5. Error surcharge

In the construction of finite-difference schemes, it was stated that the solution of the numerical scheme represented an approximation of the exact solution. The aim of this paragraph is to give an estimate of the error; this will be obtained thanks, on the one hand, to the properties of discrete coercivity established previously and, on the other hand, to the notion of truncation error; this is defined as the difference between the two members of the numerical scheme where the approximate solution is replaced by the exact solution at the discretization points. For the Poisson problem, with homogeneous Dirichlet boundary conditions and for the three schemes considered to approximate this problem in the unit segment, unit square and unit cube respectively, we can therefore formalize this definition.

• To begin with, let's consider the case of...