High-Accuracy Finite Element Methods for Solution of Discrete Spectrum Problems

A symbolic-numeric algorithm implemented in the Maple system for constructing multivariable Hermitian finite elements are presented. The basis functions of finite elements are high-order polynomials determined from a specially constructed set of values of the polynomials themselves, their partial derivatives and their derivatives along the normals to the boundaries of the finite elements. Such a choice of polynomials makes it possible to construct a piecewise polynomial basis continuous across the boundaries of the elements together with the derivatives up to a given order, which is used to solve elliptic boundary-value problems (BVPs) using the high-accuracy finite element method (FEM). The efficiency and accuracy order of the FEM, the algorithm and the program are demonstrated by test examples of exactly solvable Helmholtz problems on a triangle, square and four-dimensional hypercube, depending on the number of finite elements of the domain partition, the number of piecewise polynomial basis functions, and the dimension of eigenvectors of the corresponding algebraic eigenvalue problems. Comparison of lower parts spectra of quadrupole-octupole-vibrational collective model obtained by FEM and finite difference method (FDM) in solving the two dimensional BVP with the numerical tabular coefficients is also given.