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来求解关于演化的偏微分方程组(PDE)，即依赖于时间的曲面，该曲面耦合到封闭散体上的偏微分方程组。体-表面耦合问题在流体力学和生物应用中有着广泛的应用。该项目对数值方法的发展、这些方法的分析、它们的有效实现和它们的验证做出了贡献。在这一过程中，中心挑战是开发一个局部质量守恒的表面偏微分方程组的格式，同时允许可能表现出各向异性甚至拓扑变化的强烈变形。我们将使用曲面的隐式描述，它允许简单地处理几何图形更改。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.