Different full field models exist for the simulation of anisotropic grain growth in polycrystalline structures. In the present work, two full field methods i.e. level set (LS) and multi-phase-field (MPF) methods are implemented into finite element (FE) formulations. The model formulation and simulation results are compared on FE unstructured mesh for the two methods. Relationships between the parameters in both models representing same grain boundary properties are derived. Using equivalent model parameters, both methods can provide similar numerical accuracy in the case of different theoretical test of 2D anisotropic grain growth. In addition, two 2D large scale grain growth simulations are performed with respectively isotropic and anisotropic grain boundary energies by level set method to illustrate the performance of this full field approach.