Author(s)

VLADIMIR SLADEK, MIROSLAV REPKA, JAN SLADEK

Abstract

A novel discretization method is proposed and developed for numerical solution of boundary value problems governed by partial differential equations. The spatial variation of field variables is approximated by using Lagrange finite elements for interpolation without discretization of the analysed domain into the mesh of finite elements. Only the net of nodal points is used for discrete degrees of freedom on the analysed domain and its boundary. The governing equations are considered at interior nodal points while the boundary conditions at nodal points on the boundary. The finite elements are created for each nodal point properly instead of using fixed finite elements like in standard Finite Element Method. In this way, we can eliminate interfaces between elements as well as the difficulties with continuity of derivatives of field variables on such interfaces. Both the strong and weak formulations are implemented for governing equations. The reliability (accuracy and efficiency) of the new method has been verified in numerical simulations for 2D problems of heat conduction in solids with possible continuous gradation of the heat conduction coefficient.

Keywords

Lagrange finite element, bi-quadratic and bi-cubic approximation, strong and weak formulations