Topics in the analysis of finite elements

有限元分析主题

• 批准号: 1318108
• 负责人: Johnny Guzman
• 金额: $ 21万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1318108&HistoricalAwards=false
• 关键词: Topics; analysis; finite; elements; Topics; analysis; finite; elements

Three distinct projects that study the behavior of finite element methods (FEM) will be considered. The first project is studying the pollution effects of immersed boundaries in the immersed boundary finite element method. A sharp error analysis will be given that measures how far one has to be from the immersed boundary to obtain optimal convergence. The second project will involve adaptive Discontinuous Galerkin (DG) methods. Contraction properties of weakly penalized DG methods will be proved. The final project is max-norm stability analysis of inf-sup stable finite element methods for the Stokes problem. A Fortin projection that is exponentially decaying will be constructed for the lowest-order Taylor-Hood element in three dimensions. Exponentially decaying projections will be an important tool to prove max-norm stability estimates. FEM are widely used to simulate a variety of problems in engineering and science. Users of these methods rely on theoretical results that give them some guarantee of their reliability. The P.I. will use mathematical analysis to describe the behavior of FEM for three important FE methods. In particular, the P.I. will mathematically study the behavior of the immeresed boundary FEM which is a method especially suited for fluid-solid interactions. For example, these methods have been used to simulate blood flow and animal locomotion, to name a few. The results of this investigation will give users theoretical guidance on where to put more computational effort which in turn will make their simulations more accurate for imporant applications.

将考虑研究研究有限元方法行为（FEM）的三个不同的项目。第一个项目是研究浸入边界有限元方法中浸入边界的污染效应。将给出急剧的误差分析，该分析衡量了一个从浸入边界必须有多远才能获得最佳收敛。第二个项目将涉及自适应不连续的Galerkin（DG）方法。将证明弱惩罚DG方法的收缩特性。最终项目是针对Stokes问题的Inf-Sup稳定有限元方法的最大总体稳定性分析。在三个维度上，将为最低级的泰勒矿元素构建一个指数衰减的堡垒投影。指数衰减的预测将是证明最大值稳定性估计值的重要工具。 FEM被广泛用于模拟工程和科学方面的各种问题。这些方法的用户依赖于理论结果，可以保证其可靠性。 P.I.将使用数学分析描述三种重要的FE方法的FEM的行为。特别是P.I.数学上会研究沉浸式边界FEM的行为，这是一种特别适合流体固定相互作用的方法。例如，这些方法已用于模拟血流和动物的运动，仅举几例。这项调查的结果将为用户提供有关在何处进行更多计算工作的理论指导，从而使其模拟更加准确，以实现无罪雄厚的应用程序。