The polynomial approximation in the finite element method

In the analysis of a finite element method (FEM) we can describe the shape of a given element by a set of elementary functions known as shape functions. The approaches describing these functions are quite different ones. In the plane (x1,x2), these functions are a product of Lagrangian polynomials when the coordinate system can be chosen with the axes parallel to the sides of the element, otherwise a system of barycentric coordinates (sometimes called area coordinates) could be introduced. m of this paper is the description and the representation of shape functions when the element has triangular shape (the simplest). The representation has been done by using two algorithmic schemes: Neville-Aitken and De Casteljau. For these schemes we have deduced very important properties.