We propose new computational schemes and algorithms of the finite element method for solving elliptic multidimensional boundary value problems with variable coefficients at derivatives in a polyhedral d-dimensional domain, aimed at describing collective models of atomic nuclei. The desired solution is sought in the form of an expansion in the basis of piecewise polynomial functions constructed in an analytical form by joining Hermite interpolation polynomials and their derivatives on the boundaries of neighboring finite elements having the form of d-dimensional parallelepipeds. Calculations of the spectrum, quadrupole momentum and electric transitions of standard boundary value problems for the geometric collective model of atomic nuclei are analyzed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall (1982).
I. S. Berezin and N. P. Zhidkov, Computing Methods, Pergamon Press, Oxford (1965).
R. A. Lorentz, Multivariate Birkhoff interpolation, Springer, Berlin (1992).
F. Lekien and J. Marsden, “Tricubic interpolation in three dimensions,” Int. J. Num. Meth. Eng. 63, 455–471 (2005).
U. Vandandoo et al, High-Order Finite Difference and Finite-Element Methods for Solving Some Partial Differential Equations, Springer, Charm (2024).
G. Chuluunbaatar et al, “Construction of multivariate interpolation Hermite polynomials for finite element method,” EPJ Web of Conferences 226, Article No. 02007 (2020).
A. A. Gusev et al, “Hermite interpolation polynomials on parallelepipeds and FEM applications,” Math. Comput. Sci. 17, Article No. 18 (2023).
A. A. Gusev et al, “Symbolic-numerical solution of boundary-value problems with selfadjoint second-order differential equation using the finite element method with interpolation Hermite polynomials,” Lect. Notes Comput. Sci. 8660, 138–154 (2014)
M. Moshinsky and Y. F. Smirnov, The Harmonic Oscillator in Modern Physics, Harwood Acad. Publ., Chur (1996).
D. Troltenier, J. A. Maruhn, W. Greiner, and P. O. Hess, “A general numerical solution of collective quadrupole surface motion applied to microscopically calculated potential energy surfaces,” Z. Phys. A. Hadrons and Nuclei 343, 25–34 (1992).
D. Troltenier, J. A. Maruhn, and P. O. Hess, “Numerical application of the geometric collective model,” In: Computational Nuclear Physics. Vol. 1, pp. 105–128, Springer, Berlin (1991).
A. Deveikis et al, “Symbolic-numeric algorithm for calculations in geometric collective model of atomic nuclei,” Lect. Notes Comput. Sci. 13366, 103–123 (2022).
A. Dobrowolski, K. Mazurek, and A. G´o´zd´z, “Rotational bands in the quadrupole-octupole collective model,” Phys. Rev. C 97, 024321, 11 p. (2018).
M. J. Ermamatov and P. O. Hess, “Microscopically derived potential energy surfaces from mostly structural considerations,” Ann. Phys. 37, 125–158 (2016).
A. A. Gusev, O. Chuluunbaatar, S. I. Vinitsky, and A. G. Abrashkevich, “KANTBP 3.0: New version of a program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel adiabatic approach,” Comput. Phys. Commun. 185, 3341–3343 (2014).
O. Chuluunbaatar et al, “Solution of quantum mechanical problems using finite element method and parametric basis functions,” Bull. Russ. Acad. Sci., Phys. 82, 654–660 (2018).
P. M. Krassovitskiy and F. M. Pen’kov, “Features of scattering by a nonspherical potential,” Phys. Part. Nucl. 53, 247–250 (2022).
E. V. Mardyban, E. A. Kolganova, T. M. Shneidman, and R. V. Jolos, “Evolution of the phenomenologically determined collective potential along the chain of Zr isotopes,” Phys. Rev. C 105, Article ID 024321 (2022).
A. A. Gusev et al, “Finite element method for solving the collective nuclear model with tetrahedral symmetry,” Acta Phys. Pol. B 12, 589–594 (2019).
J. M. Eisenberg and W. Greiner, Nuclear Theory. Vol. 1: Nuclear Models. Collective and Single-Particle Phenomena, North-Holland, Amsterdam etc. (1970).
J. M. Eisenberg and W. Greiner, Nuclear Theory. Vol. 2: Excitation Mechanisms of Nucleus Electromagnetic and Weak Interactions, North-Holland, Amsterdam etc. (1970).
D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, Nauka, Leningrad (1975); World Scientific Publ. Co., Singapore (1988).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Batgerel, B., Vinitsky, S.I., Chuluunbaatar, O. et al. Schemes of Finite Element Method for Solving Multidimensional Boundary Value Problems. J Math Sci 279, 738–755 (2024). https://doi.org/10.1007/s10958-024-07056-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07056-6