A method of discretization for a non linear singularly perturbed boundary value problem is considered. It involves a certain number of steps, one of them including the application of Petrov-Galerkin finite element methods. The resulting scheme is called adjoint method scheme and is in some way related to Niijima's scheme (cf. ). It is proved that this discretization provides existence and uniqueness of solution for a problem defined by the Lagerström- Cole model equation. Finally some numerical experiments compare the results obtained when the adjoint method scheme is used, as well as when Niijima's scheme or a direct finite element discretization are applied.