Topics in Finite Element Analysis

这个项目的首要目标是设计和开发数值方法来解决工程和生物科学中出现的重要问题。具体而言，本研究将探讨三个重要问题。首先是提出有效的数值方法，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将开发数值方法来高效准确地解决这个问题。第二个问题是分析求解一类椭圆界面问题的鲁棒数值方法。界面问题在流体流动模拟、流固耦合建模中有着广泛的应用。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L^1范数的最小化问题。PI将采用的方法是考虑L^p范数中的最小化问题，并使p趋于1。简单来说，就是对原问题进行正则化。L^p最小化问题将采用p-拉普拉斯特征值问题的形式。这种方法的优点是可以得到相应方程的正则性结果。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性。最后，PI将研究二维和三维不可压缩欧拉方程使用逆风通量的H（div）符合和不连续伽辽金方法。PI将证明数值方法的最佳误差估计，从而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.