In this article we study reaction-diffusion systems containing self and cross-diffusion using a free energy approach. This framework naturally leads to the formulation of an energy law, and to a numerical method respecting a discrete version of the latter (a linear second order finite-element method with adaptive mesh refinement). Applying this numerical scheme to a Gray-Scott system augmented with self and cross-diffusion terms, we verify numerically the energy law, and unveil patterns distinct from those obtained with linear diffusion.