Adaptive Discontinuous Galerkin Methods with Applications

许多工程上感兴趣的问题都是在复杂的、可能是非光滑的和三维域中提出的。它们通常涉及多个物理模型和比例，以及高度变化或非线性的材料属性。需要先进的数值技术才能做出有意义的预测。为了获得更好的和高精度的模拟输出，我们将重点开发和实施间断伽辽金(DG)有限元方法，并结合自适应细化策略。近年来，这些方法成为计算流体力学、结构力学和电磁学中各种问题的稳健和灵活的模拟技术。它们自然适合于自适应计算，并为模拟三维空间中的耦合多物理模型提供了一个统一的计算平台。 这项提案的目标有两个。首先，我们计划设计和分析自适应间断Galerkin(DG)有限元方法，用于三维空间中各种类型的偏微分方程组的数值逼近。我们特别感兴趣的是实现全自动精化策略，该策略允许在基本网格大小和近似程度上进行各向异性精化，并且当感兴趣的现象中存在奇点或层时，在自由度数上产生高阶或指数收敛速度。此外，我们希望开发快速高效的谱/HP元素区域分解求解器。第二个目标是将这些方法应用于耦合不可压缩流体流动问题的进一步发展和有效实施。我们打算继续我们关于导电流体与磁场相互作用的快速求解器的工作，并研究用于非等温不可压缩流动问题的计算机模拟的类似技术。

Many problems of engineering interest are posed in complex, possibly non-smooth and three-dimensional domains. They often involve multiple physical models and scales, as well as highly varying or non-linear material properties. Advanced numerical techniques are required for meaningful predictions. To achieve improved and high-order accurate simulation outputs, we shall focus on the development and implementation of discontinuous Galerkin (DG) finite element methods, combined with adaptive refinement strategies. In recent years, these methods established themselves as robust and flexible simulation techniques for wide classes of problems in computational fluid dynamics, structural mechanics and electro-magnetics. They are naturally suited for adaptive computations, and for providing a unified computational platform for simulating coupled multi-physics models in three space dimensions. The objectives of this proposal are twofold. First, we plan to design and analyse adaptive discontinuous Galerkin (DG) finite element methods for the numerical approximation of partial differential equations of various types in three space dimensions. We are particularly interested in realizing fully automatic refinement strategies, which allow for anisotropic refinements in both the elemental mesh sizes and approximation degrees, and which yield high-order or exponential convergence rates in the numbers of degrees of freedom, also when singularities or layers are present in the phenomena of interest. In addition, we wish to develop fast and efficient spectral/hp element domain decomposition solvers. The second objective is the further advancement and efficient implementation of these methods as applied to coupled incompressible fluid flow problems. We intend to continue our work on fast solvers for electrically conducting fluids interacting with magnetic fields, and to investigate similar techniques for the computer simulation of non-isothermal incompressible flow problems.