High-Accuracy Finite Element Method for Solving Boundary-Value Problems for Elliptic Partial Differential Equations

A new computational scheme of the finite element method of a high order of accuracy for solving boundary value problems for an elliptic partial differential equation that preserves the continuity of the derivatives of the approximate solution in a bounded domain of a multidimensional Euclidean space is proposed. A piecewise continuous basis of the finite element method is generated using interpolation Hermite polynomials of several variables and ensures the continuity of not only the approximate solution but also its derivatives up to a given order on the boundaries of finite elements, depending on the smoothness of the variable coefficients of the equation and the boundary of the domain. The efficiency and accuracy order of the computational scheme, algorithm and program are demonstrated by the example of an exactly solvable boundary-value problem for a triangular membrane depending on the number of finite elements of the partition of the domain and the dimension of the eigenvector of the algebraic problem. It was shown that, in order to achieve a given accuracy of the approximate solution, for schemes of the finite element method with Hermite interpolation polynomials the dimension of the eigenvector is approximately two times smaller than for schemes with Lagrange interpolation polynomials that preserve on the boundaries of finite elements only the continuity of the approximate solution. The high-accuracy computational scheme of the finite element method is oriented to calculations of the spectral and optical characteristics of quantum-mechanical systems.