非定常Stokes方程的稳定化CN有限体积元格式
A Stabilized Crank-Nicolson Finite Volume Element Formulation for Non-Stationary Stokes Equation

作者: 李 宏 , 赵智慧 :内蒙古大学数学科学学院,呼和浩特; 罗振东 :华北电力大学数理学院,北京;

关键词: Stokes方程稳定化Crank-Nicolson有限体积元格式误差估计Stokes Equation Stabilized Crank-Nicolson Finite Volume Element Formulation Error Estimate

摘要:

建立二维非定常Stokes方程的时间二阶精度的稳定化Crank-Nicolson (CN)有限体积元格式, 并给出其稳定化CN有限体积元解的误差估计。数值实验说明时间二阶精度的稳定化CN有限体积元格式比时间一阶精度格式更优越, 从而表明稳定化CN有限体积元格式对于求解非定常Stokes方程的数值解是有效可行的。

Abstract: A stabilized Crank-Nicolson (CN) finite volume element formulation with time second-order accuracy is established for two-dimensional non-stationary Stokes equation. The error estimates of its numerical solutions are provided. Some numerical experiments are presented illustrating that the stabilized CN finite volume element formulation with time second-order accuracy is far more advantageous than that with time first-order accuracy, thus validating that the stabilized CN finite volume element formulation is feasible and efficient for finding the numerical solutions for two- dimensional non-stationary Stokes equation.

文章引用: 李 宏 , 赵智慧 , 罗振东 (2013) 非定常Stokes方程的稳定化CN有限体积元格式。 流体动力学, 1, 26-33. doi: 10.12677/IJFD.2013.12005

参考文献

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[18] Heywood J G, Rannacher R. Finite element approximation of the non-stationary Navier–Stokes problem part IV: error analysis for second–order time discretization [J]. SIAM Journal on Numerical Analysis, 1990, 27(2): 353–384.

[19] Brezzi F, Douglas Jr J. Stabilized mixed method for the Stokes problem [J]. Numerische Mathematik, 1988, 53: 225–235.

[20] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods [M]. New York: Springer–Verlag, 1991.

[21] 罗振东. 混合有限元法基础及其应用[M]. 北京: 科学出版社, 2006.

[22] Cai Z, McCormick S. On the accuracy of the finite volume element method for diffusion equations on composite grid [J]. SIAM Journal on Numerical Analysis, 1990, 27: 636–655.

[23] Suli E. Convergence of finite volume schemes for Poisson's equation on no-nuniform meshes [J]. SIAM Journal on Numerical Analysis, 1991, 28 (5): 1419–1430.

[24] Jones W P, Menziest K R. Analysis of the cell-centered finite volume method for the diffusion equation [J]. Journal of Computational Physics, 2000, 165: 45–68.

[25] Bank R E, Rose D J. Some error estimates for the box methods[J]. SIAM Journal on Numerical Analysis, 1987, 24(4): 777–787.

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[27] Chatzipantelidis P, Lazarrov R D, Thomee V. Error estimates for a finite volume element method for parabolic equations in convex in polygonal domains [J]. Numerical Methods for Partial Differential Equations, 2004, 20: 650–674.

[28] Ye X. On the relation between finite volume and finite element methods applied to the Stokes equations [J]. Numerical Methods for Partial Differential Equations, 2001, 17: 440–453.

[29] Yang M, Song H L. A post processing finite volume method for time-dependent Stokes equations [J]. Applied Numerical Mathematics, 2009, 59: 1922–1932.

[30] Li J, Chen Z X. A new stabilized finite volume method for the stationary Stokes equations [J]. Adv. Comput. Math., 2009, 30: 141–152.

[31] 安静, 孙萍, 罗振东, 黄晓鸣. 非定常Stokes方程的稳定化全离散有限体积元格式[J]. 计算数学, 2011, 33(2): 213–224.

[32] Adams R A. Sobolev Space [M]. New York: Academic Press, 1975.

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