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

Part of the Lecture Notes in Computer Science book series (LNTCS,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

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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.

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Gusev, A.A. et al. (2018). Symbolic-Numerical Algorithms for Solving Elliptic Boundary-Value Problems Using Multivariate Simplex Lagrange Elements. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2018. Lecture Notes in Computer Science(), vol 11077. Springer, Cham. https://doi.org/10.1007/978-3-319-99639-4_14

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  • DOI: https://doi.org/10.1007/978-3-319-99639-4_14

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