Symbolic-Numerical Algorithm for Generating Interpolation Multivariate Hermite Polynomials of High-Accuracy Finite Element Method

Abstract

A symbolic-numerical algorithm implemented in Maple for constructing Hermitian finite elements is 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 directions of the normals to the boundaries of finite elements. Such a choice of the polynomials allows us to construct a piecewise polynomial basis continuous across the boundaries of elements together with the derivatives up to a given order, which is used to solve elliptic boundary value problems using the high-accuracy finite element method. The efficiency and the accuracy order of the finite element scheme, algorithm and program are demonstrated by the example of the exactly solvable boundary-value problem for a triangular membrane, depending on the number of finite elements of the partition of the domain and the number of piecewise polynomial basis functions.