Calculation Schemes for Solving Sturm- Liouville Problem by Finite-Element Method with Interpolating Hermite Polynomials

Calculation schemes for solving Sturm-Liouville problem with first-, second-and third-type boundary conditions by finite-element method holding a continuity of derivatives of a required solution in its approximated solution are constructed. Recurrence relations for the calculation in analytical form of the interpolating Hermite polynomials with nodes of arbitrary multiplicity are derived. Using the interpolating Hermite polynomials, the basis piecewise-polynomial functions on finite-element grid with nonuniform step, approximating desired solution of the original problem are constructed and used for reduction to a generalized algebraic eigenvalue problem with banded stiffness and mass matrices. The stiffness and mass matrices are formed by sums of integrals containing the given coefficient and potential functions of the original self-adjoint second-order differential equation and the calculated interpolating Hermite polynomials and their derivatives on the finite element grid. The integrals are calculated using Gauss quadratures and in special cases, including the piecewise continuous polynomial coefficient and potential functions in analytical form. The efficiency and rate of convergence of the proposed calculation schemes and elaborated algorithms and programs implemented in Maple and Fortran is proved by benchmark calculations of exactly solvable Sturm-Liouville problems with continuous and piecewise continuous potential functions.

Sturm-Liouville problem, calculation scheme, finite element method, interpolation Hermite polynomials