Error Estimates for a Finite Element Discretization of a Two‐Scale Phase Field Model
Christof Eck · Multiscale Modeling and Simulation · 2007
We present finite element error estimates for discretizations of a phase field model and a two‐scale model. Both models describe solidification with equiaxed dendritic microstructure. The two‐scale model is a homogenization limit of the phase field model for periodic initial conditions of scale $\varepsilon > 0$ 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 $\varepsilon^{5/4}$ in an asymptotic sense.