随着计算机的性能不断提高，大规模流动问题的高效求解算法成为当前计算数学界和工程界的前沿研究热点之一。有限元方法在偏微分方程相关的问题中得到了广泛的应用，而不同类型的问题又往往需要设计特殊的有限元格式。对于具有边界层效应的流动问题(飞机外部绕流问题、圆柱绕流问题等等)，设计非常准确且高效的数值算法对实际应用有重要意义。本项目根据流动的特性，拟使用杂交网格技术，结合四边形和六面体上的约束线性元构造方法，利用深入研究过的基于局部高斯积分的稳定化有限元思想，构建能够处理边界层效应的流动问题的低阶有限元求解方法。然后通过基于区域分解的Newton-Krylov-Schwarz方法，设计能够高效求解的可扩展并行算法，从而为难以求解但又有着重要工程应用的带有边界层效应的粘性流问题提供有效的算法及其理论支持。

Along with the continuous improvement of the performance of computers, the research on efficient algorithms for solving the large-scale flow problems has become one of the forefront hot topics. Finite element methods have been proved to be efficient for many problems of partial differential equations, but different problems usually need special treatments. There are very important in practical applications to design accurate and efficient numerical algorithms for the flow problems with the boundary layers, like the problems of flow passing an aircraft or a cylinder. In this project, by taking advantage of the special features of the flow, we propose to use the hybrid meshes, on which the constrained linear elements are utilized, and construct a lower order stabilized finite element method on the hybrid meshes based on the well-studied stabilized technique of two local Gauss integrations, then develope an extensible parallel algorithm using the domain-decompositon-based Newton-Krylov-Schwarz method. Thus we hope to provide some efficient algorithms and theorical justifications for the numerical computations of flow problems with the boundary layers, which are difficult to solve but have wide applications in engineering.