Overview of the finite element method

To numerically solve a PDE, the finite element method is often used in industry. This method involves transforming the problem to be solved into an equivalent problem. This phase corresponds to putting the PDE problem into variational form. In the latter formulation, the problem is posed in an infinite-dimensional space. The finite element method consists in posing an analogous problem in finite dimension, starting from a "triangulation" of the domain Ω where the PDE is defined, which requires the definition of basis functions whose choice is such that the discretization matrix is as hollow as possible (cf. the articles and

In this article, we show how to construct the basis functions associated with any "triangulation" of the domain under consideration. We begin by giving the expressions of the basis functions associated with any triangles and squares, and then show the invariance of these functions when an affine transformation is used, which makes it easy to determine the discretization matrix of the problem. Finally, we give some results on error majorization, as well as indications on the use of numerical quadrature formulas when implementing the finite element method.

This article is the third and final part of a set of three articles dealing with the finite element method:

• [AF 503] Variational approach for the finite element method ;

• [AF 504] Introduction to the finite element method ;

• [AF 505] Overview of the finite element method.

For further information on the general concept of finite elements, please refer to references [5] to [18].