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## Archive of Issues

Russia Izhevsk
Year
2016
Volume
26
Issue
2
Pages
160-168
 Section Mathematics Title Independence of interpolation error estimates by polynomials of $2k+1$ degree on angles in a triangle Author(-s) Bazhenov V.S.a, Latypova N.V.a Affiliations Udmurt State Universitya Abstract The paper considers Birkhoff-type triangle-based interpolation of two-variable function by polynomials of $2k+1$ degree by set of two variables. Similar estimates are automatically transferred to error estimates of related finite element method. The approximation error estimates of derivatives for the given finite elements depend only on the decomposition diameter, and do not depend on triangulation angles. We show that obtained approximation error estimates for a function and its partial derivatives are unimprovable. Unimprovability is understood in a following sense: there exists a function from the given class and there exist absolute positive constants independent of triangulation such that for any nondegenerate triangle estimates from below are valid. In this work, a system of specific functions is offered for interpolation conditions. These functions allow to obtain corresponding error estimates for definite partial derivatives. Keywords error of interpolation, piecewise polynomial function, triangulation, finite element method UDC 517.518 MSC 41A05 DOI 10.20537/vm160202 Received 29 February 2016 Language Russian Citation Bazhenov V.S., Latypova N.V. Independence of interpolation error estimates by polynomials of $2k+1$ degree on angles in a triangle, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 2, pp. 160-168. References Ciarlet P.G., Raviart P.A. General Lagrange and Hermite interpolation in ${\mathbb R}$$n$ with applications to finite element methods, Arch. Ration. Mech. Anal., 1972, vol. 46, no. 3, pp. 177-199. Subbotin Yu.N. Multidimensional multiple polynomial interpolation, Metody Approksim. Interpol. (Methods of Approximation and Interpolation: Transactions), Novosibirsk: Computing Centre, Academy of Sciences of the USSR, 1981, pp. 148-152. Subbotin Yu.N. Dependence of the estimates of approximation by interpolating polynomials of 5-th degree upon geometric properties of triangle, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 1992, vol. 2, pp. 110-119 (in Russian). Subbotin Yu.N. A new cubic element in the FEM, Proceedings of the Steklov Institute of Mathematics, 2005, suppl. 2, pp. S176-S187. Baidakova N.V. On some interpolation process by polynomials of degree $4m+1$ on the triangle, Russian Journal of Numerical Analysis and Mathematical Modelling, 1999, vol. 14, no. 2, pp. 87-107. Baidakova N.V. A method of Hermite interpolation by polynomials of the third degree on a triangle, Proceedings of the Steklov Institute of Mathematics, 2005, suppl. 2, pp. S49-S55. Latypova N.V. Error estimates of approximation by polynomials of degree $4k+3$ on the triangle, Proceedings of the Steklov Institute of Mathematics, 2002, suppl. 1, pp. S190-S213. Latypova N.V. Error of interpolation by piecewise cubic polynomial on triangle, Vestn. Udmurt. Univ. Mat., 2003, pp. 3-18 (in Russian). Baidakova N.V. New estimates of the error of approximation of derivatives under interpolation of a function on a triangle by polynomials of the third degree, Izv. Sarat. Univ. (N.S.), Ser. Mat. Mekh. Inform., 2013, vol. 13, no. 1 (2), pp. 15-19 (in Russian). Latypova N.V. Error of interpolation by a piecewise parabolic polynomial on a triangle, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2009, no. 3, pp. 91-97 (in Russian). Latypova N.V. Independence of error estimates of interpolation by cubic polynomials from the angles of a triangle, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2011, vol. 17, no. 3, pp. 233-241 (in Russian). Latypova N.V. Independence of interpolation error estimates by fourth-degree polynomials on angles in a triangle, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 3, pp. 64-74 (in Russian). Latypova N.V. Independence of interpolation error estimates by fifth-degree polynomials on angles in a triangle, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 3, pp. 53-64 (in Russian). Latypova N.V. Error of interpolation by sixth-degree polynomials on a triangle, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 4, pp. 79-87 (in Russian). Berezin I.S., Zhidkov N.P. Metody vychislenii (Computing Methods), vol. 1, Moscow: Fizmatgiz, 1962, 464 p. Fadeev D.K., Sominskii I.S. Sbornik zadach po vysshei algebre (Collection of problems on higher algebra), Moscow: Nauka, 1977, 288 p. Full text