A Posteriori Error Analysis and Adaptive Algorithms for Variational Inequalities

变分不等式的后验误差分析和自适应算法

• 批准号: 0106781
• 负责人: Weimin Han
• 金额: $ 13.7万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0106781&HistoricalAwards=false
• 关键词: Posteriori; Error; Analysis; Adaptive; Algorithms

The objective of this project is to develop a systematic theory on a posteriori error estimation and adaptive algorithms for finite element solutions of variational inequalities. A posteriori error estimates play an essential role in assessing the reliability of numerical solutions and in developing efficient adaptive algorithms. Although several standard techniques have been developed to derive and analyze a posteriori error estimates for finite element solutions of differential equation problems, they do not work directly for a posteriori error analysis of numerical solutions of variational inequalities. In the proposed research, the duality theory of nonlinear analysis will be employed as a basic mathematical tool for the development of a posteriori error estimates of numerical solutions of variational inequalities.Variational inequalities form an important family of nonlinear problems arising in a wide range of applications. A partial list of the applications where variational inequalities arise include contact mechanics, plasticity, non-Newtonian fluid flows, metal forming, metal extrusion, financial mathematics. A systematic approach will be developed to derive a posteriori error estimates and to form and test efficient, reliable adaptive solution algorithms for finite element approximations of variational inequalities. The research results will have a direct impact on many areas of science and technology.

本项目的目标是发展一套系统的后验误差估计理论和变分不等式有限元解的自适应算法。后验误差估计在评估数值解的可靠性和开发有效的自适应算法方面起着至关重要的作用。虽然已经发展了几种标准技术来推导和分析微分方程有限元解的后验误差估计，但它们并不直接用于变分不等式数值解的后验误差分析。在所提出的研究中，非线性分析的对偶理论将被用作发展变分不等式数值解的后验误差估计的基本数学工具。变分不等式是一类重要的非线性问题，具有广泛的应用。出现变分不等的部分应用包括接触力学、塑性、非牛顿流体流动、金属成形、金属挤压、金融数学。将开发一种系统的方法来推导后验误差估计，并形成和测试用于变分不等式有限元逼近的有效、可靠的自适应求解算法。研究成果将对许多科学技术领域产生直接影响。