• 论文 • 上一篇    下一篇

定常Navier-Stokes问题低次等阶稳定有限体积元算法研究

杨建宏   

  1. 宝鸡文理学院数学与信息科学学院, 宝鸡 721013
  • 收稿日期:2016-04-27 出版日期:2017-06-15 发布日期:2017-07-18
  • 基金资助:

    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).

杨建宏. 定常Navier-Stokes问题低次等阶稳定有限体积元算法研究[J]. 数值计算与计算机应用, 2017, 38(2): 91-104.

Yang Jianhong. ANALYSIS ON A STABILIZED FINITE VOLUME ELEMENT METHOD FOR THE STATIONARY NAVIER-STOKES EQUATIONS[J]. Journal of Numerical Methods and Computer Applications, 2017, 38(2): 91-104.

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

Yang Jianhong   

  1. Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, China
  • Received:2016-04-27 Online:2017-06-15 Published:2017-07-18
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fH1时,稳定有限体积元方法与稳定有限元方法之间具有O(|logh|1/2h2)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 pairP1-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|1/2h2) is found between the finite element solution and finite volume element solution for the N-S equations with fH1. 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 accurary and efficiency. There is the poor numerical accurary for the Newton scheme, but the Picard 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.

MR(2010)主题分类: 

()
[1] Li J, He Y N. A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations[J]. Appl Numer Math., 2008, 58(10):1503-1514.

[2] Girault V and Raviart P A. Finite Element Method for Navier-Stokes Equations:Theory and Algorithms[M]. Berlin, Heidelberg:Springer-Verlag, 1987.

[3] Li J and He Y N. A stabilized finite element method based on two local Gauss integral technique for the stationary Stokes equations[J]. J Comp Appl Math., 2008, 214(1):58-65.

[4] Li J, He Y N and Chen Z X. A new stabilized finite element method for the transient Navier-Stokes equations[J]. Comp Meth Appl Mech Eng., 2007, 197(4):22-35.

[5] Qin Y M, Feng M F, Luo K and Wu K T. Local projection stabilized finite element method for the Navier-Stokes equations[J]. Appl Math Mech., 2010, 31(5):651-664. (in chinese)

[6] Li J, Zhao X and Wu J H. Numerical Study of Stabilization of the Lower Order Finite Volume Methods for the Incompressible Flows[J]. Acta Math Sin., 2013, 56(1):15-26. (in chinese)

[7] Chou S H, Li Q. Error estimates in, and in co-volume methods for elliptic and parabolic problems:a unified approach[J]. Math Comp., 2000, 229(69):103-120.

[8] Ye X. On the relationship between finite volume and finite element methods applied to the Stokes equations[J]. Numer Methods Partial Diff Equ., 2001, 17(5):440-453.

[9] Li J, Chen Z X. A New Stabilized Finite Volume Method for the Stationary Stokes Equations[J]. Adv Comput Math., 2009, 30(2):141-152.

[10] Xu J C. A novel two-grid method for semilinear elliptic equations[J]. SIAM J Sci Comput., 1994, 15(1):231-237.

[11] Xu J C. Two-grid finite element discretization techniques for linear and nonlinear PDE[J]. SIAM J Numer Anal., 1996, 33(5):1759-1777.

[12] Niemisto A. FE-approximation of unconstrained optimal control like problems[R]. University of Jyvaskyla, Report 70, 1995.

[13] Layton W, Lenferink W. Two-level Picard-defect corrections for the Navier-Stokes equations at high Reynolds number[J]. Appl. Math. Comp., 1995, 202(69):263-274.

[14] Layton W, Lenferink W. A multi-level mesh independence principlefor the Navier-Stokes equations[J]. SIAM J Numer Anal., 1996, 33(2):17-30.

[15] Layton W, Lee H K and Peterson J. Numerical solution of the stationary Navier-Stokes equations using a multilevel finite element method[J]. SIAM J Sci Comput., 1998, 20(1):1-12.

[16] Girault V, Lions J. L. Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra[J]. Portug. Math., 2001, 58(1):25-57.

[17] Chen C, Liu W. Two-grid finite volume element methods for semilinear parabolic problems[J]. Appl Numer Math., 2010, 60(1-2):10-18.

[18] Li J, Shen L and Chen Z X. Convergence and Stability of a stabilized finite volume method for Stationary Navier-Stokes equations[J]. BIT Numer Math., 2010, 50(4):823-842.

[19] Li R, Zhu P, Generalized difference methods for second order elliptic partial differential equations (I)-triangle grids[J]. Numer Math J Chinese Universities, 1982, (4):140-152.

[20] Rachid A, Bahaj M, and Ayoub N, Two-level stabilized finite volume methods for stationary Navier-Stokes equations[J]. Adv Numer Anal., 2012, 2012(1):1-14.

[21] Yang J H, Lei G and Yang J W. Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow[J]. Adv Appl Math Mech., 2014, 6(5):663-679.
[1] 谢和虎. 子空间扩展算法及其应用[J]. 数值计算与计算机应用, 2020, 41(3): 169-191.
[2] 王芹, 马召灿, 白石阳, 张林波, 卢本卓, 李鸿亮. 三维半导体器件漂移扩散模型的并行有限元方法研究[J]. 数值计算与计算机应用, 2020, 41(2): 85-104.
[3] 葛志昊, 李婷婷, 王慧芳. 双资产欧式期权定价问题的特征有限元方法[J]. 数值计算与计算机应用, 2020, 41(1): 27-41.
[4] 李瑜, 谢和虎. 基于特征线法的群体平衡系统的数值模拟[J]. 数值计算与计算机应用, 2019, 40(4): 261-278.
[5] 邓维山, 徐进. 一种泊松-玻尔兹曼方程稳定算法的高效有限元并行实现[J]. 数值计算与计算机应用, 2018, 39(2): 91-110.
[6] 余涛, 张镭. 线性弹性问题的局部正交分解方法[J]. 数值计算与计算机应用, 2018, 39(1): 10-19.
[7] 周宇, 李秋齐. 基于降基多尺度有限元的PGD方法及其在含参数椭圆方程中的应用[J]. 数值计算与计算机应用, 2017, 38(2): 105-122.
[8] 曹济伟, 葛志昊, 刘鸣放. Stokes方程基于多尺度函数的稳定化有限元方法[J]. 数值计算与计算机应用, 2017, 38(1): 68-80.
[9] 周志强, 吴红英. 分数阶对流-弥散方程的移动网格有限元方法[J]. 数值计算与计算机应用, 2014, 35(1): 1-7.
[10] 黄文艳, 魏剑英, 葛永斌. 三维非定常不可压涡量——速度Navier-Stokes方程组的有限差分法[J]. 数值计算与计算机应用, 2012, 33(4): 301-311.
[11] 杨建宏. 定常Navier-Stokes方程的三种两层稳定有限元算法计算效率分析[J]. 数值计算与计算机应用, 2011, 32(2): 117-124.
[12] 程俊霞, 任健. 含曲率的水平集方程在非结构四边形网格上的数值离散方法[J]. 数值计算与计算机应用, 2011, 32(1): 33-40.
[13] 王同科. 一类二维粘性波动方程的交替方向有限体积元方法[J]. 数值计算与计算机应用, 2010, 31(1): 64-75.
阅读次数
全文


摘要