Topics of Immersed Finite Element Methods

界面问题无处不在。当模拟涉及多种材料或多物理场时，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现实问题都是由三维界面问题建模的。浸入有限元法（IFEM）是一类求解界面非拟合网格界面问题的数值方法。本研究计划将调查两个相互关联的问题。第一个问题是设计一个基于后验误差估计的自适应IFEM。第二个问题侧重于三维ifm的发展、实施和分析。第一个问题是研究基于残差和基于恢复的各种浸入式有限元离散误差估计。其中包括符合、不符合和不连续伽辽金框架的浸入式有限元逼近。对有限元法的可靠性和效率误差估计进行了严格的数学分析。第二个问题是三维空间的界面问题。本文旨在开发一种创新的方法来高效地构建三维浸入式有限元函数。这些浸入式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ifm方案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细化功能的三维有限元软件包。

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.