Topics of Immersed Finite Element Methods

界面问题无处不在。当模拟涉及多个材料或多个物理时，就会出现界面问题。流体力学、材料科学、机械工程和生物医学工程中的许多实际问题都是由三维界面问题来模拟的。浸没有限元方法(IFEM)是一类求解界面不拟合网格上界面问题的数值方法。在本研究项目中，将研究两个相互关联的问题。第一个问题旨在设计一种基于后验误差估计的自适应IFEM。第二个问题是关于三维IFEM的发展、实现和分析。第一个问题涉及各种浸没有限元离散的基于残差和基于恢复的误差估计的研究。其中包括协调、非协调和间断Galerkin框架中的浸没有限元近似。将对IFEM的可靠性和效率误差估计进行严格的数学分析。第二个问题集中在三维空间的界面问题上。它的目的是开发一种创新的方法来有效地构造三维浸没有限元函数。这些沉浸的有限元函数将在三维界面问题的各种数值格式中实现。理论上，将对新的IFEM格式进行先验和后验误差估计。在计算上，将开发一个具有自适应网格加密功能的三维IFEM软件包。

Interface problems are ubiquitous. When simulations involve multiple materials or multi-physics, interface problems arise. Many real-world problems in fluid mechanics, material science, mechanical engineering, and biomedical engineering are modeled by three-dimensional interface problems. The immersed finite element methods (IFEM) are a class of numerical methods for solving interface problems on interface-unfitted meshes. Two interrelated problems will be investigated in this research project. The first problem aims to design a self-adaptive IFEM based on the a posteriori error estimation. The second problem focuses on development, implementation, and analysis of three-dimensional IFEM.The first problem concerns the study of both residual-based and recovery-based error estimation for various immersed finite element discretizations. These include the immersed finite element approximation in conforming, nonconforming and discontinuous Galerkin frameworks. Rigorous mathematical analysis will be carried out for the reliability and efficiency error estimates of IFEM. The second problem focuses on interface problems of three spatial dimensions. It aims to develop an innovative approach to efficiently construct the three-dimensional immersed finite element functions. These immersed finite element functions will be implemented in various numerical schemes for three-dimensional interface problems. Theoretically, both a priori and a posteriori error estimates will be conducted for new IFEM schemes. Computationally, a three-dimensional IFEM software package will be developed with the feature of adaptive mesh refinement.