This paper studies the combination of the Full-Multi-Grid (FMG) algorithm with an anisotropic metric- based mesh adaptation algorithm. For the sake of simplicity, the case of an elliptic two-dimentional Partial Differential Equation (PDE) is studied. Meshes are unstructured and non-embedded, defined through the metric-based parametrisation. A rather classical MG preconditionner is applied, in combination with a quasi-Newton fixed point. An anisotropic metric-based mesh adaptation loop is introduced inside the FMG algorithm. FMG convergence stopping test is re-visited. Applications to a few 2D continuous and discontinuous-coefficient elliptic model problems show the efficiency of this combination.