In this work, we consider, from the numerical point of view, a boundary-initial value problem for non-simple porous elastic materials. The mechanical problem is written as a coupled hyperbolic linear system in terms of the displacement and porosity fields. The resulting variational formulation is used to approximate the solution by the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the numerical scheme is deduced under adequate regularity conditions. Finally, some numerical simulations are presented to show the accuracy of the finite element scheme studied previously, the evolution of the discrete energy and the behavior of the solution.