Topics in Finite Element Analysis

该项目的总体目标是设计和开发数值方法来解决工程和生物科学中的重要问题。特别是，三个重要的问题将在研究中进行调查。首先是提出有效的数值方法，可以以平衡的方式将一个区域一分为二。这个问题被称为Cheeger问题，PI将开发有效和准确地解决这个问题的数值方法。 第二个问题是分析求解一类椭圆界面问题的稳健数值方法。界面问题在流体流动模拟、流固耦合建模等方面有着广泛的应用。 最后，PI将分析新的有前途的数值方法的重要和经典的流体流动问题。对于第一个项目，问题归结为解决一个最小化问题的L^1范数。PI将采取的方法是考虑L^p范数下的最小化问题，并让p趋于1。简单地说，一个是对原始问题进行正则化。 L^p最小化问题将采取p-Laplacian特征值问题的形式。 这种方法的优点是，正则性的结果是可用于相应的方程。 在第二个问题中，PI将研究二阶椭圆界面问题的浸入边界有限元方法。该项目的目标是证明全通量近似的估计，以揭示该方法的收敛行为。 最后，PI将研究H（div）协调和不连续伽辽金方法使用迎风通量不可压缩欧拉方程在二维和三维。PI将证明数值方法的最优误差估计，以便它们可以为计算模拟提供可靠的指导。

The over-arching goal of this project is to design and develop numerical methods to solve important problems arising from engineering and biological sciences. In particular, three important problems will be investigated in the research. The first is to come up with efficient numerical methods that can split a region in two in a balanced way. This problem is known as Cheeger's problem, and the PI will develop numerical methods that do this efficiently and accurately. The second problem is to analyze robust numerical methods to solve a class of elliptic interface problems. Interface problems have a numerous applications in fluid flow simulation, fluid-structure interaction modeling. Finally, the PI will analyze new promising numerical methods for important and classical fluid flow problems.For the first project, the problem boils down to solving a minimization problem for the L^1 norm. The approach the PI will take is to consider the minimization problem in L^p norm and let p tend to 1. In simple terms, one is taking a regularization of the original problem. The L^p minimization problem will take the form of a p-Laplacian eigenvalue problem. The advantage of this approach is that regularity results are available for the corresponding equations. In the second problem, the PI will study immersed boundary finite element methods for second order elliptic interface problems. The goal of the project is to prove estimates for the full flux approximation in order to reveal the convergence behavior of the method. Finally, the PI will study H(div) conforming and discontinuous Galerkin methods using upwinded fluxes for incompressible Euler's equations in two and three dimensions. The PI will prove optimal error estimates for the numerical methods so that they can provide a reliable guidance for computational simulations.