Local and Direct discontinuous Galerkin methods: New algorithms and applications

该项目的目标是设计、分析和实现新的间断Galerkin（DG）有限元方法，解决物理和工程中产生的偏微分方程。DG方法是一种高精度的数值方法，具有处理复杂几何形状的优势，并在应用中采用h-p自适应策略。PI将专注于开发两种不连续Galerkin方法。1)局部间断Galerkin方法：提出一种直接求解Hamilton-Jacobi方程的DG方法。由于Hamilton-Jacobi方程是一个非线性偏微分方程，其离散差分方法的设计存在概念上的困难。保守派？form.当应用于水平集相关的问题时，该方法可以敏锐地捕捉界面，并在两相流问题的进一步实际应用中具有很大的潜力。为了求解静态Hamilton-Jacobi方程，将LDG方法与快速扫描方法相结合。PI和她的合作者将继续研究其他非线性波动方程的LDG方法。2)直接间断Galerkin方法（DDG）：提出一种求解扩散型方程的新DG方法.该方法的新奇在于找出哪些项本质上对不连续处的解导数贡献最大。研究了一类容许的数值通量，并进行了稳定性和误差分析。界面修正项的引入，以获得DDG方法的精度的最佳顺序。此外，PI将研究DDG方法对不可压缩Navier-Stokes方程的涡量流函数公式。随着DDG方法的成功开发，可以设计高效和精确的DG方法来解决计算流体力学中出现的问题。拟议的活动在于其全面覆盖算法开发，分析和实现。本项目所研究的非线性问题有着丰富的应用，涉及到许多有趣的物理现象。许多流体问题涉及多组分，例如气泡/液滴，射流，波和薄膜。这些问题有界面来分离不同的材料，需要特殊的数值技术来精确地处理界面。间断Galerkin方法在应用高阶多项式近似时产生很小的耗散误差，因此它是一种有吸引力的界面捕捉数值方法。预计拟议的活动将对广泛的应用领域作出积极贡献，包括（但不限于）流体动力学、计算机视觉、最佳控制、半导体器件模拟和天气预报等。此外，研究人员将把该项目与研究生计算数学教育整合起来，以便在更广泛的背景下进行交流。

This proposal is awarded using funds made available by the American Recoveryand Reinvestment Act of 2009 (Public Law 111-5).The goal of the project is to design, analyze and implement new discontinuous Galerkin(DG) finite element methods solving partial differential equations arising from physics and engineering. DG method is a highly accurate numerical method with the advantage to handle complicated geometries, and apply h-p adaptive strategies in applications. The PI will focus on the development of two discontinuous Galerkin methods. 1)Local discontinuous Galerkin methods: a new DG method is proposed to directly solve Hamilton-Jacobi equations. There is a concept difficulty to design DG methods for Hamilton-Jacobi equation, because it is a nonlinear partial differential equation not in a ?conservative? form. When applied to level set related problems, the method can sharply capture the interface, and has a great potential for further practical applications on two-phase flow problems. To solve static Hamilton-Jacobi equations, the LDG method coupled with fast sweeping method will be developed. The PI and her collaborators will continue to study LDG methods for other nonlinear wave equations. 2)Direct discontinuous Galerkin methods(DDG): a new DG method is proposed to solve diffusion type equations. The novelty of the method is to figure out what terms essentially contribute the most to the solution derivative at the discontinuity. A class of admissible numerical fluxes will be studied and stability and error analysis will be carried out. Interface correction terms are introduced to obtain optimal order of accuracy for the DDG method. Furthermore, the PI will investigate DDG methods on incompressible Navier-Stokes equations in vorticity stream-function formulation. With the successful development of DDG method, efficient and accurate DG methods can be designed to solve problems arising from computational fluid dynamics.The proposed activity lies in its comprehensive coverage of algorithm development, analysis and implementation. The nonlinear problems studied in this project have rich applications and involve interesting physical phenomena. Many fluid problems involve multi-component, examples include bubble/drops, jets, waves and films. These problems have interfaces to separate different materials, and special numerical techniques are required to treat the interface accurately. Discontinuous Galerkin methods produce very small dissipation errors when applying with high order polynomial approximations, thus it is an attractive numerical method for interface capturing. The proposed activity is expected to make positive contributions to broad areas of applications, including (but not limited to) fluid dynamics, computer vision, optimal control, semiconductor device simulation and weather forecasting, among many others. In addition, the investigator will integrate the project with graduate computational mathematics education in order to communicate in a broader context.