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

演化曲面偏微分方程和耦合面体问题的不拟合有限元方法

基本信息

批准号：
1717516
负责人：
Maxim Olshanskiy
金额：
$ 15.45万
依托单位：
University of Houston
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1717516&HistoricalAwards=false
关键词：
Unfitted Finite Element Methods Partial

项目摘要

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.

自然界和工程应用中的许多重要过程都是沿着表面进行的。例如，细胞膜中脂质和蛋白质分子的相互作用，或化学分散剂在油水界面上的作用，同时帮助清理溢油。这些过程和其他过程都是借助曲面上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们通过进行硅实验来更好地理解表面现象提供了工具。本项目旨在发展这种准确可靠的数值方法，以供学术界和工程界使用。本研究计划发展一种新的高阶非拟合有限元方法来求解演化曲面上的偏微分方程（PDEs）。所开发的数值方法使用与时间无关的背景网格。在PDE的弱公式中，该方法在重构的演化物理域上使用标准有限元空间的轨迹。由此产生的方法将达到最佳精度，处理隐式定义的表面，允许表面经历拓扑变化，并扩展到表面-体耦合问题。为了实现项目目标，将构建高阶时空非拟合有限元方法并进行全收敛分析，将该方法和分析扩展到耦合的表面-体输运-扩散问题，开发一种新的混合方法来求解演化表面上的偏微分方程，使用时间导数的有限差分近似，将新方法扩展到流形流体方程。