Topics of Immersed Finite Element Methods

接口问题无处不在。当模拟涉及多种材料或多物理时，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通过三维界面问题来建模的。浸入有限元法是一类在界面非拟合网格上求解界面问题的数值方法。本研究计画将探讨两个相关的问题。第一个问题是设计一种基于后验误差估计的自适应IFEM。第二个问题集中在三维IFEM的开发、实现和分析上。第一个问题涉及各种浸入式有限元离散的基于残差和基于恢复的误差估计的研究。这些方法包括在协调、非协调和不连续Galerkin框架中的浸入有限元近似。对IFEM的可靠性和有效性误差估计进行了严格的数学分析。第二个问题集中在三个空间维度的接口问题。它旨在开发一种创新的方法来有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方案中实现。理论上，对新的IFEM格式进行了先验和后验误差估计。在计算方面，将开发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.