Unfitted Finite Element Methods for Partial Differential Equations on Evolving Surfaces and Coupled Surface-Bulk Problems

自然界和工程应用中的许多重要过程都发生在表面上。例如，在帮助清理漏油时，细胞膜中的脂类和蛋白质分子相互作用，或化学分散剂在油水界面上的作用。这些过程和其他过程是借助曲面上提出的微分方程进行数学描述的。求解这些方程的数值方法为科学家提供了通过进行电子实验来更好地理解表面现象的工具。本课题旨在发展高精度、高可靠性的数值方法，以造福于学术界和工程界。本研究项目发展了求解演化曲面上偏微分方程组的新的高阶不适应有限元方法。所开发的数值方法使用与时间无关的背景网格。在PDE的弱形式中，该方法在重建的演化物理域上使用标准有限元空间的踪迹。所得到的方法将是最精确的，处理隐式定义的曲面，允许曲面经历拓扑变化，并扩展到表面-整体耦合问题。通过构造高阶时空不适应有限元方法并进行完全收敛分析，将该方法和分析扩展到表面-体积输运-扩散耦合问题，发展了一种新的混合方法来求解演化曲面上的偏微分方程组，并将新的方法扩展到流形上的流体方程。

Many important processes in nature and in engineering applications take place along surfaces. Examples include interaction of lipid and protein molecules in cell membranes or the action of a chemical dispersant on the oil-water interface, while helping to clean-up an oil spill. These and other processes are described mathematically with the help of differential equations posed on surfaces. Numerical methods for solving these equations provide tools for scientists to understand better surface phenomena by doing in silico experiments. The present project aims to develop such accurate and reliable numerical methods for the benefit of academic and engineering community.This research project develops new higher order unfitted finite element methods for partial differential equations (PDEs) posed on evolving surfaces. The developed numerical approach uses time-independent background meshes. In a weak formulation of a PDE, the method employs traces of standard finite element spaces on reconstructed evolving physical domains. The resulting methods will be optimally accurate, handle implicitly defined surfaces, allow a surface to undergo topological changes, and extend to surface-bulk coupled problems. The project goals will be met by constructing a higher order space-time unfitted finite element method and performing a full convergence analysis, extending the method and analysis to coupled surface-bulk transport-diffusion problems, developing a new hybrid method for PDEs on evolving surfaces that uses finite difference approximations of time derivatives, extending the new approach to fluid equations posed on manifolds.