Adaptive Finite Element Method for Interface Problems

界面问题的自适应有限元方法

• 批准号: 1115097
• 负责人: Xiu Ye
• 金额: $ 23.11万
• 依托单位: University of Arkansas Little Rock
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115097&HistoricalAwards=false
• 关键词: Adaptive; Finite; Element; Method; Interface

The objective of the project is to develop and to analyze a robust adaptive finite element method for the Darcy-Stokes-Brinkman model with extremely variable or discontinuous parameters. The proposed research include: developing a robust adaptive H(div) finite element method; conducting a priori error analysis; deriving efficiency and reliability bounds for a posteriori error estimators; proving convergence of adaptive mesh refinement procedures; developing a computer program for implementation of the method.The Darcy-Stokes-Brinkman model has wide range of applications such as surface and sub-surface water interaction in geoscience, blood circulation in health science and, fuel cells, filtration problems in environment science. Solving these problems has attracted a lot of attention from mathematicians and engineers, and many works have been done in developing and analyzing numerical algorithms for Darcy-Stokes-Brinkman model. Majority of these works treat viscosity and permeability as either constant or near zero jump. However, for the practical relevant problems, these physical parameters are either discontinuous or highly variable. Singularity caused by the highly varied physical parameters creates tremendous difficulty in development of numerical algorithms. There is a great interest in obtaining efficient and robust numerical methods to simulate these real word problems.

该项目的目标是开发和分析一个强大的自适应有限元方法的达西-斯托克斯-布林克曼模型的极端可变或不连续的参数。拟议的研究包括：发展一种稳健的自适应H（div）有限元方法;进行先验误差分析;推导后验误差估计的效率和可靠性界限;证明自适应网格细化过程的收敛性; Darcy-Stokes-Brinkman模型具有广泛的应用，如在地球科学中的地表和次地表水相互作用，健康科学中的血液循环，以及环境科学中的燃料电池和过滤问题。这些问题的求解引起了数学家和工程师们的广泛关注，人们在Darcy-Stokes-Brinkman模型的数值算法研究方面做了大量的工作。大多数这些作品处理粘度和渗透率为恒定或接近零的跳跃。然而，对于实际相关问题，这些物理参数要么是不连续的，要么是高度可变的。由物理参数高度变化引起的奇异性给数值算法的发展带来了巨大的困难。有很大的兴趣，获得有效的和强大的数值方法来模拟这些真实的字的问题。