Error estimates for a finite element discretization of a two-scale phase field model

We present finite element error estimates for discretizations of a phase field model and a two-scale model. Both models describe solidi. cation with equiaxed dendritic microstructure. The two-scale model is a homogenization limit of the phase field model for periodic initial conditions of scale epsilon>0 under the assumption of a suitable scaling of physical parameters of the model. The scaling corresponds to a model with fast heat and slow solute diffusion, where the temperature is microscopically well mixed and the solute diffusion arises on the scale of single or few equiaxed crystals only. A comparison of the error estimates for both models is presented, showing that the two-scale model is superior if the mesh size for the macroscopic model is larger than epsilon(5/4) in an asymptotic sense.