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Symbolic-Numerical Algorithm for Generating Interpolation Multivariate Hermite Polynomials of High-Accuracy Finite Element Method

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Book cover Computer Algebra in Scientific Computing (CASC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10490))

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Abstract

A symbolic-numerical algorithm implemented in Maple for constructing Hermitian finite elements is presented. The basis functions of finite elements are high-order polynomials, determined from a specially constructed set of values of the polynomials themselves, their partial derivatives, and their derivatives along the directions of the normals to the boundaries of finite elements. Such a choice of the polynomials allows us to construct a piecewise polynomial basis continuous across the boundaries of elements together with the derivatives up to a given order, which is used to solve elliptic boundary value problems using the high-accuracy finite element method. The efficiency and the accuracy order of the finite element scheme, algorithm and program are demonstrated by the example of the exactly solvable boundary-value problem for a triangular membrane, depending on the number of finite elements of the partition of the domain and the number of piecewise polynomial basis functions.

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References

  1. Ames, W.F.: Numerical Methods for Partial Differential Equations. Academic Press, London (1992)

    MATH  Google Scholar 

  2. Argyris, J.H., Buck, K.E., Scharpf, D.W., Hilber, H.M., Mareczek, G.: Some new elements for the matrix displacement method. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics (2nd), Wright-Patterson Air Force Base, Ohio, 15–17 October 1968

    Google Scholar 

  3. Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice Hall, Englewood Cliffs/New York (1982)

    Google Scholar 

  4. Bell, K.: A refined triangular plate bending element. Int. J. Numer. Methods Eng. 1, 101–122 (1969)

    Article  Google Scholar 

  5. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008). doi:10.1007/978-0-387-75934-0

    Book  MATH  Google Scholar 

  6. Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978)

    MATH  Google Scholar 

  7. Dhatt, G., Touzot, G., Lefrançois, E.: Finite Element Method. Wiley, Hoboken (2012)

    Book  MATH  Google Scholar 

  8. Gasca, M., Sauer, T.: On the history of multivariate polynomial interpolation. J. Comp. Appl. Math. 122, 23–35 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gusev, A.A., Chuluunbaatar, O., Vinitsky, S.I., Derbov, V.L., Góźdź, A., Le Hai, L., Rostovtsev, V.A.: Symbolic-numerical solution of boundary-value problems with self-adjoint second-order differential equation using the finite element method with interpolation hermite polynomials. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2014. LNCS, vol. 8660, pp. 138–154. Springer, Cham (2014). doi:10.1007/978-3-319-10515-4_11

    Google Scholar 

  10. Gusev, A.A., Hai, L.L., Chuluunbaatar, O., Vinitsky, S.I.: KANTBP 4M: Program for Solving Boundary Problems of the System of Ordinary Second Order Differential Equations. http://wwwinfo.jinr.ru/programs/jinrlib/indexe.html

  11. Habib, A.W., Goldman, R.N., Lyche, T.: A recursive algorithm for Hermite interpolation over a triangular grid. J. Comput. Appl. Math. 73, 95–118 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ladyzhenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Applied Mathematical Sciences, vol. 49. Springer, New York (1985). doi:10.1007/978-1-4757-4317-3

    MATH  Google Scholar 

  13. Lekien, F., Marsden, J.: Tricubic interpolation in three dimensions. Int. J. Numer. Meth. Eng. 63, 455–471 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Logg, A., Mardal, K.-A., Wells, G.N. (eds.): Automated Solution of Differential Equations by the Finite Element Method (The FEniCS Book). Springer, Heidelberg (2012). doi:10.1007/978-3-642-23099-8

    MATH  Google Scholar 

  15. www.maplesoft.com

  16. McCartin, B.J.: Laplacian Eigenstructure of the Equilateral Triangle. Hikari Ltd., Ruse, Bulgaria (2011)

    MATH  Google Scholar 

  17. Mitchell, A.R., Wait, R.: The Finite Element Method in Partial Differential Equations. Wiley, Chichester (1977)

    MATH  Google Scholar 

  18. Pockels, F.: Über die Partielle Differential-Gleichung \(\Delta u+k^2u=0\) und deren Auftreten in der Mathematischen Physik. B.G. Teubner, Leipzig (1891)

    MATH  Google Scholar 

  19. Ramdas Ram-Mohan, L.: Finite Element and Boundary Element Aplications in Quantum Mechanics. Oxford University Press, New York (2002)

    MATH  Google Scholar 

  20. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs/New York (1973)

    MATH  Google Scholar 

  21. Zienkiewicz, O.C.: Finite elements. The background story. In: Whiteman, J.R. (ed.) The Mathematics of Finite Elements and Applications, p. 1. Academic Press, London (1973)

    Google Scholar 

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Authors and Affiliations

  1. Joint Institute for Nuclear Research, Dubna, Russia

    A. A. Gusev, V. P. Gerdt, O. Chuluunbaatar, G. Chuluunbaatar & S. I. Vinitsky

  2. RUDN University, 6 Miklukho-Maklaya St., Moscow, 117198, Russia

    V. P. Gerdt & S. I. Vinitsky

  3. Institute of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia

    O. Chuluunbaatar

  4. N.G. Chernyshevsky Saratov National Research State University, Saratov, Russia

    V. L. Derbov

  5. Institute of Physics, University of M. Curie-Skłodowska, Lublin, Poland

    A. Góźdź

Authors
  1. A. A. Gusev
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  2. V. P. Gerdt
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  3. O. Chuluunbaatar
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  4. G. Chuluunbaatar
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  5. S. I. Vinitsky
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  6. V. L. Derbov
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  7. A. Góźdź
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Corresponding author

Correspondence to A. A. Gusev .

Editor information

Editors and Affiliations

  1. Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Gusev, A.A. et al. (2017). Symbolic-Numerical Algorithm for Generating Interpolation Multivariate Hermite Polynomials of High-Accuracy Finite Element Method. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2017. Lecture Notes in Computer Science(), vol 10490. Springer, Cham. https://doi.org/10.1007/978-3-319-66320-3_11

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  • DOI: https://doi.org/10.1007/978-3-319-66320-3_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66319-7

  • Online ISBN: 978-3-319-66320-3

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