Topics in Finite Element Analysis

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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.