Advertisement

Symbolic-Numerical Algorithms for Solving Elliptic Boundary-Value Problems Using Multivariate Simplex Lagrange Elements

  • A. A. GusevEmail author
  • V. P. Gerdt
  • O. Chuluunbaatar
  • G. Chuluunbaatar
  • S. I. Vinitsky
  • V. L. Derbov
  • A. Góźdź
  • P. M. Krassovitskiy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11077)

Abstract

We propose new symbolic-numerical algorithms implemented in Maple-Fortran environment for solving the self-adjoint elliptic boundary-value problem in a d-dimensional polyhedral finite domain, using the high-accuracy finite element method with multivariate Lagrange elements in the simplexes. The high-order fully symmetric PI-type Gaussian quadratures with positive weights and no points outside the simplex are calculated by means of the new symbolic-numerical algorithms implemented in Maple. Quadrature rules up to order 8 on the simplexes with dimension \(d=3-6\) are presented. We demonstrate the efficiency of algorithms and programs by benchmark calculations of a low part of spectra of exactly solvable Helmholtz problems for a cube and a hypercube.

Keywords

Elliptic boundary-value problem Finite element method Multivariate simplex lagrange elements High-order fully symmetric Gaussian quadratures Helmholtz equation for cube and hypercube 

Notes

Acknowledgment

The work was partially supported by the RFBR (grant No. 16-01-00080 and 18-51-18005), the MES RK (Grant No. 0333/GF4), the Bogoliubov-Infeld program, the Hulubei–Meshcheryakov program, the RUDN University Program 5-100 and grant of Plenipotentiary of the Republic of Kazakhstan in JINR. The authors are grateful to prof. R. Enkhbat for useful discussions.

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)zbMATHGoogle Scholar
  2. 2.
    Akishin, P.G., Zhidkov, E.P.: Some symmetrical numerical integration formuas for simplexes. Communications of the JINR 11–81-395, Dubna (1981). (in Russian)Google Scholar
  3. 3.
    Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice Hall, Englewood Cliffs (1982)Google Scholar
  4. 4.
    Bériot, H., Prinn, A., Gabard, G.: Efficient implementation of high-order finite elements for Helmholtz problems. Int. J. Numer. Meth. Eng. 106, 213–240 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978)zbMATHGoogle Scholar
  6. 6.
    Cui, T., Leng, W., Lin, D., Ma, S., Zhang, L.: High order mass-lumping finite elements on simplexes. Numer. Math. Theor. Meth. Appl. 10(2), 331–350 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dobrowolski, A., Mazurek, K., Góźdź, A.: Consistent quadrupole-octupole collective model. Phys. Rev. C 94, 054322-1–054322-20 (2017)Google Scholar
  8. 8.
    Dunavant, D.A.: High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int. J. Numer. Meth. Eng. 21, 1129–1148 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gusev, A.A., et al.: Symbolic-numerical algorithm for generating interpolation multivariate hermite polynomials of high-accuracy finite element method. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 134–150. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66320-3_11CrossRefGoogle Scholar
  10. 10.
    Gusev, A.A., et al.: Symbolic-numerical algorithms for solving the parametric self-adjoint 2D elliptic boundary-value problem using high-accuracy finite element method. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 151–166. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66320-3_12CrossRefGoogle Scholar
  11. 11.
    Gusev, A.A., et al.: Symbolic algorithm for generating irreducible rotational-vibrational bases of point groups. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 228–242. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-45641-6_15CrossRefGoogle Scholar
  12. 12.
    Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2, 164–168 (1944)MathSciNetCrossRefGoogle Scholar
  13. 13.
  14. 14.
    Marquardt, D.: An algorithm for least squares estimation of parameters. J. Soc. Ind. Appl. Math. 11, 431–441 (1963)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Maeztu, J.I., Sainz de la Maza, E.: Consistent structures of invariant quadrature rules for the \(n\)-simplex. Math. Comput. 64, 1171–1192 (1995)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mead, D.G.: Dissection of the hypercube into simplexes. Proc. Am. Math. Soc. 76, 302–304 (1979)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mysovskikh, I.P.: Interpolation Cubature Formulas. Nauka, Moscow (1981). (in Russian)zbMATHGoogle Scholar
  18. 18.
    Papanicolopulos, S.-A.: Analytical computation of moderate-degree fully-symmetric quadrature rules on the triangle. arXiv:1111.3827v1 [math.NA] (2011)
  19. 19.
    Sainz de la Maza, E.: Fórmulas de cuadratura invariantes de grado 8 para el simplex 4-dimensional. Revista internacional de métodos numéricos para cálculo y diseño en ingeniería 15(3), 375–379 (1999)Google Scholar
  20. 20.
    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)zbMATHGoogle Scholar
  21. 21.
    Zhang, L., Cui, T.: Liu. H.: A set of symmetric quadrature rules on triangles and tetrahedra. J. Comput. Math. 27, 89–96 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • A. A. Gusev
    • 1
    Email author
  • V. P. Gerdt
    • 1
    • 2
  • O. Chuluunbaatar
    • 1
    • 3
  • G. Chuluunbaatar
    • 1
    • 2
  • S. I. Vinitsky
    • 1
    • 2
  • V. L. Derbov
    • 4
  • A. Góźdź
    • 5
  • P. M. Krassovitskiy
    • 1
    • 6
  1. 1.Joint Institute for Nuclear ResearchDubnaRussia
  2. 2.RUDN UniversityMoscowRussia
  3. 3.Institute of MathematicsNational University of MongoliaUlaanbaatarMongolia
  4. 4.N.G. Chernyshevsky Saratov National Research State UniversitySaratovRussia
  5. 5.Institute of PhysicsUniversity of M. Curie–SkłodowskaLublinPoland
  6. 6.Institute of Nuclear PhysicsAlmatyKazakhstan

Personalised recommendations