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，气动设计工程师逐渐调整了他们的工作方式，以利用基于欧拉和纳维-斯托克斯方程解的新的CFD功能所提供的可能性。物理模型和求解算法的重大改进与计算机能力的巨大增加一样重要，以便能够在飞机开发的所有阶段进行数值模拟。然而，尽管CFD取得了进展，但在用户时间和计算资源方面，对复杂外形周围的粘性流动进行大型气动模拟仍然是非常昂贵的。在很短的周转时间内以足够的精度可靠地计算结果的要求对CFD的应用提出了严格的限制。近年来，高阶离散化方法的发展引起了人们的极大兴趣，这种方法允许改进对临界流动现象的预测，例如边界层、尾迹和涡旋，以及力系数，例如阻力、升力、力矩，同时与经典的(有限体积)方法相比，利用的自由度明显较少。基于有限元框架的一类非常有前途的高阶格式是间断Galerkin(简称DG)方法。事实上，发展数值逼近Euler和Navier-Stokes方程的DG方法是一个非常令人兴奋的研究课题，目前正由世界各地的一些小组开发。尽管DG方法具有优势和能力，但该方法还不成熟，目前的实现因其在大规模工业问题中的应用而受到很大限制。特别是，关键问题之一是设计有效的策略来求解由DG方法产生的方程组。在这个方案中，我们的目标是发展一类新的多层Schwarz型预条件算子，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，网格聚合将基于利用一类新的有限元方法，称为复合有限元(CFE)，它特别适合于以计算领域或微结构中的小细节为特征的问题。CFES的核心思想是利用基本基函数在其上只是局部分段光滑的一般成形单元域。具体地，CFE内的单元域可以由存在于标准有限元方法内的相邻单元的集合组成，其中CFE的基函数被构造为在标准有限元子域上定义的基函数的线性组合。通过这种方式，CFE提供了一个理想的数学和实用框架，可以在其中定义(粗)聚集网格上的有限元解。到目前为止，CFE的应用仅限于使用最低阶(分段线性)单元的简单模型问题的标准协调有限元逼近。在这个建议中，我们的目标是在高阶DG方法的背景下对CFE进行彻底的数学分析，包括将它们扩展到包含悬挂节点的一般非结构混合网格。在这里，将特别强调适当的聚集策略的设计，这允许基本的DG CFE方法被用作Schwarz型预条件策略中的粗网格求解器。这一研究将对CFD高阶DG方法的理论和实践发展起到重要的推动作用。

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.