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

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.

将考虑三个研究有限元法（FEM）行为的不同项目。第一个项目是研究浸入边界有限元法中浸入边界的污染效应。一个尖锐的误差分析将给出措施多远，从浸入边界，以获得最佳收敛。第二个项目将涉及自适应不连续Galerkin（DG）方法。弱惩罚DG方法的收缩性质将被证明。最后一个项目是Stokes问题的下超稳定有限元方法的极大范数稳定性分析。对于三维空间中的最低阶泰勒-胡德元，将构造一个指数衰减的福廷投影。指数衰减投影将是证明最大范数稳定性估计的重要工具。 有限元法被广泛用于模拟工程和科学中的各种问题。这些方法的用户依赖于理论结果，这些理论结果为他们的可靠性提供了一定的保证。私家侦探将使用数学分析来描述三种重要的有限元方法的有限元行为。特别是，P.I.将在数学上研究不固定边界有限元法的行为，这是一种特别适合于流固相互作用的方法。例如，这些方法已被用于模拟血流和动物运动，仅举几例。这项调查的结果将为用户提供理论指导，在哪里投入更多的计算工作，这反过来又将使他们的模拟更准确的重要应用。