定常Navier-Stokes问题低次等阶稳定有限体积元算法研究  

ANALYSIS ON A STABILIZED FINITE VOLUME ELEMENT METHOD FOR THE STATIONARY NAVIER-STOKES EQUATIONS

作　　者：杨建宏[1] Yang Jianhong(Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, China)

机构地区：[1]宝鸡文理学院数学与信息科学学院,宝鸡721013

出　　处：《数值计算与计算机应用》2017年第2期91-104,共14页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金项目(11371031);宝鸡市科技计划项目(15RKX-1-5-10);宝鸡文理学院科研项目(ZK16011)

摘　　要：通过有限元空间和有限体积元空间的一种双射投影得到了不可压缩流问题低次等阶稳定有限体积元方法.该方法采用低次等阶元P1-P1(或Q1-Q1)对Navier-Stokes(N-S)方程进行数值求解,利用局部压力投影技术进行稳定化处理.通过有限元和有限体积元方法的等价性进行有限体积元方法的理论分析.发现不可压缩流N-S问题在f∈H^1时,稳定有限体积元方法与稳定有限元方法之间具有O(|logh|^(1/2)/h^2)阶超收敛逼近结果.将稳定有限体积算法的三种两重网格格式进行了比较分析,发现当粗、细网格尺度比例选取适当时,两重算法具有传统算法相同的收敛速度,而两重算法具有明显的效率优势,并且Simple格式速度最快,Picard格式更适合较小粘性系数问题的数值求解.In this paper, a new stabilized finite volume element method is considered for the stationary Navier-Stokes equations through a bijective projection that from the finite element space to the finite volume element space. This method is established on the local pressure projection techniqure and uses the lower equal-order finite element pair P1- P1 （or Q1- Q1） which do not satisfy the inf-sup condition. Based on the relationship between the finite element method and finite volume element method, a supercovergence result O（｜logh｜^1/2h^2） is found between the finite element solution and finite volume element solution for the N-S equations with f ∈ H^1. the performance of three kinds iterative scheme of two-level stabilized methods are compared in efficiency and precision aspects by a series of numerical experiments. We discover that the Simple scheme is better than two others on accuraryand efficiency. There is the poor numerical accurary for the Newton scheme , but the Pi- card scheme is more suitable to incompressible flow with low viscosity coefficient Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.

关 键 词：Navier—Stokes方程 有限元方法 有限体积元方法 低次等阶元 局部压力投影 

分 类 号：O241.8[理学—计算数学]