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

计算流体力学（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，气动设计工程师们逐渐调整了他们的工作方式，以利用基于Euler和Navier-Stokes方程解的新型CFD功能所提供的可能性。物理建模和求解算法的重大改进与计算机能力的巨大增长同样重要，从而使飞机开发的各个阶段都能进行数值模拟。然而，尽管CFD取得了很大的进步，但从用户时间和计算资源的角度来看，复杂结构周围粘性流动的大型气动模拟仍然非常昂贵。在较短的周转时间内以足够的精度可靠地计算结果的要求严重限制了CFD的应用。近年来，人们对高阶离散化方法的发展非常感兴趣，这种方法可以改进对临界流动现象的预测，例如边界层、尾迹和漩涡，以及力系数，例如阻力、升力、力矩，同时与经典（有限体积）方法相比，利用的自由度要少得多。基于有限元框架的一类极有前途的高阶方案是不连续伽辽金（DG，简称）方法。事实上，欧拉方程和纳维-斯托克斯方程数值逼近的DG方法的发展是一个非常令人兴奋的研究课题，目前世界各地的许多研究小组都在开发这个课题。尽管DG方法具有优势和能力，但该方法尚不成熟，目前的实施对其应用于大规模工业问题有很强的局限性。特别是，其中一个关键问题是设计有效的策略来解决由DG方法生成的方程组。在本提案中，我们旨在开发一类新的多级schwarz型预调节器，用于二维和三维可压缩流体流动问题的高阶DG离散化。在这里，网格聚合将基于开发一类新的有限元方法，称为复合有限元（CFEs），它特别适合于以计算领域或微观结构中的小细节为特征的问题。CFEs的核心思想是利用一般形元域，在这些域上元素基函数仅是局部分段光滑的。特别是，CFE中的元素域可以由标准有限元方法中存在的邻近元素的集合组成，而CFE的基函数被构造为标准有限元子域上定义的那些元素的线性组合。通过这种方式，CFEs提供了一个理想的数学和实用框架，在这个框架中可以定义（粗）聚合网格上的有限元解。迄今为止，CFEs的应用仅限于采用最低阶（分段线性）单元的简单模型问题的标准符合有限元近似。在本提案中，我们的目标是在高阶DG方法的背景下对CFEs进行彻底的数学分析，包括将其扩展到包含悬挂节点的一般非结构化混合网格。在这里，特别强调将致力于设计适当的聚合策略，这允许底层DG CFE方法被用作schwarz型预处理策略中的粗网格求解器。该研究将在CFD应用的高阶DG方法的理论和实践发展方面取得重大进展。

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.