Recent advances in the development of adaptive grid methods for the 2D fluid simulation of tokamak edge plasmas are presented. Based on previous experiences on the solution of scalar model problems using both finite elements (FE) and finite volumes (FV), the first extension of the technique to a simplified version of the full system of Braginskii equations is presented here. The performances of adaptive grid technique are discussed and compared to the traditional fixed meshes on model problems representing a steady-state detachment and a transient evolution.