Phase field models provide a well-established framework for the mathematical description of free boundary problems for phase transitions. The diffuse interface is represented by the level sets of a function, called order parameter or phase field, whose value identifies the phases at particular points in space and time. The evolution of the order parameter is described by non-linear parabolic differential equations as obtained by minimizing a suitable, non-convex total free energy.
This project is devoted to the construction, analysis and practical application of adaptive multigrid methods for the efficient and reliable simulation of phase transition and phase separation. In particular, we will concentrate on the vector-valued Allen-Cahn equation, on the Cahn-Hilliard equation and on the coupling of Cahn-Hilliard equations with linear elasticity. Possible applications include the diffusional coarsening in microelectronic solders or, perspectively, the formation of clouds. We will focus on the construction of robust multigrid algorithms. Robustness means that convergence behavior should be insensitive not only with respect to discretization parameters such as mesh size or time step, but also with respect to relevant parameters of the continuous problem, such as the amount of interfacial energy or temperature.