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

数学科学：斯托克斯问题的四面体网格上的高阶混合单元

• 批准号: 9625907
• 负责人: Shangyou Zhang
• 金额: $ 5.4万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625907&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Order; Mixed; Elements

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.

张 9625907 研究人员研究四面体网格上的高阶混合有限元方法，解决 3D 稳态 Stokes 问题，以及由此产生的离散线性方程组的迭代求解器。 研究者所做的初步工作在一些宏元素类型网格上发现了一系列稳定的混合元素。 该结果表明可以找到速度的连续分段多项式的最小多项式次数，其中使用少一级分段多项式来近似压力并且混合单元是稳定的；十多年来，这一直是一个悬而未决的问题。 研究者研究了各种增强拉格朗日方法（包括著名的Uzawa方法和迭代惩罚方法），其中结合有效的多重网格方法或/和域分解方法来实现最佳计算阶数。 混合单元和迭代方法的设计难点在于处理不可压缩条件。 研究人员为不可压缩流体动力学中的一些著名方程开发了计算机算法。 不可压缩计算流体动力学问题在整个工业和科学领域中不断出现。 开发这些方法面临着困难的挑战，涉及计算机内存和速度的限制，以及方法本身的准确性和鲁棒性。 该算法预计将应用于机翼设计等领域以及涉及复杂形状流体流动的制造工艺（例如化学工业）。