Least-Squares Finite Element Methods and Optimization-Based Domain Decomposition Methods for Partial Differential Equations

偏微分方程的最小二乘有限元方法和基于优化的域分解方法

• 批准号: 9806358
• 负责人: Max Gunzburger
• 金额: $ 10.26万
• 依托单位: Iowa State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9806358&HistoricalAwards=false
• 关键词: Least; Squares; Finite; Element; Methods

9806358 Gunzburger In the last few years, the engineering and mathematical communities have shown increasing interest in least-squares finite element methods for solving a variety of problems in fluids, electromagnetics, elasticity, and other applications. The great promise of least-squares methods arises from the fact that, when compared to other discretization schemes, they lead to discrete problems that are much easier to solve on a computer. In the past the PI has studied numerous facets of least-squares finite element methods. These include: the use of mesh-dependent weights in least-squares functionals in order to achieve optimally accuracy, the solution of practical implementation issues that needed to be addressed in order to make these methods practical and competitive, and the application of these methods to problems with discontinuous coefficients that arise, e.g., from inhomogeneous media properties. The PI plans to apply least-squares finite element methodologies to optimization and control problems and to develop, analyze, and implement domain decomposition algorithms in the least-squares setting. Domain decomposition methods have attracted even more attention due to their usefulness in a parallel processing environment. The PI has developed novel non-overlapping domain decomposition methods based on optimization or optimal control ideas that posses numerous desirable features, the most important perhaps being that they are easily extended to nonlinear problems. The PI plans to introduce preconditioners to speed-up the performance of the methods, to look at different functionals and optimization parameters on which to base the decomposition into subdomains so that again more efficient algorithms are obtained, to apply and analyze algorithms to the solution of optimization problems for partial differential equations, to develop algorithms for time-dependent problems, and to implement algorithms on parallel computers consisting of clusters of Pentium processors.

小行星9806358 在过去的几年中，工程和数学界对最小二乘有限元法越来越感兴趣，用于解决流体，电磁学，弹性和其他应用中的各种问题。最小二乘方法的巨大希望来自于这样一个事实，即与其他离散化方案相比，它们导致的离散问题更容易在计算机上解决。在过去，PI研究了最小二乘有限元法的许多方面。其中包括：在最小二乘泛函中使用网格相关权重以实现最佳精度，解决需要解决的实际实现问题以使这些方法实用和具有竞争力，以及将这些方法应用于出现的具有不连续系数的问题，例如，不均匀的介质特性。PI计划将最小二乘有限元方法应用于优化和控制问题，并在最小二乘设置中开发，分析和实施区域分解算法。 区域分解方法由于其在并行处理环境中的有用性而引起了更多的关注。PI已经开发了基于优化或最优控制思想的新型非重叠区域分解方法，这些方法具有许多理想的功能，最重要的可能是它们很容易扩展到非线性问题。PI计划引入预处理器来加速方法的性能，查看不同的泛函和优化参数，以将其分解为子域，从而再次获得更有效的算法，应用和分析算法来解决偏微分方程的优化问题，开发时间相关问题的算法，并在由奔腾处理器集群组成的并行计算机上实现算法。