Discontinuous galerkin methods for flow and electro-magnetic problems

间断伽辽金方法是一种专门用于计算机模拟科学和工程中复杂现象的数值方法。虽然第一个间断Galerkin方法是在1973年设计的，但这些方法在90年代经历了重大的发展，使它们成为计算流体力学的主流。近年来，这些方法被应用于许多其他工程问题，包括电磁学和结构力学。不连续Galerkin方法最近的成功可以归功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组，以及在计算区域内类型变化的方程。它们非常适合于多物理应用和复杂几何中具有高度不同材料属性的问题。此外，间断Galerkin方法可以很容易地处理不规则的精细网格和可变的逼近度；这一特性被称为hp适应性。该提议的长期目标是发展、分析和实施间断Galerkin有限元方法，以改进计算流体力学和电磁学问题的模拟。特别是，我们计划解决当前不连续Galerkin方法发展中的一些最相关的问题：具有较少全局耦合自由度的新方法的设计，精确无发散方法的发展，新的和稳健收敛的自适应算法的分析，以及新的求解器和实现技术的设计。我们还计划为不可压缩流体流动和电磁学问题开发一个灵活的模拟代码，它结合了计算数学中的现代发展，如HP自适应、高效迭代求解器和高阶时间步长。

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.