Theory and Algorithm of Adaptive Methods for Numerical Methods

这项建议是关于偏微分方程(PDE)的高级数值方法的研究，这些方法来自于科学和工程应用。本文的研究主题是多层自适应有限元方法的发展、应用和分析。与计算网格的均匀加密相比，网格自适应自适应有限元方法更倾向于局部增加感兴趣区域的网格密度，从而节省计算机资源。网格自适应策略可以分为两类：H-方法和R-方法。PI建议在这两种方法中研究几个新的想法，并将它们结合起来，以开发一种更高效、更完整、更灵活的方法来处理大类偏微分方程。更准确地说，对于r-方法，PI利用最优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偏微分方程组的数值求解。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一种新的二等分网格分解的细化和粗化算法。此外，这两种方法将自然地结合在一起，从而产生更多层次的网格自适应策略，其中h方法将主要用作局部光滑器，而粗网格将利用从细网格到服务器的信息移动作为粗网格校正。PI希望为自适应和多层求解的结合使用发展更完整的理论基础和现代技术。本文开发和研究的多层自适应方法有望对一大类实际问题的数值解产生更广泛的影响。这项工作的特殊目标应用是对流占优的问题和斑图形成的数值模拟。对流为主的对流扩散问题对于实际应用中的几个流动问题特别重要，例如汽车工业(内燃机中的流动)、电镀工业(电化学反应流动与电极边界处的传质)、航空航天(高雷诺数流动)等。图案的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规模结构。通过开发改进的多层数值技术来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准确的解，该项目将为探索物理和生物模型提供强大的工具。此外，全面参与本科生和研究生的计算数学教育是该项目的一个组成部分。通过开发一个MatLab软件包(IFEM)，PI将能够设计一门新的面向项目的多层自适应有限元方法课程。

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.