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)


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.


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



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.


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© 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

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