Abstract
Explicit formulas are derived for 21 Zlamal–Zenisek basis interpolating polynomials of degree 5 in each triangle of the triangulation. Their use significantly reduces the number of arithmetic operations in the FEM because otherwise 21 systems with 21 unknowns should be solved in each triangle to find all the 21 coefficients of each of the basis interpolating polynomials of degree 5. The formulas are also presented for interpolation operators with the use of these basis polynomials and for the integral representation of the remainder term of the approximation of differentiable functions by these operators.
Similar content being viewed by others
References
I. V. Sergienko and V. S. Deineka, Systems Analysis [in Russian], Naukova Dumka, Kyiv (2013).
I. V. Sergienko, V. K. Zadiraka, and O. M. Lytvyn, Elements of the General Theory of Optimal Algorithms and Related Issues [in Ukrainian], Naukova Dumka, Kyiv (2012).
Yu. M. Matsevityi, Inverse Heat Conduction Problems [in Russian], Vol. 2: Applications, Naukova Dumka, Kyiv (2003).
M. Zlamal, “On the finite element method,” Numer. Math., 12, 394–409 (1968).
M. Zlamal, “On some finite element procedures for solving second order boundary value problems,” Numer. Math., 14, 42–48 (1969).
M. Zlamal, “A finite element procedure of the second order accuracy,” Numer. Math., 12, 394–402 (1970).
A. Zenisek, “Interpolation polynomials on the triangle,” Numer. Math., 15, 283–296 (1970).
M. Zlamal and A. Zenisek, “Mathematical aspect of the finite element method,” in: V. Kolar et al. (eds.), Technical, Physical and Mathematical Principles of the Finite Element Method, Acad. VED, Praha, (1971), pp. 15–39.
R. Varga, Functional Analysis and Approximation Theory in the Numerical Analysis [Russian translation], Mir, Moscow (1974).
I. Babushka and A. K. Aziz, “On the engle condition in the finite element method,” SIAM J. Numer. Anal., 13, No. 2, 214–226 (1976).
J. H. Bramble and M. Zlamal, “Triangular elements in the finite element method,” Math. Comput., 24, 809–820 (1970).
P. G. Ciarlet and P. A. Raviart, “General Lagrange and Hermite interpolation in Lp with application in finite element methods,” Arch. Rat. Mech. and Anal., 46, No. 3, 177–179 (1972).
Yu. N. Subbotin, “Multidimentional piecewise-polynomial interpolation,” in: Yu. A. Kuznetsov (ed.), Approximation and Interpolation Methods [in Russian], VTsN, Novosibirsk (1981), pp. 148–153.
Yu. N. Subbotin, “Dependence of the estimates of multidimentional piecewise polynomial approximation on geometrical characteristics of triangulation,” in: Tr. MIAN SSSR, 189, 117–137 (1989).
Yu. N. Subbotin, “Dependence of the estimates of approximation by interpolating polynomials of degree five on geometrical characteristics of triangle,” in: Tr. Inst. Matematiki i Mekhaniki UrO RAN, 2, 110–119 (1992).
Yu. N. Subbotin, “Analysis of the properties of monotonicity and convexity in local approximation,” Zh. Vych. Mat. Mat. Fiz., 33, No. 7, 996–1003 (1993).
Yu. N. Subbotin, “A new cubic element in the FEM,” in: Tr. Inst. Matematiki i Mekhaniki UrO RAN, Ekaterinburg, Theory of Functins, 11, No. 2, 120–130 (2005).
N. V. Baidakova, “On some interpolation process by polynomials of degree 4 1 m+ on the triangle,” Russ. J. Numer. Anal. Math. Modelling, 14, No. 2, 87–107 (1999).
N. V. Latypova, “Error estimates for approximation by polynomials of degree 4 1 m+ on the triangle,” Proc. of the Steklov Institute of Mathematics, Suppl. 1, 190–213 (2002).
N. V. Latypov, “Error of piecewise-cubic interpolation on a triangle,” Vest. Udmurt. Univ., Ser. Matem., (2003), pp. 3–10.
A. Zenisek, “Maximum-angle condition and triangular finite elements of Hermite type,” Math. Comput., 64, No. 211, 929–941 (1995).
N. V. Baidakova, “A method of Hermitian interpolation by polynomials of the third degree on a triangle,” in: Tr. Inst. Matematiki i Mekhaniki UrO RAN, Ekaterinburg, Theory of Functions, 11, No. 2, 47–52 (2005).
A. Zenisek and J. Hoderova–Zlamalova, “Semiregular hermite tetrahedral finite elements,” Appl. of Math., No. 4, 295–315 (2001).
Yu. V. Kupriyanova, “On the estimate of directional derivative of the Hermitian spline on a triangle,” Matematika, Mekhanika, Issue 8, 59–61 (2006).
Yu. V. Kupriyanova, “Estimate of the derivative of the Hermitian spline on three-dimensional simplex,” in: Abstr. of Papers Read at the 13th Saratov Winter School on Modern Problems of the Function Theory and their Application, Nauchnaya Kniga, Saratov (2006).
Yu. V. Kupriyanova, “Approximation of directional derivatives of interpolating polynomial on a triangle,” in: Proc. Conf. Modern Methods of the Function Theory and Related Problems, Voronezh (2007), pp. 120–121.
Yu. V. Kupriyanova, “On a theorem in spline theory,” Comp. Math. Math. Phys., 48, No. 2, 195–200 (2008).
Yu. V. Matveeva, “Interpolation by cubic polynomials of the third degree on a triangle with the use of mixed derivatives,” Izv. Saratov. Univ., Ser. Matematika, Mekhanika, Informatika, 7, Issue 1, 28–32 (2007).
Yu. V. Matveeva, “Approximation of functions by polynomials on a triangular mesh,” Author’s Abstr. of PhD Theses, Saratov (2008).
S. M. Nikol’skii, A Course in Mathematical Analysis [in Russian], Fizmatlit, Moscow (2001).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2014, pp. 25–33.
Rights and permissions
About this article
Cite this article
Sergienko, I.V., Lytvyn, O.M., Lytvyn, O.O. et al. Explicit Formulas for Interpolating Splines of Degree 5 on the Triangle. Cybern Syst Anal 50, 670–678 (2014). https://doi.org/10.1007/s10559-014-9657-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10559-014-9657-x