4.2.2 Discretization of the Domain

The domain Ω=(0,L) of interest is divided into a collection of finite elements. The set of subintervals in a domain is called the finite element mesh of the domain. The mesh depends on the geometry of the domain and on the desired accuracy of the solution [125]. A linear element Ω_i is located between the nodes k and k+1, with coordinates x_k and x_k+1, respectively, with a length h_i=x_k+1-x_k. The discretization of the domain allows one to calculate approximate solutions over each subdomain rather than over the entire domain. The solutions on the subdomains are represented by continuous functions at the nodes of these subdomains. In the present 1D case, the discretization of the straight line is obtained by constructing the proper finite element mesh and by defining the elements and the nodes of the domain in order to seek an approximation of the solution to the governing DE over each subdomain. For simplicity, a fine element mesh of two elements (Ω₁ and Ω₂) and three nodes (1,2, and 3) is proposed, as shown in (4.1).

Figure 4.2: Schematic view of the finite element mesh of the domain Ω=(0,L), with the definition of the boundary conditions at the endpoints.