Local and Direct discontinuous Galerkin methods: New algorithms and applications

本提案使用2009年美国复苏和再投资法案（公法111-5）提供的资金。该项目的目标是设计，分析和实现新的不连续伽辽金（DG）有限元方法来求解物理和工程中的偏微分方程。DG方法是一种精度高的数值方法，具有处理复杂几何图形和应用惠普自适应策略的优点。PI将重点讨论两种不连续伽辽金方法的发展。1)局部不连续Galerkin方法：提出了一种新的直接求解Hamilton-Jacobi方程的DG方法。Hamilton-Jacobi方程是一个非保守的非线性偏微分方程，在设计DG方法时存在概念上的困难。的形式。当应用于水平集相关问题时，该方法可以清晰地捕捉到界面，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，将采用LDG法与快速扫描法相结合的方法。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)直接不连续伽辽金方法（Direct discontinuous Galerkin methods， DDG）：提出了一种求解扩散型方程的新方法。该方法的新颖之处在于找出哪些项对不连续处的解导数贡献最大。本文将研究一类可容许的数值通量，并进行稳定性和误差分析。为了获得DDG法的最优精度顺序，引入了界面修正项。此外，PI将研究涡旋流函数公式中不可压缩Navier-Stokes方程的DDG方法。随着DDG方法的成功发展，可以设计出高效、准确的DG方法来解决计算流体力学中的问题。所提议的活动在于其对算法开发、分析和实现的全面覆盖。本课题研究的非线性问题具有丰富的应用价值，涉及到有趣的物理现象。许多流体问题涉及多组分，例如气泡/液滴、射流、波浪和薄膜。这些问题具有分离不同材料的界面，需要特殊的数值技术来精确处理界面。不连续伽辽金方法在应用高次多项式近似时产生非常小的耗散误差，因此是一种有吸引力的界面捕获数值方法。拟议的活动预计将对广泛的应用领域做出积极贡献，包括（但不限于）流体动力学、计算机视觉、最优控制、半导体器件模拟和天气预报等。此外，研究者将该项目与研究生计算数学教育相结合，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.