Hermite interpolation polynomials on parallelepipeds and FEM applications. (English) Zbl 07734148

Summary: We implement in Maple and Mathematica an algorithm for constructing multivariate Hermitian interpolation polynomials (HIPs) inside a \(d\)-dimensional hypercube as a product of \(d\) pieces of one-dimensional HIPs of degree \(p'\) in each variable, that are calculated analytically using the authors’ recurrence relations. The piecewise polynomial functions constructed from the HIPs have continuous derivatives and are used in implementations of the high-accuracy finite element method. The efficiency of our finite element schemes, algorithms and GCMFEM program implemented in Maple and Mathematica are demonstrated by solving reference boundary value problems (BVPs) for multidimensional harmonic and anharmonic oscillators used in the Geometric Collective Model (GCM) of atomic nuclei. The BVP for the GCM is reduced to the BVP for a system of ordinary differential equations, which is solved by the KANTBP 5 M program implemented in Maple.


68-XX Computer science
Full Text: DOI


[1] Berezin, IS; Zhidkov, NP, Computing Methods (1965), Oxford: Pergamon Press, Oxford · Zbl 0122.12903
[2] Lorentz, RA, Multivariate Birkhoff Interpolation (1992), Berlin: Springer, Berlin · Zbl 0760.41002 · doi:10.1007/BFb0088788
[3] Lekien, F.; Marsden, J., Tricubic interpolation in three dimensions, Int. J. Num. Meth. Eng., 63, 455-471 (2005) · Zbl 1140.76423 · doi:10.1002/nme.1296
[4] Chuluunbaatar, G., Gusev, A.A., Chuluunbaatar, O., Gerdt, V.P., Vinitsky, S.I., Derbov, V.L., Góźdź, A., Krassovitskiy, P.M., Hai, L.L.: Construction of multivariate interpolation Hermite polynomials for finite element method. EPJ Web Conf. 226, 02007 (2020) · Zbl 1353.81007
[5] Gusev, A.A., Chuluunbaatar, O., Vinitsky, S.I., Derbov, V.L., Góźdź, A., Hai, L.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. LNSC 8660, 138-154 (2014) · Zbl 1416.68218
[6] Gusev, A.A., Chuluunbaatar, G., Chuluunbaatar, O., Gerdt, V.P., Vinitsky, S.I., Hai, L.L., Lua, T.T., Derbov, V.L., Góźdź, A.: Algorithm for calculating interpolation Hermite polynomials in \(d\)-dimensional hypercube in the analytical form. In “Computer algebra” Conference Materials, Moscow, June 17-21, 2019 / ed. S.A. Abramov, L.A. Sevastianov. - Peoples’ Friendship University of Russia, 119-128 http://www.ccas.ru/ca/_media/ca-2019.pdf
[7] Troltenier, D.; Maruhn, JA; Hess, PO; Langanke, K.; Maruhn, JA; Konin, SE, Numerical application of the geometric collective model, Computational Nuclear Physics, 105-128 (1991), Berlin: Springer-Verlag, Berlin · doi:10.1007/978-3-642-76356-4_6
[8] Chuluunbaatar, G.; Gusev, A.; Derbov, V.; Vinitsky, S.; Chuluunbaatar, O.; Hai, LL; Gerdt, V., A Maple implementation of the finite element method for solving boundary-value problems for systems of second-order ordinary differential equations, Commun. Comput. Inform. Sci., 1414, 152-166 (2021)
[9] Gusev, A., Vinitsky, S., Chuluunbaatar, O., Chuluunbaatar, G., Gerdt, V., Derbov, V., Gozdz, A., Krassovitskiy, P.: Interpolation Hermit polinomials for finite element method. EPJ Web Conf. 173, 03009 (2018) · Zbl 1453.65406
[10] Bathe, KJ, Finite Element Procedures in Engineering Analysis (1982), NY: Eng. Cliffs, NY
[11] Walker, P.: Quadcubic interpolation: a four-dimensional spline method, preprint (2019), available at http://arxiv.org/abs/1904.09869v1; Walker, P., Krohn, U. and Carty, D.: ARBTools: A tricubic spline interpolator for three-dimensional scalar or vector fields. Journal of Open Research Software, 7(1), p12. (2019)
[12] Schwarz H. R.: Methode der finiten Elemente. 2-nd edn. B.G. Teubner, Stuttgart (1984)
[13] Schwarz, HR, FORTRAN-Programme zur methode der finiten Elemente (1991), Fachmedien Wiesbaden: Springer, Fachmedien Wiesbaden · Zbl 0711.65095 · doi:10.1007/978-3-663-10784-2
[14] Troltenier, D.; Maruhn, JA; Greiner, W.; Hess, PO, A general numerical solution of collective quadrupole surface motion applied to microscopically calculated potential energy surfaces, Z. Phys. A. Hadrons Nuclei, 343, 25-34 (1992) · doi:10.1007/BF01291593
[15] Abramowitz, M.; Stegun, IA, Handbook of Mathematical Functions (1972), New York: Dover, New York · Zbl 0543.33001
[16] Deveikis, A.; Gusev, AA; Vinitsky, SI; Blinkov, YA; Góźdź, A.; Pȩdrak, A.; Hess, PO, Symbolic-numeric algorithm for calculations in geometric collective model of atomic nuclei, Comput. Sci., 13366, 103-123 (2022) · Zbl 1517.81098
[17] Deveikis, A., Gusev A.A., Vinitsky S.I., Góźdź, A., Pȩdrak, A., Burdik, Č., Pogosyan, G.S.: Symbolic-numeric algorithm for computing orthonormal basis of \(O(5)\times SU(1,1)\) group. CASC 2020. LNCS 12291, 206-227 (2020) · Zbl 07635831
[18] Moshinsky, M., The harmonic oscillator in modern physics and Smirnov (1996), Y.F.: HAP, Y.F. · Zbl 0865.00015
[19] Yannouleas, C.; Pacheco, JM, An algebraic program for the states associated with the \(U(5) \supset O(5)\supset O(3)\) chain of groups, Comput. Phys. Commun., 52, 85-92 (1988) · Zbl 0798.22010 · doi:10.1016/0010-4655(88)90175-0
[20] Yannouleas, C.; Pacheco, JM, Algebraic manipulation of the states associated with the \(U(5) \supset O(5)\supset O(3)\) chain of groups: orthonormalization and matrix elements, Comput. Phys. Commun., 54, 315-328 (1989) · Zbl 0798.22010 · doi:10.1016/0010-4655(89)90094-5
[21] Varshalovitch, D.A., Moskalev, A.N., and Hersonsky, V.K.: Quantum theory of angular momentum Leningrad Nauka. (1975); Singapore: World Scientific (1988)
[22] Bohr, A. and Mottelson, B.R.: Nuclear Structure. N Y, Amsterdam: W A Bejamin Inc, Vol 2, (1970) · Zbl 0051.44305
[23] Eisenberg, J.M., Greiner W.: Nuclear theory. Vol. 1: Nuclear models. Collective and single-particle phenomena. Amsterdam, London, North-Holland Publ. Co. (1970); Moscow, Atomizdat (1975)
[24] Dobrowolski, A.; Mazurek, K.; Góźdź, A., Consistent quadrupole-octupole collective model, Phys. Rev. C, 94 (2016) · doi:10.1103/PhysRevC.94.054322
[25] Dobrowolski, A.; Mazurek, K.; Góźdź, A., Rotational bands in the quadrupole-octupole collective model, Phys. Rev. C, 97 (2018) · doi:10.1103/PhysRevC.97.024321
[26] Ermamatov, MJ; Hess, Peter O., Microscopically derived potential energy surfaces from mostly structural considerations, Ann. Phys., 37, 125-158 (2016) · doi:10.1016/j.aop.2016.04.010
[27] Rohoziński, SG; Dobaczewski, J.; Nerlo-Pomorska, B.; Pomorski, K.; Srebrny, J., Microscopic dynamic calculations of collective states in xenon and barium isotopes, Nucl. Phys. A, 292, 66-87 (1977) · doi:10.1016/0375-9474(77)90358-X
[28] Mardyban, EV; Kolganova, EA; Shneidman, TM; Jolos, RV, Evolution of the phenomenologically determined collective potential along the chain of Zr isotopes, Phys. Rev. C, 105 (2022) · doi:10.1103/PhysRevC.105.024321
[29] Hess, PO; Ermamatov, M., In search of a broader microscopic underpinning of the potential energy surface in heavy deformed nuclei, J. Phys.: Conf. Ser., 876 (2017)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.