Four different finite element level-set (FE-LS) formulations are compared for the modeling of grain growth in the context of polycrystalline structures and, moreover, two of them are presented for the first time using anisotropic grain boundary (GB) energy and mobility. Mean values and distributions are compared using the four formulations. First, we present the strong and weak formulations for the different models and the crystallographic parameters used at the mesoscopic scale. Second, some Grim Reaper analytical cases are presented and compared with the simulation results, here the evolutions of individual multiple junctions are followed. Additionally, large scale simulations are presented. Anisotropic GB energy and mobility are respectively defined as functions of the misorientation/inclination and disorientation. The evolution of the disorientation distribution function (DDF) is computed and its evolution is in accordance with prior works. We found that the formulation called "Anisotropic" is the more physical one but it could be replaced at the mesoscopic scale by an Isotropic formulation for simple microstructures presenting an initial Mackenzie-type DDF.