Mathematical Sciences: High-Order Mixed Elements on Tetrahedral Meshes for the Stokes Problem

Zhang 9625907研究了三维定常Stokes问题的四面体网格上的高阶混合有限元方法，以及由此产生的离散线性方程组的迭代求解器。研究人员所做的初步工作是在一些宏单元型网格上找到了一族稳定的混合单元族。这一结果表明，有可能找到速度的连续分段多项式的最小多项式次数，其中用一次以下的分段多项式来逼近压力，并且混合元素是稳定的；这是一个十多年来一直悬而未决的问题。研究人员研究了各种增广拉格朗日方法(包括著名的尤扎瓦方法和迭代惩罚方法)，其中有效的多重网格法或/和区域分解方法相结合以获得最优的计算顺序。混合单元和迭代法设计的难点是处理不可压缩条件。研究人员开发了不可压缩流体动力学中一些著名方程的计算机算法。不可压缩计算流体动力学的问题出现在整个工业和科学中。开发这些方法有困难的挑战，涉及计算机内存和速度的限制，以及方法本身的准确性和稳健性。该算法有望应用于翼型设计等领域，以及涉及复杂形状流体流动的制造过程，例如化学工业。

Zhang 9625907 The investigator studies high order mixed finite element methods on tetrahedral meshes for the 3D stationary Stokes problems, and the iterative solvers for the resulting discrete linear systems of equations. Preliminary work done by the investigator finds a family of stable mixed elements on some macro-element type grids. This result indicates the possibility of finding the minimal polynomial degree of continuous piecewise polynomials for the velocity where one degree less piecewise polynomials are employed to approximate the pressure and the mixed elements are stable; this has been an open problem for more than a decade. The investigator studies various augmented Lagrangian methods (including the well-known Uzawa method and the iterative penalty method) where the effective multigrid method or/and domain decomposition method are combined to achieve the optimal order of computation. The difficulty in the design of both the mixed elements and the iterative methods is to treat the incompressible condition. The investigator develops computer algorithms for some well-known equations in incompressible fluid dynamics. Problems of incompressible computational fluid dynamics arise throughout industry and science. There are difficult challenges in developing these methods involving limits of computer memory and speed, as well as accuracy and robustness of the method itself. The algorithm is expected to have applications to areas such as airfoil design and to manufacturing processes involving fluid flow in complex shapes, for example, the chemical industry.