带非线性阻尼项定常Stokes方程的有限元迭代算法

doi:10.19596/j.cnki.1001-246x.8878

摘要/Abstract

摘要：

针对带非线性阻尼项定常不可压缩Stokes问题, 提出两种基于有限元离散的数值迭代算法, 其基本思想是: 首先使用有限元法求解Stokes问题, 得到迭代初始解; 其次, 运用Stokes迭代算法或Oseen迭代算法求解带非线性阻尼项定常不可压缩Stokes问题, 得到有限元近似解。证明算法的收敛性和稳定性, 并给出了相应的误差估计; 通过4个数值实验, 验证了理论分析的正确性和算法的有效性。研究结果表明: 当方程满足稳定性条件时, 两种数值迭代算法都是可行的。

关键词: Stokes方程, 阻尼项, 迭代法, 稳定性, 误差估计

Abstract:

In the fields of ocean engineering and aerospace, the motion state of fluids has a significant impact on the performance and stability of systems. Stokes equations with damping terms are commonly used to describe the flow behavior of fluids under damping, such as the fluids in porous media. Two numerical iterative algorithms based on finite element discretization are proposed for the steady incompressible Stokes questions with the nonlinear damping term. The basic idea is to first use the finite element method to solve the Stokes problem and obtain the initial iterative solution. Secondly, it uses the Stokes iterative algorithm or Oseen iterative algorithm to solve the steady incompressible Stokes problem with nonlinear damping term and obtain approximate finite element solutions. Convergence and stability of the proposed algorithms are analyzed. Error estimates of the obtained approximate solutions are derived. Some numerical results are also given to show correctness of theoretical analysis and effectiveness of the algorithms. The results show that when the equation satisfies the stability condition, both numerical iterative algorithms are feasible.

Key words: Stokes equation, damping term, iterative method, stability, error estimation

徐彤, 尚月强. 带非线性阻尼项定常Stokes方程的有限元迭代算法[J]. 计算物理, 2025, 42(2): 211-223.

Tong XU, Yueqiang SHANG. Finite Element Iterative Algorithms for Steady Stokes Equations with Nonlinear Damping Term[J]. Chinese Journal of Computational Physics, 2025, 42(2): 211-223.

图/表 9

参考文献 27

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算法	1/h	CPU/s	it	‖▽(u-u_hⁿ)‖	‖p-p_hⁿ‖	UH1	PL2
Stokes迭代	20	0.335	3	2.095 04 × 10^-4	7.905 69 × 10^-4
	30	0.745	3	9.342 65 × 10^-5	3.513 64 × 10^-4	1.991 71	2
	40	1.43	3	5.261 72 × 10^-5	1.976 42 × 10^-4	1.995 72	2
	50	2.418	3	3.369 47 × 10^-5	1.264 91 × 10^-4	1.997 38	2
	60	3.659	3	2.340 67 × 10^-5	8.784 10 × 10^-5	1.998 23	2
	70	5.409	3	1.720 01 × 10^-5	6.453 63 × 10^-5	1.998 72	2
Oseen迭代	20	0.236	2	2.095 04 × 10^-4	7.905 69 × 10^-4
	30	0.538	2	9.342 65 × 10^-5	3.513 64 × 10^-4	1.991 71	2
	40	1.033	2	5.261 72 × 10^-5	1.976 42 × 10^-4	1.995 72	2
	50	1.748	2	3.369 47 × 10^-5	1.264 91 × 10^-4	1.997 38	2
	60	2.64	2	2.340 67 × 10^-5	8.784 1 × 10^-5	1.998 23	2
	70	4.012	2	1.720 01 × 10^-5	6.453 63 × 10^-5	1.998 72	2

算法	1/h	CPU/s	it	‖▽(u-u_hⁿ)‖	‖p-p_hⁿ‖	UH1	PL2
Stokes迭代	20	0.335	3	2.095 04 × 10^-4	7.905 69 × 10^-4
	30	0.745	3	9.342 65 × 10^-5	3.513 64 × 10^-4	1.991 71	2
	40	1.43	3	5.261 72 × 10^-5	1.976 42 × 10^-4	1.995 72	2
	50	2.418	3	3.369 47 × 10^-5	1.264 91 × 10^-4	1.997 38	2
	60	3.659	3	2.340 67 × 10^-5	8.784 10 × 10^-5	1.998 23	2
	70	5.409	3	1.720 01 × 10^-5	6.453 63 × 10^-5	1.998 72	2
Oseen迭代	20	0.236	2	2.095 04 × 10^-4	7.905 69 × 10^-4
	30	0.538	2	9.342 65 × 10^-5	3.513 64 × 10^-4	1.991 71	2
	40	1.033	2	5.261 72 × 10^-5	1.976 42 × 10^-4	1.995 72	2
	50	1.748	2	3.369 47 × 10^-5	1.264 91 × 10^-4	1.997 38	2
	60	2.64	2	2.340 67 × 10^-5	8.784 1 × 10^-5	1.998 23	2
	70	4.012	2	1.720 01 × 10^-5	6.453 63 × 10^-5	1.998 72	2

算法 ν	Stokes迭代				Oseen迭代
算法 ν	CPU/s	it	‖▽(u-u_hⁿ)‖	‖p-p_hⁿ‖	CPU/s	it	‖▽(u-u_hⁿ)‖	‖p-p_hⁿ‖
10^-2	2.422	3	3.369 47 × 10^-5	1.264 91 × 10^-4	2.015	2	3.369 47 × 10^-5	1.264 91 × 10^-4
10^-3	3.073	4	3.369 47 × 10^-5	1.264 91 × 10^-4	2.43	3	3.369 47 × 10^-5	1.264 91 × 10^-4
10^-4	5.584	8	3.369 47 × 10^-5	1.264 91 × 10^-4	4.227	6	3.369 47 × 10^-5	1.264 91 × 10^-4

算法 ν	Stokes迭代				Oseen迭代
算法 ν	CPU/s	it	‖▽(u-u_hⁿ)‖	‖p-p_hⁿ‖	CPU/s	it	‖▽(u-u_hⁿ)‖	‖p-p_hⁿ‖
10^-2	2.422	3	3.369 47 × 10^-5	1.264 91 × 10^-4	2.015	2	3.369 47 × 10^-5	1.264 91 × 10^-4
10^-3	3.073	4	3.369 47 × 10^-5	1.264 91 × 10^-4	2.43	3	3.369 47 × 10^-5	1.264 91 × 10^-4
10^-4	5.584	8	3.369 47 × 10^-5	1.264 91 × 10^-4	4.227	6	3.369 47 × 10^-5	1.264 91 × 10^-4

算法 α	Stokes迭代				Oseen迭代
算法 α	CPU/s	it	‖▽(u-u_hⁿ)‖	‖p-p_hⁿ‖	CPU/s	it	‖▽(u-u_hⁿ)‖	‖p-p_hⁿ‖
10^-2	1.898	2	3.369 47 × 10^-5	1.264 91 × 10^-4	1.824	2	3.369 47 × 10^-5	1.264 91 × 10^-4
10^-1	2.481	3	3.369 47 × 10^-5	1.264 91 × 10^-4	1.834	2	3.369 47 × 10^-5	1.264 91 × 10^-4
1	3.139	4	3.369 47 × 10^-5	1.264 91 × 10^-4	2.464	3	3.369 47 × 10^-5	1.264 91 × 10^-4
10	5.985	8	3.369 47 × 10^-5	1.264 91 × 10^-4	4.221	6	3.369 47 × 10^-5	1.264 91 × 10^-4
10²	不收敛				13.762	21	3.369 47 × 10^-5	1.264 91 × 10^-4