非定常Stokes方程的混合有限元法的高精度逼近  

High-precision approximation of mixed finite element method for unsteady Stokes equation

作　　者：丁晓 陈璐 江山 DING Xiao;CHEN Lu;JIANG Shan(School of Mathematics and Statistics,Nantong University,Nantong 226019,China;Nantong Tianjiabing Junior Middle School,Nantong 226001,China)

机构地区：[1]南通大学数学与统计学院,江苏南通226019 [2]南通市田家炳初级中学,江苏南通226001

出　　处：《南通大学学报（自然科学版）》2024年第3期82-88,共7页Journal of Nantong University(Natural Science Edition)　

基　　金：国家自然科学基金面上项目(11771224);南通市基础科学研究指令性项目(JC2021123)。

摘　　要：针对二维非定常Stokes方程,采用Taylor-Hood混合有限元法进行数值模拟。首先,利用虚功原理将方程转化为变分形式;其次,将求解区域均匀划分为有限个三角形单元,在每个逻辑单元建立与等参单元之间的联系,为速度u选取二次元基函数,为压力p选取线性基函数,从而构造空间尺度的有限元空间;然后,构造时间尺度的全离散θ-型隐格式,选取θ=0.5的Crank-Nicolson六点对称有限差分格式;最后,原问题转化为常微分方程求其数值解,并通过数值算例检验该方法的可行性与有效性。理论构造和数值结果均验证,非定常问题在空间层、时间层离散分别使用线性有限元和二次有限元计算均可得到稳定的一致收敛结果,且二次有限元的精度更高、收敛更快。为了更直观生动地展示实验结果,文章最后给出了精确解与有限元解的三维误差图。For the two-dimensional unsteady Stokes equations,the Taylor-Hood mixed finite element method is employed for numerical simulation.Firstly,the equation is transformed into the variational form by using the variational principle.Secondly,the domain is uniformly divided into finite triangular elements,and the connection is built among the logical elements and the isoparametric ones.The quadratic basis function is selected for velocity u,and the linear basis function is selected for pressure p,so it establishes a finite element space in spatial scale.Then it is further combined with the finite difference method in temporal scale to construct a fully discrete-type implicit scheme,for selecting the Crank-Nicolson six-point symmetric finite difference scheme is applied.Finally,the equation is transformed into ordinary differential equations for solving its numerical solution,through the numerical example we testify the feasibility and effectiveness of our method.Theoretical constructions and numerical results both validate that the unsteady problem under the space and time discretization it obtains consistent convergent results,which are processed with the linear finite element and the quadratic finite element.And the results of quadratic finite element are much accurate and faster convergence.In order to present the experimental results more intuitively and vividly,this study ends up with several three-dimensional error figures between the exact solution and the finite element solution.

关 键 词：二维非定常Stokes方程 混合有限元法 有限差分格式 一致收敛 

分 类 号：O241.82[理学—计算数学]