Theory and Algorithm of Adaptive Methods for Numerical Methods

数值方法自适应方法理论与算法

• 批准号: 0811272
• 负责人: Long Chen
• 金额: $ 15万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0811272&HistoricalAwards=false
• 关键词: Theory; Algorithm; Adaptive; Methods; Numerical

This proposal is on the study of advanced numerical methods for partial differential equations (PDEs) that arise from scientific and engineering applications. The theme of research is on the development, application and analysis of multilevel adaptive finite element methods. Comparing with the uniform refinement of the computational grid, adaptive finite element methods through mesh adaptation are more preferred to locally increase mesh densities in the regions of interest, thus saving the computer resources. The strategies of mesh adaptation can fall into two categories: h-method and r-method. The PI proposes to study several novel ideas in both methods and combine them to develop a more efficient, integrated, and flexible method for a large class of PDEs. More precisely, for r-method, the PI proposes a new energy using the concept of Optimal Delaunay Triangulation (ODT) and will develop related fast optimization methods and apply to the numerical solution of PDEs. For h-method, the PI will design and analyze multigrid methods, gradient recovery schemes, and refinement and coarsening algorithms based on a novel decomposition of bisection grids. Furthermore, these two methods will be naturally incorporated to result a more multilevel mesh adaptation strategy, in which h-method will be mainly used as a local smoother while the coarse mesh will be moved using the information from fine grids to severs as a coarse grid correction. The PI hopes to develop a more complete theoretical foundation and modern techniques for the combined use of adaptivity and multilevel solvers.The multilevel adaptive methods developed and studied in this work are expected to have a broader impact on the numerical solutions of a large class of practical problems. Special target applications for this work are the convection-dominated problems and numerical simulation of pattern formation. The convection-dominated convection diffusion problems are particularly important to several flow problems in the real applications, for example, automotive industry (flow in combustion engines), plating industry (electro-chemically reacting flows with mass transfer at the electrode boundaries), and aerospace (high Reynolds number flow) among many others. Pattern formation occurs in diverse physical, chemical, and biological systems, from Drosophila embryo to the large-scale structure of the universe. By developing improved multilevel numerical techniques to reduce the computer time required to solve the underlying equations, and at the same time producing more accurate solutions through the use of adaptive finite element methods, this project will provide powerful tools for the exploration of models in physics and biology. In addition, a fully integrated involvement in undergraduate and graduate computational mathematics education is an integral part of the project. By developing a MATLAB package (iFEM), the PI will be able to design a new project-oriented course on multilevel adaptive finite element methods.

该提案是关于科学和工程应用中产生的偏微分方程（PDE）的高级数值方法的研究。研究的主题是多层自适应有限元方法的发展，应用和分析。与均匀加密计算网格相比，通过网格自适应的自适应有限元方法更有利于局部增加感兴趣区域的网格密度，从而节省计算机资源。网格自适应的策略可以分为两类：h方法和r方法。PI建议研究这两种方法中的几个新想法，并将它们联合收割机结合起来，为一大类偏微分方程开发一种更有效、更集成、更灵活的方法。更确切地说，对于r-方法，PI使用最优Delaunay三角剖分（ODT）的概念提出了一种新的能量，并将开发相关的快速优化方法并应用于PDE的数值求解。对于h-方法，PI将设计和分析多重网格方法，梯度恢复方案，以及基于对分网格的新分解的细化和粗化算法。此外，这两种方法将自然地结合起来，从而产生一个更多级的网格自适应策略，其中h-方法将主要用作局部平滑，而粗网格将使用从细网格到服务器的信息作为粗网格校正。PI希望开发一个更完整的理论基础和现代技术相结合的自适应性和多级solvers.The多级自适应方法的开发和研究在这项工作中，预计将有更广泛的影响，一个大类的实际问题的数值解。这项工作的特殊目标应用是对流为主的问题和模式形成的数值模拟。对流主导的对流扩散问题对于真实的应用中的一些流动问题特别重要，例如，汽车工业（内燃机中的流动）、电镀工业（在电极边界处具有质量传递的电化学反应流动）和航空航天（高雷诺数流动）等。模式形成发生在不同的物理、化学和生物系统中，从果蝇胚胎到宇宙的大尺度结构。通过开发改进的多级数值技术，以减少求解基本方程所需的计算机时间，同时通过使用自适应有限元方法产生更精确的解，该项目将为物理和生物模型的探索提供强大的工具。 此外，在本科生和研究生计算数学教育的完全集成的参与是该项目的一个组成部分。通过开发MATLAB软件包（iFEM），PI将能够设计一门关于多级自适应有限元方法的新项目导向课程。