Discontinuous Galerkin Methods for Partial Differential Equations

偏微分方程的间断伽辽金法

• 批准号: 0712955
• 负责人: Bernardo Cockburn
• 金额: $ 36.62万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0712955&HistoricalAwards=false
• 关键词: Discontinuous; Galerkin; Methods; Partial; Differential

This project is devoted to devising and studying, theoretically as well as computationally, efficient methods for numerically solving problems modeled by partial differential equations. We propose to investigate how to overcome several difficulties raised by the application of the so-called discontinuous Galerkin method to a variety of problems of practical interest. The applications are to the evolution of surfaces and to navigation of robots in arbitrary terrains (Hamilton-Jacobi equations), to incompressible fluid flow (Navier-Stokes equations), to structural mechanics with special emphasis on shells (elasticity equations), and to Darcy flow in porous media (second-order elliptic equations).For each of the above problems, we consider the problem of how to obtain highly accurate computer simulations. In many of these problems (fluid flow and structural mechanics), the emphasis will be put on the development of a recently discovered class of discontinuous Galerkin methods more robust and efficient than previously known methods. In others (evolution of surfaces and navigation of robots), the effort will be concentrated in the efficient control of the quality of the simulation. Indeed, since the exact solution of these complex problems is not known, to guarantee a given accuracy of the simulation, special techniques have to be devised in order to assess its quality. Moreover, these techniques can be employed to automatically let the computer know when and where to increase or decrease the computational effort to obtain the simulation; in this way, the efficiency of its computationis significantly enhanced. Discontinuous Galerkin methods are particularly well-suited for this approach.

这个项目致力于设计和研究，从理论上和计算上，有效的方法来数值求解偏微分方程建模的问题。 我们建议研究如何克服几个困难所提出的所谓的不连续Galerkin方法的应用程序的各种问题的实际利益。 应用程序是表面的演变和导航的机器人在任意地形（Hamilton-Jacobi方程），不可压缩的流体流动（Navier-Stokes方程），结构力学，特别强调壳（弹性方程），并在多孔介质中的达西流（二阶椭圆方程）。对于上述每一个问题，我们考虑如何获得高精度的计算机模拟的问题。 在许多这些问题（流体流动和结构力学），重点将放在最近发现的一类非连续Galerkin方法比以前已知的方法更强大和有效的发展。 在其他方面（表面的演变和机器人导航），努力将集中在有效控制模拟的质量。 事实上，由于这些复杂问题的确切解决方案是未知的，以保证给定的精度的模拟，特殊的技术必须设计，以评估其质量。 此外，这些技术可以用来自动让计算机知道何时以及在何处增加或减少计算工作量以获得模拟;以这种方式，其计算效率显着提高。 不连续Galerkin方法特别适合这种方法。