Finite Element Iterative Algorithms for Steady Stokes Equations with Nonlinear Damping Term

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

Abstract

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

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.

Figures/Tables 9

References 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