Adaptive Multigrid Methods for Phase Field Models

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Please 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. Driven by their practical relevance, considerable efforts have been invested into the anaylses of phase field models and their numerical discretizations. In particular, implicit schemas are available which, in contrast to explicit time discretizations, avoid any stability restrictions on the time step. However, efficinet and reliable solution of the resulting large-scale, non-linear and often non-smooth algebraic systems still seems to be in its infancy. 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 vectorvalued 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. Construction and convergence analysis of such methods will heavily rely on techniques from constrained optimization. In particular, restrictions, prolongations and smoothers will be associated with local minimization problems.