Topics in the analysis of finite elements

将考虑三个研究有限元方法行为的截然不同的项目。第一个项目是研究浸没边界有限元方法中浸没边界的污染效应。将给出一个尖锐的误差分析，以衡量一个人必须离浸没边界有多远才能获得最优收敛。第二个项目将涉及自适应不连续Galerkin(DG)方法。证明了弱惩罚DG方法的压缩性质。最后对求解Stokes问题的inf-sup稳定有限元方法进行了极大范数稳定性分析。对于三维最低阶泰勒-胡德单元，将构造一个指数衰减的Fortin投影。指数衰减投影将是证明极大范数稳定性估计的重要工具。有限元被广泛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，这些结果为他们的可靠性提供了一定的保证。P.I.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.将从数学上研究浸没边界有限元的行为，这是一种特别适合于流固相互作用的方法。例如，这些方法被用来模拟血液流动和动物运动，仅举几例。这项研究的结果将为用户提供关于在哪里投入更多计算精力的理论指导，这反过来将使他们的模拟对于重要的应用更加准确。

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.