Intuitive definition of Finite Elements in 1D

In comparison with the finite difference method, The finite element (FE) method is a more mathematical approach. It results from the analytic point of view that as a member of a function space, the solution must have a representation in terms of the bases of such function space.

Let us start with a simple 1D example. In this case, our domain \(\Omega\) is simply an open interval \(\Omega := (a, b)\). The discretization of an interval is done by selecting \(N\) internal points \(x_1, \dots, x_N\) such that

(1)\[\begin{equation} a =: x_0 < x_1 < \dots < x_N < x_{N+1} := b. \end{equation}\]

By calling each subdomain \(K_j = (x_{j-1}, x_j)\) (also called elements), we then call the set of all elements

\[\begin{equation*} \mathcal{I}_h = \{ K_j \}_{j=1}^{N+1} \end{equation*}\]

a partition (of \(\Omega\)).

The parameter \(h\) indicates how fine a mesh is. Having defined the element diameters

\[\begin{equation*} h_j = x_j - x_{j-1} \qquad j=1, \dots, N+1, \end{equation*}\]

then \(h = \max_j h_j\).