In this paper we construct and approximate breathers in the DNLS model
starting from the continuous limit: such periodic solutions are obtained as
perturbations of the ground state of the NLS model in $H^1(\RR^n)$, with
$n=1,2$. In both the dimensions we recover the Sievers-Takeno (ST) and the Page
(P) modes; furthermore, in $\RR^2$ also the two hybrid (H) modes are
constructed. The proof is based on the interpolation of the lattice using the
Finite Element Method (FEM).