Adaptive Multigrid Methods for Phase Field Models

场模型为相变自由边界问题的数学描述提供了一个良好的框架。漫反射界面由称为序参数或相场的函数的水平集表示，其值标识空间和时间中特定点处的相。序参数的演化用非线性抛物型微分方程组来描述，通过最小化适当的非凸总自由能得到。在实际应用的驱动下，相场模型的分析及其数值离散化已经投入了大量的努力。特别是，隐式模式是可用的，与显式时间离散化相比，它避免了对时间步长的任何稳定性限制。然而，对于由此产生的大规模、非线性且经常是非光滑的代数系统的有效和可靠的解似乎仍处于起步阶段。本项目致力于自适应多重网格法的构建、分析和实际应用，以有效和可靠地模拟相变和相分离。特别是，我们将集中于向量值Allen-Cahn方程、Cahn-Hilliard方程以及Cahn-Hilliard方程与线弹性的耦合。可能的应用包括微电子焊料中的扩散粗化，或者从角度来看，云的形成。我们将专注于构造健壮的多重网格算法。稳健性意味着收敛行为不仅对离散参数如网格大小或时间步长不敏感，而且对连续问题的相关参数也不敏感，如界面能量或温度。这种方法的构造和收敛分析将在很大程度上依赖于约束优化的技术。特别是，限制、延长和平滑将与局部最小化问题相关联。

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.