Higher order Unfitted Finite Element Methods for moving domain problems

在物理学、生物学、化学和工程学中，仿真科学的许多应用都涉及复杂和不断发展的几何形状。在许多重要的问题中，这些几何形状表现出拓扑变化或强烈的变形，这使得数值处理非常具有挑战性。高精度和高效率的数值解的偏微分方程在这些不断发展的领域是一个具有挑战性的任务。在过去的十年里，关于这个问题的研究已经开始。特别是使用与计算网格分离的几何描述的方法，与传统的一致网格描述相比，该方法提供了更灵活的几何处理，近年来已经变得非常流行。虽然取得了很大的进展，但仍有一些重要问题需要进一步研究。特别是数值方法，这是灵活的几何配置，鲁棒性和高阶精度在同一时间失踪。这个项目的目标就是填补这个空白。在本项目中，将开发和分析适合于与单独的几何描述集成并具有可证明的高阶误差界的有限元方法。这涉及到几个问题。首先，需要几何的高阶表示。为此，通常使用带有指示函数的隐式描述。虽然几何描述通常工作良好，但描述的实际可用性带来了问题。对于合适的有限元公式积分的细胞被切断是必需的。由于这些域仅被隐式地描述，因此精确的数值积分是困难的。虽然对于二阶精度的鲁棒和建立的方法存在，扩展到更高阶的精度需要新的方法在离散化。另一个重要的问题是几何演化的处理。与隐式几何描述的几何形状的演变，可以处理，而无需调整计算网格，但新的挑战出现相对于时间离散化的问题上不断发展的领域。在本项目中，将开发解决这些问题的离散化方法。相关研究的一个重要组成部分是严格的误差分析，提供最佳阶误差界的离散化方法。项目中考虑的方法应基于开源C++软件工具箱DUNE进行实施和数值研究。

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.