Problems in mathematical foundations of adaptive finite element methods

自适应有限元方法的数学基础问题

• 批准号: 1518925
• 负责人: Alan Demlow
• 金额: $ 16.68万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1518925&HistoricalAwards=false
• 关键词: Problems; mathematical; foundations; adaptive; finite

A posteriori error estimates and adaptive finite element methods (AFEM) are widely-used tools for solving partial differential equations (PDEs) arising in science and engineering applications. A posteriori estimates provide computable bounds on discretization errors, while AFEM are efficient solution techniques which accurately reflect solution properties via automatic local mesh grading. The goals of this project are to better understand the mathematical underpinnings of AFEM and to provide new a posteriori error estimates and adaptive algorithms in several specific application areas. A major part of the project is devoted to development and analysis of a posteriori error estimates and AFEM for PDEs on surfaces. Specific projects concern Eulerian formulations of parabolic PDEs on evolving surfaces, solution of elliptic PDEs on surfaces for which the only available information is a discrete approximation, and elliptic eigenvalue problems. Another emphasis is fine properties of FEM, in particular the development of a priori and a posteriori error estimates in nonstandard norms. The PI will develop new a priori error estimates in such norms on the types of highly graded meshes typically seen in practice, prove new a posteriori maximum-norm bounds for elliptic interface problems, and integrate similar error analysis into his study of surface eigenvalue problems. A wide variety of applications in science and engineering give rise to partial differential equations (PDEs) which must be solved in order to obtain accurate predictions about the physical world. PDEs are typically solved approximately on computers in modern applications, and there is a tradeoff between the quality of the approximate solution and the investment of computational resources. The PI will study the mathematical underpinnings of adaptive algorithms which automatically generate more accurate solutions while efficiently employing the computing power at hand. Part of the project is aimed at enriching mathematical understanding of existing algorithms, and part to developing new and mathematically well-justified adaptive algorithms for various applications.

后验错误估计和自适应有限元方法（AFEM）是广泛使用的工具，用于解决科学和工程应用中出现的部分微分方程（PDE）。 后验估计值在离散误差上提供了可计算的界限，而AFEM是有效的解决方案技术，可以通过自动局部网格分级准确地反映解决方案属性。 该项目的目标是更好地了解AFEM的数学基础，并在几个特定的​​应用领域提供新的后验错误估计和自适应算法。 该项目的主要部分致力于对表面上PDE的后验错误估计和AFEM的开发和分析。 特定的项目涉及欧拉（Eulerian）在不断发展的表面上对抛物线PDE的配方，椭圆形PDE在表面上的解决方案，唯一可用的信息是离散近似和椭圆特征值问题。 另一个重点是FEM的良好特性，尤其是在非标准规范中的先验发展和后验误差估计。 PI将在此类规范中开发新的先验误差估计，以实践中通常看到的高度分级网格的类型，证明新的椭圆界面问题的后验最大范围，并将相似的误差分析整合到他对表面特征值问题的研究中。为了获得有关物理世界的准确预测，必须解决科学和工程中的各种应用，引起了部分微分方程（PDE）。 PDE通常在现代应用程序中大约解决了计算机，并且在近似解决方案的质量与计算资源的投资之间存在一个权衡。 PI将研究自适应算法的数学基础，该算法会自动生成更准确的解决方案，同时有效地利用手头的计算能力。 该项目的一部分旨在丰富对现有算法的数学理解，部分是为各种应用开发新的和数学良好的自适应算法。