Introduction to the finite element method

We saw in the article [AF 503] that an elliptic partial differential equation can be expressed in various equivalent formulations, in the sense that any solution of one formulation is a solution of another formulation and vice versa. The strong formulation of the problem is of interest insofar as the use of the finite-difference method is envisaged to approximate the problem. The equivalent formulation of the problem is based on the formulation of an optimization problem associated with the functional $v\to J\left(v\right)$ , with $J\left(v\right)$ defined by : $J\left(v\right)=\frac{1}{2}a\left(v,\mathrm{\hspace{0.17em}}v\right)-L\left(v\right)$

requires the bilinear form $a\left(u,\mathrm{\hspace{0.17em}}v\right)$ to be symmetrical, which in itself is restrictive insofar as certain phenomena are modeled using non-self-adjoint operators. However, when a(.,.) is symmetrical, this formulation of the problem leads to the Ritz method; numerically, the idea is to seek to minimize J(.) no longer on the entire set E, but on a subspace of E constructed from easily calculable functions; the unknown function that achieves the minimum is represented as a linear combination of form functions (or any other physically admissible family) and the coefficients of this linear combination are the unknowns of the problem. J(.) is then transformed into a quadratic functional, and determining the minimum of this new functional then amounts to canceling its partial derivatives with respect to these unknowns, which classically leads to the solution of a linear system. We will not develop this method further.

The other equivalent formulation highlighted is the weak or variational formulation based on the virtual work theorem; this equivalent expression of the problem contains all the information relating to it, i.e. the partial differential equation and the boundary conditions; moreover, it offers great scope for the effective calculation of approximate...