1. Introduction#
The finite element method is used in many scientific fields to solve partial differential equations. It makes it possible to build a simple approximation of the unknowns to transform these continuous equations into a system of equations of finite dimension, which can be written schematically in the following form:
where \(\left\{U\right\}\) is the vector of unknowns, \(\mathrm{[}A\mathrm{]}\) is a matrix, and \(\left\{L\right\}\) is a vector.
First, we transform the partial differential equations into an integral formulation (or strong formulation of the problem), often this first integral form is modified (weakened) using Green’s formula (we then obtain a weak formulation). The approximate solution is sought as a linear combination of given functions. These functions must be simple but general enough to be able to « properly » approach the solution. In particular, they must make it possible to generate a finite dimensional space that is as close as you want to the function space in which the solution is found. Based on this ancient idea (weighted residue method), the various ways of choosing these functions give rise to different numerical methods (collocation, spectral methods, finite elements, etc.).
The originality of the finite element method is that it takes polynomials as approximation functions that are zero over almost the entire domain, and therefore participate in the calculation only in the vicinity of a particular point. Thus, the matrix \([A]\) is very sparse, containing only the interaction terms between « neighboring points », which reduces the calculation time and the memory space required for storage. In addition, the matrix \([A]\) and the vector \(\left\{L\right\}\) can be constructed by assembling elementary matrices and vectors, calculated locally.