Recovery Type a Posteriori Estimation and Mesh Refinement for the Finite Element Method on Anisotropic Meshes

对于具有强各向异性特征的问题，只要网格对齐且保持合适的纵横比，基于各向异性网格的有限元方法可以比基于各向同性网格的有限元方法更有效。然而，对于各向异性网格的有限元分析还很少，大多数计算工作都局限于线性单元，基于启发式调整和特殊处理，本项目的目的是为各向异性网格上的有限元分析提供数值分析，并为自适应网格加密开发有效的控制方法。本项目的主要目标包括：(I)分析各向异性网格上有限元的各种恢复型误差估计器；(Ii)开发可靠的网格度量和网格质量措施以控制各向异性网格的加密过程；(Iii)开发用于有限元解的后处理的软件工具箱，包括误差估计、网格度量构造和网格质量评估。(4)为科学与工程专业的研究生提供自适应有限元计算方面的研究培训。误差估计和自适应网格加密是实际有限元模拟中的两个主要组成部分。前者评估解决方案的准确性，并在交付时提供质量保证。后者是在未达到所需精度时改进解决方案的不可或缺的工具。对于各向异性有限元，允许单元长而窄，以更有效地适应下划线问题。它们可能在大小、方向和纵横比上有所不同。这个项目的目的是为了更好地了解各向异性有限元的行为，并开发更有效的有限元模拟技术。由于恢复型误差估计技术简单、稳健，是工程上最常用的方法，因此它将是本研究的重点。该项目的成功完成将为航空航天工程、材料加工和生物医学研究中的实际有限元建模和仿真提供严格的数学理论和更有效的算法。

For problems exhibiting strong anisotropic features the finite element method (FEM) based on anisotropic meshes can be much more efficient than the one based on isotropic meshes, provided the meshes are properly aligned and the right aspect ratios are maintained. However, there has been little rigorous analysis about it, and most computational work has been restricted to linear elements, based on heuristic justification and ad-hoc treatments.The aim of this project is to provide the numerical analysis for the FEM on anisotropic meshes and develop effective controls for the adaptivemesh refinement. The main objectives of this project include:(i) Analyze various recovery type error estimators for the FEM on anisotropic meshes; (ii) Develop reliable mesh metrics and mesh quality measures to control the anisotropic mesh refinement process; (iii) Develop a software tool box for the post-processing of thefinite element solutions, including error estimation, mesh metrics construction, and mesh quality evaluation. (iv) Provide research training to graduate students on adaptive finite element computation in science and engineering.Error estimation and adaptive mesh refinement are two major components in practical finite element simulations. The former assesses the accuracy of the solution and provides the quality assurance when it is delivered. The latter is an indispensable tool for improving the solution when the desired accuracy has not been reached. For anisotropic FEM, the elements are allowed to be long and narrow to fit the underlining problems moreeffectively. They may vary in sizes, orientations, and aspect ratios. This project is aimed at providing better understanding of the behavior of anisotropic FEM and developing more efficient techniques for the FE simulations. Since the recovery type error estimation techniques are simple, extraordinary robust, and most commonly used in engineering, it will be the main focus of this research. Successful completion of this project will provide rigorous mathematical theory and more efficient algorithms for practical FE modeling and simulations in aerospace engineering, material processing, and biomedical researches.