Problem discretization. Case of the one-dimensional problem
Finite-difference method for stationary PDEs

To introduce finite-difference techniques, let's consider the case of a one-dimensional Poisson problem with homogeneous Dirichlet boundary conditions. This problem can be illustrated in physical terms as follows: let's consider an elastic string of unit length, attached at each of its ends; we act on this string with a force f (x ), perpendicular to the string, in the plane containing the string. We propose to determine, for all x ∊ Ω = [0, 1], the displacement u (x ) of the string subjected to the force f (x ), a displacement compatible with the boundary conditions u (0) = u (1) = 0. Under the assumption of small displacements, we show that the preceding elasticity problem is modeled by the following partial differential equation: