引用本文:张琪慧, 尚月强.Navier-Stokes方程的亚格子模型后处理混合有限元方法[J].西南大学学报(自然科学版),2019,41(3):67~74
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Navier-Stokes方程的亚格子模型后处理混合有限元方法
张琪慧, 尚月强
西南大学 数学与统计学院, 重庆 400715
摘要:
提出并考察了3种基于亚格子模型的后处理混合有限元方法,其主要思想是:第一步在粗网格上求解带有亚格子模型稳定项的Navier-Stokes方程,得到最后时刻T的有限元解uH;第二步在最后时刻T,对第一步所得解uH进行后处理,主要通过在细网格上(或用高阶元)分别求解带有亚格子模型稳定项的Stokes问题、Newton问题或者Ossen问题.实验结果表明:在选取适当的稳定化参数和网格尺寸的条件下,3种稳定化的后处理有限元方法提高了稳定化的混合有限元解的精确度,并且收敛阶较标准的有限元方法明显提高了一阶.从计算时间看,除ν=1以外,在其它情况下稳定化的Newton型后处理花费的时间相对较多,而稳定化的Ossen型后处理花费的时间相对较少.从精确度来看,Newton型后处理和Ossen型后处理方法所得速度的H1-范误差和压力的L2-范误差比Stokes型后处理方法更有效.
关键词:  Navier-Stokes方程  后处理  亚格子稳定化  有限元方法
DOI:10.13718/j.cnki.xdzk.2019.03.010
分类号:O241.82
基金项目:重庆市基础科学与前沿技术研究专项项目(cstc2016jcyjA0348).
A Subgrid Stabilizing Postprocessed Mixed Finite Element Method for the Navier-Stokes Equations
ZHANG Qi-hui, SHANG Yue-qiang
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Abstract:
In this paper, we mainly study three postprocessed mixed finite element methods for the incompressible Navier-Stokes equations, which are based on a subgrid model. These methods consist of two steps. The first step is to solve a subgrid stabilized nonlinear Navier-Stokes problem on a coarse grid to obtain an approximate solution uH at time T. The second step is to postprocess uH on a finer grid (or by high-order finite elements), by solving a stabilized Stokes problem, a stabilized Newton-Type problem, or a stabilized Ossen problem. The numerical results show that under the conditions of selecting appropriate stabilizing parameters and grid sizes, the postprocessed finite element method can improve the precision of the mixed finite-element solution, and the order of convergence is obviously improved by one unit compared with the standard subgrid stabilized method. From the point of the computational time, in addition to ν=1, the stabilized Newton-type postprocessed method takes a relatively more time than the others, while the stabilized Ossen-type postprocessed method takes the least time among the three methods. And from the point of precision of the computed solutions, the Newton and Oseen-type postprocessed methods are better than the Stokes-type postprocessed method.
Key words:  Navier-Stokes equations  postprocessing  subgrid stabilization  finite element method
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