Topics in the analysis of finite elements

研究有限元方法（FEM）行为的三个不同的项目将被考虑。第一个项目是在浸入边界有限元法中研究浸入边界的污染效应。一个尖锐的误差分析将给出测量距离浸入边界有多远才能获得最优收敛。第二个项目将涉及自适应间断伽辽金（DG）方法。证明了弱惩罚DG方法的收缩性质。最后的课题是Stokes问题的中支撑稳定有限元法的最大范数稳定性分析。对于三维的最低阶泰勒胡德元素，将构造一个指数衰减的Fortin投影。指数衰减预测将是证明最大范数稳定性估计的重要工具。有限元法被广泛应用于模拟工程和科学中的各种问题。这些方法的使用者依赖于理论结果，这些理论结果给了他们一定的可靠性保证。P.I.将用数学分析来描述三种重要有限元方法的有限元行为。特别地，pi将从数学上研究浸没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液流动和动物运动，仅举几例。这项调查的结果将为用户提供理论指导，告诉他们在哪里投入更多的计算努力，从而使他们的模拟更准确地用于重要的应用。

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.