Mass-conservative coupling of bulk and surface processes on implicit, time-dependent domains

本项目提案的目的是开发一种数值方法，用于求解与封闭体上偏微分方程耦合的不断变化的（即随时间变化的）表面上的偏微分方程。在许多应用中会出现体-面耦合问题;例如在流体动力学和生物学应用中。该项目对数值方法的发展，这些方法的分析，其有效实施和验证做出了贡献。在此过程中，主要的挑战是开发一个局部质量的表面偏微分方程的方案，保守，同时允许可能表现出各向异性性质的强变形，甚至拓扑变化。我们将使用一个隐式的表面描述，它允许一个简单的处理几何形状的变化。这种表示也非常适合合并图像数据，如CT或显微镜数据。

The aim of this project proposal is the development of a numerical method for solving partial differential equations (PDEs) on evolving, i.e. time-dependent, surfaces which are coupled to PDEs on the enclosed bulk. Coupled bulk-surface problems arise in many applications; for example in fluid dynamics and biological applications.The project yields contributions to the development of numerical methods, the analysis of these methods, their efficient implementation and their validation.In the course of this, the central challenge is to develop a scheme for surface PDEs which is locally mass-conservative and at the same time allows for strong deformations that might exhibit an anisotropic nature and even topology changes. We will use an implicit description of the surface which allows for a simple handling of geometry changes. This representation is also very well suited to incorporate image data, like CT or microscopy data.