定常Navier-Stokes方程基于完全重叠型区域分解的并行稳定化有限元方法  

A parallel stabilized finite element method based on fully overlapping domain decomposition for the steady Navier-Stokes equations

作　　者：郑波 尚月强 Bo Zheng;Yueqiang Shang

机构地区：[1]西南大学数学与统计学院,重庆400715

出　　处：《中国科学：数学》2020年第8期1117-1130,共14页Scientia Sinica：Mathematica

基　　金：重庆市基础与前沿探索项目(批准号:cstc2018jcyjAX0305);中央高校基本业务费专项资金(批准号:XDJK2018B032)资助项目。

摘　　要：基于完全重叠型区域分解技巧,针对低阶P1-P1有限元,本文提出求解二维定常不可压缩Navier-Stokes方程的并行稳定化有限元方法,其稳定项是基于两局部Gauss积分的压力投影.该方法的基本思想是,使用一局部加密的多尺度网格计算给定子区域上的局部稳定化有限元解.理论分析上借助有限元解的局部先验误差估计,推导出并行稳定化方法所得速度和压力解的误差界.选取适当的算法参数比例,该方法能取得与标准稳定化有限元方法相同的收敛阶,同时减少大量的计算时间.最后给出两类数值算例验证并行稳定化方法的高效性.Based on a fully overlapping domain decomposition technique,a parallel stabilized finite element method is presented to solve the two-dimensional steady incompressible Navier-Stokes equations by using the lowest equal-order P1-P1 finite elements,where the stabilization term is the pressure projection stabilization based on two local Gauss integrations.The basic idea of this method is to use a locally refined multiscale mesh to compute a local stabilized finite element solution in the given subregion of interest.The local a priori error estimate of the finite element solutions is used in theoretical analysis.Error bounds for the velocity and the pressure solutions obtained by the parallel stabilized method are estimated.By choosing appropriate algorithmic parameter scalings,the method can yield a convergence rate of the same order as the standard stabilized finite element method with a massive reduction in computational time.Two kinds of numerical experiments are given to verify the high efficiency of the parallel stabilized algorithm.

关 键 词：NAVIER-STOKES方程 两局部Gauss积分 有限元 并行算法 稳定化方法 

分 类 号：O241.82[理学—计算数学]