Estimation of the error of calculating the functional containing higher-order derivatives on a triangular grid

A. A. Klyachin

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1856-1867.

Résumé

In the present paper, it is proved that a functional containing second derivatives can be calculated with an error of order $ O (h ^ {4m+1}) $ with a triangular mesh of $h \to 0$ if the piecewise polynomial functions of degree $ 4m+1 $ for $m\geq 1$. For $ n = 2 $ we give an example of the fact that the piecewise quadratic approximation gives the second order of accuracy for calculating the functional for a special kind of triangulation.

Détail

Keywords: piecewise polynomial function, approximation of the functional
Mots-clés : triangulation.

Bibliographie

[1] L.D. Landau, E.M. Lifshits, Theoretical physics, Textbook. allowance, v. VII, Theory of elasticity, Science. Ch. ed. physical-mat. Lit., M., 1987 | MR

[2] L. V. Kantorovich, Approximate methods of higher analysis, Nauka, M., 1950 | MR

[3] S. G. Mikhlin, Variational methods in mathematical physics, Nauka, M., 1970 | MR

[4] A. A. Klyachin, “On the uniform convergence of piecewise linear solutions to the equilibrium capillary surface equation”, J. Appl. Industr. Math., 9:3 (2015), 381–391 | DOI | MR | Zbl

[5] A. A. Klyachin, M. A. Gatsunayev, “On uniform convergence of piecewise linear solutions of the minimal surface equation”, Ufa Mathematical Journal, 6:3 (2014), 3–16 | DOI | MR | Zbl

[6] A. A. Klyachin, “Estimation of the error in the computation of integral functionals using piecewise linear functions”, Bulletin of Volgograd State University, Series 1: Mathematics. Physics, 2015, no. 1(26), 6–12 | MR

[7] V. A. Klyachin, E. A. Pabat, “$ C ^ 1 $ is an approximation of level surfaces of functions defined on irregular grids”, Siberian Journal of Industrial Mathematics, XIII:2 (2010), 69–78 | MR | Zbl

[8] A. Ženišek, “Interpolation Polynomials on the Triangle”, Numer. Math., 15 (1970), 283–296 | DOI | MR

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

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

[11] Yu.N. Subbotin, “Multidimensional piecewise polynomial interpolation”, Approximation and interpolation methods, VTsN, Novosibirsk, 1981, 148–153 | MR

[12] Yu.N. Subbotin, “The dependence of estimates of a multidimensional piecewise-polynomial approximation on the geometric characteristics of a triangulation”, Proc. Steklov Inst. Math., 189 (1990), 135–159 | MR | Zbl

[13] Yu.N. Subbotin, “The dependence of estimates of approximation by interpolation polynomials of the fifth degree on the geometric characteristics of a triangle”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2, 1992, 110–119 | MR | Zbl

[14] V. Latypova, “Piecewise-cubic error interpolation on a triangle”, Bulletin of Udmurt University. Maths., 2003, 3–10

[15] Yu. V. Matveeva, “On the Hermitian interpolation of many members of the third degree on the triangle using mixed derivatives”, Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 7:1 (2007), 23–27 | DOI | MR

[16] N. V. Baidakova, “Influence of smoothness on the error of approximation of derivatives under local interpolation on triangulations”, Proc. Steklov Inst. Math. (Suppl.), 277:1 (2012), 33–47 | DOI | MR

[17] N. V. Baidakova, “Above Estimates for the Error Values of the Approximation of Derivatives in the Finite Element of Cie–Klaf–Tocher”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 18, no. 4, 2012, 80–89

[18] N. V. Baidakova, “New Estimates of the Error Values for the Approximation of Derivatives in the Interpolation of a Function by Third Degree Polynomials on a Triangle”, Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 13:1 (2013), 15–19 | DOI | Zbl

[19] V. A. Klyachin, A. A. Shiroky, “The Delaunay triangulation for multidimensional surfaces and its approximative properties”, Russian Math. (Iz. VUZ), 56:1 (2012), 27–34 | DOI | MR | Zbl

[20] V. A. Klyachin, A. A. Shiroky, “Delone Triangulation of Multidimensional Surfaces”, Vestn. SamSU. Natural Science Ser., 2010, no. 4(78), 51–55

[21] V. A. Klyachin, D. V. Shurkaeva, “Isoperimetric Coefficient of a Simplex in the Problem of the Approximation of Derivatives”, Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 15:2 (2015), 151–160 | DOI

[22] V. A. Klyachin, “Algorithm of triangulation, based on the condition of an empty convex set”, Bulletin of VolSU. Series 1. Mathematics. Physics, 2015, no. 3(28), 27–33 | MR

[23] M. A. Gatsunaev, “Approximate calculation of surface area”, Materials of the Scientific Session (Volgograd, April 26–30, 2010), v. 6, 66–70

[24] A.F. Rasmussen, F. Atgeirr, Approximation of Curve Length and Surface Area, CMA, University of Oslo Darmstadt, 2006

[25] A.F. Rasmussen, M. S. Floater, “Approximation of Curve Length and Surface Area”, Proceedings of the SIAM conference on Geometric Design and Computing (Phoenix, 2005)

[26] N. N. Kalitkin, Numerical methods, «Nauka», M., 1978 | MR