Multilevel Preconditioners based on Composite Finite Element Methods for Fluid Flow Problems

计算流体动力学（CFD）已成为航空工业新产品开发的关键技术。在过去的十年中，空气动力学设计工程师们已经逐步调整了他们的工作方式，以利用基于欧拉和纳维尔-斯托克斯方程解的新CFD能力所提供的可能性。物理建模和求解算法的显著改进与计算机能力的巨大增长一样重要，以使飞机研制的所有阶段都能进行数值模拟。然而，尽管在计算流体动力学方面取得了进展，在用户时间和计算资源方面，复杂构型周围粘性流的大型气动模拟仍然非常昂贵。在短周转时间内以足够的精度可靠地计算结果的要求对CFD的应用施加了严格的限制。近年来，人们对高阶离散化方法的发展产生了很大的兴趣，这些方法允许改进对临界流动现象的预测，例如边界层、尾流和涡流，以及力系数，阻力、升力、力矩，同时与经典（有限体积）方法相比，利用显著更少的自由度。基于有限元框架的一类非常有前途的高阶格式是间断伽辽金（DG，简称）方法。事实上，发展DG方法的数值近似的欧拉和Navier-Stokes方程是一个非常令人兴奋的研究课题，目前正在开发的一些团体在世界各地。尽管DG的方法的优点和能力，该方法还不成熟，目前的实施受到其应用到大规模的工业问题的强烈限制。特别是，其中一个关键问题是设计有效的策略，由DG方法产生的方程组的解决方案。在这个建议中，我们的目标是开发一个新的类的多级Schwarz型预条件的高阶DG离散的二维和三维可压缩流体流动问题。在这里，网格聚合将进行基于利用一类新的有限元方法，被称为复合有限元（CFEs），这是特别适合的问题，其特征在于在计算域或微观结构的小细节。CFEs的核心思想是利用一般形状的元素域上的元素基函数只是局部分段光滑。特别地，CFE内的单元域可以由标准有限元方法内存在的相邻单元的集合组成，其中CFE的基函数被构造为标准有限元子域上定义的那些的线性组合。通过这种方式，CFE提供了一个理想的数学和实践框架，其中可以定义（粗）聚合网格上的有限元解。到目前为止，CFEs的应用已被限制到标准的符合有限元近似的简单模型问题，采用最低阶（分段线性）元素。在这项建议中，我们的目标是开发一个彻底的数学分析的CFEs的背景下，高阶DG方法，包括其扩展到一般的非结构化混合网格包含悬挂节点。在这里，特别强调将致力于适当的聚合策略的设计，它允许底层DG CFE方法被用作施瓦茨型预处理策略内的粗网格求解器。这项研究将导致显着的进步，在理论和实践的发展高阶DG方法的CFD应用。

Computational fluid dynamics (CFD) has become a key technology in the development of new products in the aeronautical industry. During the last decade aerodynamic design engineers have progressively adapted their way-of-working to take advantage of the possibilities offered by new CFD capabilities based on the solution of the Euler and Navier-Stokes equations. Significant improvements in physical modelling and solution algorithms have been as important as the enormous increase of computer power to enable numerical simulations at all stages of aircraft development. However, despite the progress made in CFD, in terms of user time and computational resources, large aerodynamic simulations of viscous flows around complex configurations are still very expensive. The requirement to reliably compute results with a sufficient level of accuracy within short turn-around times places severe constraints on the application of CFD. In recent years there has been significant interest in the development of high-order discretization methods which allow for an improved prediction of critical flow phenomena, such as boundary layers, wakes, and vortices, for example, as well as force coefficients, e.g., drag, lift, moment, while exploiting significantly fewer degrees of freedom compared with classical (finite volume) methods. One extremely promising class of high-order schemes based on the finite element framework are Discontinuous Galerkin (DG, for short) methods. Indeed, the development of DG methods for the numerical approximation of the Euler and Navier-Stokes equations is an extremely exciting research topic which is currently being developed by a number of groups all over the world. Despite the advantages and capabilities of the DG approach, the method is not yet mature and current implementations are subject to strong limitations for its application to large scale industrial problems. In particular, one of the key issues is the design of efficient strategies for the solution of the system of equations generated by a DG method. In this proposal we aim to develop a new class of multilevel Schwarz-type preconditioners for the high-order DG discretization of two- and three-dimensional compressible fluid flow problems. Here, mesh aggregation will be undertaken based on exploiting a new class of finite element methods, referred to as Composite Finite Elements (CFEs), which are particularly suited to problems characterized by small details in the computational domain or micro-structures. The key idea of CFEs is to exploit general shaped element domains upon which elemental basis functions are only locally piecewise smooth. In particular, an element domain within a CFE may consist of a collection of neighbouring elements present within a standard finite element method, with the basis function of the CFE being constructed as a linear combination of those defined on the standard finite element subdomains. In this way, CFEs offer an ideal mathematical and practical framework within which finite element solutions on (coarse) aggregated meshes may be defined. To date, the application of CFEs has been restricted to standard conforming finite element approximations of simple model problems employing lowest-order (piecewise linear) elements. In this proposal we aim to develop a thorough mathematical analysis of CFEs within the context of high-order DG methods, including their extension to general unstructured hybrid meshes containing hanging nodes. Here, particular emphasis will be devoted to the design of appropriate aggregation strategies, which allow for the underlying DG CFE method to be employed as a coarse mesh solver within Schwarz-type preconditioning strategies. This research will lead to significant advances in both the theoretical and practical development of high-order DG methods for CFD applications.