The Finite Elements Method (FEM for short) is a modern numerical tool to solve Partial Differential Equations (PDEs). Usually presented with their finite-difference and finite volume counterpart, this method remains indispensable for whoever wishes to compute numerical solutions in a competitive time. The FEM is part of the class of Direct Methods, in contrast with the multi-grid method for example, who belongs to the class of relaxation methods.
The FEM is very popular to teach in applied mathematics for at least two reasons.
The first one is that, in all its forms, the FEM always implies the numerical discretisation of a variational formulation of the PDE of concern. The latter being justified with classical arguments of functional analysis, the studend practices an abstract rigorous theoretical framework. Also, the actual resolution requires solid knowledge of numerical algebra and numerical analysis, such as quadrature, matrix factorization... Hence, several facets of applied mathematics are exercised with a unique course, making it very rewarding to learn.
The second reason is that it has many applications of paramount importance, in particular in the industrial sector.
The FEM usually relies on three fundamental approximations.
The first one is the meshing of the domain of interest. In the general case of a non-polyhedric domain, the meshing can be inexact and it adds a supplementary approximation. The control parameter is the meshing size h.
The second one is the use of a finite dimensional space. In the case of Lagrange polynomials, the control parameter is the degree of the said polynomials.
The third and last one is the quadrature formula to compute integrals in Mass and Stiffness matrix. This method is usually necessary for polynomials of degree greater than one. If the quadrature is chosen wisely, it can lead to interesting features of the Mass matrix such as mass lumping. The control parameter is the degree of the quadrature.