The reaction-diffusion equations are approximated by a fully discrete systems : a Legendre-Galerkin approximation for the space variables and a semi-implicit scheme for the time integration. The stability and the convergence of the fully discrete systems are established. It is also shown that, under a restriction on the space dimension and the growth rate of the nonlinear term, the approximate attractors of the discrete finite dimensional dynamical systems converge to the attractor of the original infinite dynamical systems. An error estimate of optimal order is derived as well without any further regularity assumption.