Adaptive Finite Element Simulation of Relaxed Models for Liquid-Solid Phase Transition

The purpose of this work is to develop time-space adaptive techniques for phase field models for liquid-solid phase transition. These models are based on coupled systems of the Cahn-Hilliard and the Allen-Cahn equation. The adaptive techniques under consideration allow for mesh adaptation with respect to a functional of interest. They are based on a discontinuous Galerkin approach for time stepping and the exploitation of a parabolic duality argument. For the Bi-Laplace operator arising in linearization of the Cahn-Hilliard equation, a non-conforming Galerkin method is employed.