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## Archive of Issues

Russia Izhevsk
Year
2012
Issue
3
Pages
53-64
 Section Mathematics Title Independence of interpolation error estimates by fifth-degree polynomials on angles in a triangle Author(-s) Latypova N.V.a Affiliations Udmurt State Universitya Abstract The paper considers several methods of Birkhoff-type triangle-based interpolation of two-variable function by fifth-degree polynomials. Similar estimates are automatically transferred to error estimates of related finite element method. The error estimates for the given elements depend only on the decomposition diameter, and do not depend on triangulation angles. We show that the estimates obtained are unimprovable. Unimprovability is understood in a following sense: there exists 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. Keywords error of interpolation, piecewise polynomial function, triangulation, finite element method UDC 517.518 MSC 41A05 DOI 10.20537/vm120306 Received 29 March 2012 Language Russian Citation Latypova N.V. Independence of interpolation error estimates by fifth-degree polynomials on angles in a triangle, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 3, pp. 53-64. References Ciarlet P.G., Raviart P.A. General Lagrange and Hermite interpolation in ${\mathbb R}$$n$ with applications to finite element methods, Arch. Rat. Mech. and 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. Subbotin Yu.N. A New Cubic Element in the FEM, Proceeding 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, Proceeding 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, Proceeding 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. 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. 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. Latypova N.V. Independence of interpolation error by fourth-degree polynomials on angles in a triangle, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 3, pp. 64-74. Berezin I.S., Zhidkov N.P. Metody vychislenii (Computing Methods), vol. 1, Moscow: Fizmatgiz, 1962. 464 p. Full text