Higher order Unfitted Finite Element Methods for moving domain problems

在物理、生物、化学和工程中，模拟科学的许多应用涉及复杂和不断变化的几何形状。在许多重要问题中，这些几何形状表现出拓扑变化或强烈变形，这给数值处理带来了极大的挑战。高精度、高效率地求解这些演化区域上的偏微分方程组是一项具有挑战性的任务。在过去的十年里，关于这一主题的研究已经开始。尤其是使用与计算网格分离的几何描述的方法，与传统的协调网格描述相比，提供了更灵活的几何处理。虽然已经取得了很大的进展，但仍有一些重要的问题需要进一步研究。尤其是几何构型灵活、同时具有较强的鲁棒性和高精度的数值方法更是缺乏。这个项目的目标就是填补这一空白。在这个项目中，将开发和分析适合于具有分离的几何描述和具有可证明的高阶误差界的积分的有限元方法。这就带来了几个问题。首先，几何图形的高阶表示是必要的。为此，通常使用带有指示器功能的隐含描述。虽然几何描述通常工作得很好，但描述的实际可用性带来了一个问题。对于合适的有限元公式，需要对被切割的单元进行积分。由于这些区域只是隐式描述的事实，精确的数值积分是困难的。虽然对于二阶精度，已有稳健和已有的方法，但扩展到高阶精度需要在离散化方面采用新的方法。另一个重要的问题是几何演化的处理。利用隐式几何描述，可以在不调整计算网格的情况下处理几何的演化，但对于演化域上的问题，时间离散化出现了新的挑战。在这个项目中，将开发解决这些问题的离散化方法。相关研究的一个重要部分是严格的误差分析，它为离散化方法提供了最优阶误差界。项目中考虑的方法将基于开源C++软件工具箱沙丘进行数值实现和研究。

In physics, biology, chemistry and engineering many applications of simulation science involve complex and evolving geometrical shapes. In many important problems these geometrical shapes exhibit topology changes or strong deformation which makes the numerical treatment very challenging. The highly accurate and efficient numerical solution of PDEs on these evolving domains is a challenging task. In the past decade research on this topic has been started. Especially methods using a geometry description which is separated from the computational mesh and in turn provide a more flexible handling of the geometry compared to traditional conforming mesh descriptions have become very popular in recent years. Although big progress has been made, there are important problems which require further research. Especially numerical methods which are flexible with respect to the geometrical configuration, robust and high order accurate at the same time are missing. The goal of this project is to fill this gap. Within this project finite element methods which are suitable for an integration with a separated geometry description and have provable higher order error bounds will be developed and analysed. This entails several issues. First, a higher order representation of the geometry is necessary. For that most often an implicit description with an indicator function is used. While the geometry description typically works well, the practical usability of the description poses a problem. For suitable finite element formulations integrals on cells which are cut are required. Due to the fact that these domains are only described implicitly, an accurate numerical integration is difficult. While for a second order accuracy robust and established methods exist, the extension to higher order accuracy requires new approaches in the discretization. Another important problem consists of the handling of the geometry evolution. With an implicit geometry description the evolution of the geometry can be handled without adapting the computational mesh, but new challenges arise with respect to the time discretization for problems on evolving domains. Within this project discretization methods will be developed which solve these problems. An important part of the related research consists of a rigorous error analysis which provides optimal order error bounds for the discretization methods. The methods considered in the project shall be implemented and investigated numerically based on the open source C++ software toolbox DUNE.