As it will be explained by Matthieu Vanicat in another talk, computing smooth branches of steady states for some reaction diffusion system of PDEs and rigorously proving the validity of those results can be done numerically by a finite number of verifications. However the computational cost of this method can be significant. Thus I will discuss how to adapt the different parameters of the algorithm such as the dimension of the Galerkin projection and the decay rate of the Fourier coefficients, as well as the way of constructing whole branches of bifurcations, in order to improve the speed of the proof. Finally, to illustrate the interest of this method I will show on one particular system the kind of results that we were able to get such as the existence of several non trivial co-existing steady states.