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.

该建议是在研究由科学和工程应用产生的部分微分方程（PDE）的高级数值方法的研究。研究的主题是关于多级自适应有限元方法的开发，应用和分析。与计算网格的均匀完善相比，通过网格适应性的自适应有限元方法更喜欢在局部增加感兴趣区域的网格密度，从而节省了计算机资源。网状改编的策略可以分为两类：H-Method和R-Method。 PI建议在方法中研究几种新思想，并将它们结合起来，以开发一种更有效，集成和灵活的方法为大型PDES开发。更确切地说，对于R方法，PI使用最佳Delaunay三角剖分（ODT）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PDES的数值解决方案。对于H方法，PI将基于新的分解网格分解，设计和分析多族方法，梯度恢复方案以及完善和粗化算法。此外，这两种方法将自然合并，以产生更多级的网格适应策略，其中H方法将主要用作局部更平滑，而粗网格将使用从细网格到SEVERS作为粗网格校正的信息移动。 PI希望为适应性和多层次求解器的联合使用开发更完整的理论基础和现代技术。在这项工作中开发和研究的多层次自适应方法有望对大量实用问题的数字解决方案产生更大的影响。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。以对流为主导的对流扩散问题对于实际应用中的几个流量问题，例如汽车行业（燃烧引擎中的流动），电镀工业（电镀工业的流动）（电力变化在电极边界处具有质量传递的反应流），以及航空航天（高雷诺数流量）。模式形成发生在从果蝇胚胎到宇宙大规模结构的各种物理，化学和生物系统中发生的。通过开发改进的多级数值技术来减少求解基础方程所需的计算机时间，同时通过使用自适应有限元方法生成更准确的解决方案，该项目将为探索物理和生物学模型的探索提供强大的工具。 此外，全面综合参与了本科和研究生计算数学教育是该项目不可或缺的一部分。通过开发MATLAB软件包（IFEM），PI将能够在多级自适应有限元方法上设计一个新的面向项目的课程。