Discontinuous galerkin methods for flow and electro-magnetic problems

间断伽辽金方法是计算机模拟科学和工程中复杂现象的专用数值方法。虽然第一个不连续Galerkin方法是在1973年设计的，但这些方法在九十年代经历了重大发展，使它们成为计算流体动力学的主流。近年来，这些方法正被应用于许多其他问题的工程利益，包括电磁学和结构力学。间断Galerkin方法最近的成功可以归因于它们的灵活性和通用性。这些方法可以稳健地处理几乎任何类型的偏微分方程，以及其类型在计算域内变化的方程。它们非常适合多物理场应用和复杂几何形状中材料特性变化很大的问题。此外，不连续伽辽金方法可以很容易地处理不规则的细化网格和可变的近似度，一个属性被称为hp-adaptivity.The长期目标，这项建议是开发，分析和实施的不连续伽辽金有限元方法的改进模拟计算流体力学和电磁学的问题。特别是，我们计划解决一些最相关的问题，目前的发展不连续Galerkin方法：设计新的方法，具有更少的全球耦合自由度，发展完全无发散的方法，分析新的和鲁棒收敛的自适应算法，并设计新的求解器和实现技术。我们还计划开发一个灵活的模拟代码的问题，在不可压缩流体流动和电磁学，它结合了计算数学的现代发展，如惠普自适应，高效的迭代求解器，高阶时间步进。

Discontinuous Galerkin methods are specialized numerical methods for the computer simulation of complex phenomena in the sciences and engineering. Although the first discontinuous Galerkin method was devised back in 1973, these methods experienced a significant development during the nineties which brought them to the mainstream of computational fluid dynamics. In recent years, these methods are being applied to many other problems of engineering interest including electro-magnetics and structural mechanics. The recent success of discontinuous Galerkin methods can be attributed to their flexibility and versatility. These methods can deal robustly with partial differential equations of almost any kind, as well as with equations whose type changes within the computational domain. They are ideally suited for multi-physics applications and for problems with highly varying material properties in complex geometries. Moreover, discontinuous Galerkin methods can easily handle irregularly refined meshes and variable approximation degrees; a property referred to as hp-adaptivity.The long-term objectives of this proposal are the development, analysis and implementation of discontinuous Galerkin finite element methods for the improved simulation of problems in computational fluid mechanics and electro-magnetics. In particular, we plan to address some of the most relevant issues of the current development of discontinuous Galerkin methods: the design of new methods that have fewer globally coupled degrees of freedom, the development of exactly divergence-free methods, the analysis of new and robustly convergent adaptive algorithms, and the devising of new solvers and implementation techniques. We further plan to develop a flexible simulation code for problems in incompressible fluid flow and electro-magnetics, which incorporates modern developments in computational mathematics, such as hp-adaptivity, efficient iterative solvers, and high-order time-stepping.