3 is suggested. For the proposed method, the sine of the minimum angle is absent in the denominators of error bounds for any derivatives of the function up to the third order, which makes it possible to weaken our requirements on the triangulation. 
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N. V. Baidakova. A method of Hermite interpolation by polynomials of the third degree on a triangle. Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 47-52. http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a3/

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

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

[3] 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

[4] Synge J. L., The hypercircle in mathematical physics, Cambridge Univ. Press, Cambridge, 1957 | MR | Zbl

[5] Zlámal 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

[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., “Mnogomernaya kusochno polinomialnaya interpolyatsiya”, Metody approksimatsii i interpolyatsii, eds. A. Yu. Kuznetsov, VTsN, Novosibirsk, 1981, 148–153 | MR

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

[9] Subbotin Yu. N., “Pogreshnost approksimatsii interpolyatsionnymi mnogochlenami malykh stepenei na $n$-simpleksakh”, Mat. zametki, 48:4 (1990), 88–99 | MR | Zbl

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

[11] 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

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

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

[14] Subbotin Yu. N., “Novyi kubicheskii element v MKE”, Tr. IMM, 11, no. 2, 2005, 120–130

[15] Berezin I. S., Zhidkov N. P., Metody vychislenii, T. 1, Fizmatgiz, M., 1962