定常不可压Navier-Stokes方程的两水平grad-div稳定化有限元方法

doi:10.19596/j.cnki.1001-246x.8731

摘要/Abstract

摘要：

使用标准的混合有限元方法数值求解定常不可压Navier-Stokes方程所得速度解的精度常常受压力的影响。为了克服或减弱压力对速度精度的影响，本文将grad-div稳定化方法和两水平有限元方法相结合，提出数值求解定常不可压Navier-Stokes方程的两水平grad-div稳定化有限元方法。首先在粗网格上求解grad-div稳定化的非线性Navier-Stokes问题，然后在细网格上分别求解grad-div稳定化的Stokes型、Newton型和Oseen型的线性问题。最后给出数值算例验证两水平grad-div稳定化有限元方法的高效性。

关键词: Navier-Stokes方程, grad-div稳定化, 两水平方法, 有限元方法

Abstract:

Accuracy of the approximate velocity of the steady incompressible Navier-Stokes equations computed by the standard mixed finite element methods is often affected by the pressure. In order to circumvent or weaken the influence of pressure on the accuracy of the computed velocity, by combining grad-div stabilized method with two-level finite element method, this paper presents a kind of two-level grad-div stabilized finite element methods for solving the steady incompressible Navier-Stokes equations numerically. The basic idea of the methods is to first solve a grad-div stabilized nonlinear Navier-Stokes problems on a coarse grid, and then solve, respectively, Stokes-linearized, Newton-linearized and Oseen-linearized Navier-Stokes problem with grad-div stabilization on a fine grid. Numerical examples are given to verify the high efficiency of the two-level grad-div stabilized finite element methods.

Key words: Navier-Stokes equations, grad-div stabilization, two-level method, finite element method

中图分类号:

O241.82

王雅莉, 郑波, 尚月强. 定常不可压Navier-Stokes方程的两水平grad-div稳定化有限元方法[J]. 计算物理, 2024, 41(4): 418-425.

Yali WANG, Bo ZHENG, Yueqiang SHANG. Two-level Grad-div Stabilized Finite Element Methods for Steady Incompressible Navier-Stokes Equations[J]. Chinese Journal of Computational Physics, 2024, 41(4): 418-425.

图/表 5

参考文献 20

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算法	1/h	1/H	CPU/s	‖▽(u-u^h)‖₀	‖p-p^h‖₀	速度收敛阶	压力收敛阶
单水平算法	8		0.077	1.364 7×10^-1	2.468 2×10^-2
	27		0.715	3.977 0×10^-3	2.031 3×10^-3	2.906 64	2.053 12
	64		3.071	3.025 5×10^-4	3.596 1×10^-4	2.984 79	2.006 16
	125		14.536	4.071 3×10^-5	9.419 0×10^-5	2.996 16	2.001 27
标准算法	8	4	0.181	9.069 4×10^-1	2.497 8×10^-2
	27	9	0.305	2.424 1×10^-2	2.031 3×10^-3	2.977 67	2.062 92
	64	16	1.65	1.832 2×10^-3	3.596 1×10^-4	2.992 35	2.006 15
	125	25	7.258	2.465 5×10^-4	9.419 0×10^-5	2.996 12	2.001 27
算法1	8	4	0.063	1.521 3×10^-1	2.471 8×10^-2
	27	9	0.296	3.978 5×10^-3	2.031 3×10^-3	2.995 61	2.054 31
	64	16	1.664	3.025 6×10^-4	3.596 1×10^-4	2.985 23	2.006 16
	125	25	7.494	4.071 3×10^-5	9.419 0×10^-5	2.996 17	2.001 27
算法2	8	4	0.065	1.510 8×10^-1	2.471 4×10^-2
	27	9	0.319	3.978 1×10^-3	2.031 3×10^-3	2.989 99	2.05417
	64	16	1.811	3.025 6×10^-4	3.596 1×10^-4	2.985 13	2.00616
	125	25	8.058	4.071 3×10^-5	9.419 0×10^-5	2.996 17	2.00127
算法3	8	4	0.062	1.364 4×10^-1	2.468 3×10^-2
	27	9	0.29	3.977 0×10^-3	2.031 3×10^-3	2.906 42	2.053 15
	64	16	1.62	3.025 5×10^-4	3.596 1×10^-4	2.984 79	2.006 16
	125	25	7.275	4.071 3×10^-5	9.419 0×10^-5	2.996 16	2.001 27

算法	1/h	1/H	CPU/s	‖▽(u-u^h)‖₀	‖p-p^h‖₀	速度收敛阶	压力收敛阶
单水平算法	8		0.077	1.364 7×10^-1	2.468 2×10^-2
	27		0.715	3.977 0×10^-3	2.031 3×10^-3	2.906 64	2.053 12
	64		3.071	3.025 5×10^-4	3.596 1×10^-4	2.984 79	2.006 16
	125		14.536	4.071 3×10^-5	9.419 0×10^-5	2.996 16	2.001 27
标准算法	8	4	0.181	9.069 4×10^-1	2.497 8×10^-2
	27	9	0.305	2.424 1×10^-2	2.031 3×10^-3	2.977 67	2.062 92
	64	16	1.65	1.832 2×10^-3	3.596 1×10^-4	2.992 35	2.006 15
	125	25	7.258	2.465 5×10^-4	9.419 0×10^-5	2.996 12	2.001 27
算法1	8	4	0.063	1.521 3×10^-1	2.471 8×10^-2
	27	9	0.296	3.978 5×10^-3	2.031 3×10^-3	2.995 61	2.054 31
	64	16	1.664	3.025 6×10^-4	3.596 1×10^-4	2.985 23	2.006 16
	125	25	7.494	4.071 3×10^-5	9.419 0×10^-5	2.996 17	2.001 27
算法2	8	4	0.065	1.510 8×10^-1	2.471 4×10^-2
	27	9	0.319	3.978 1×10^-3	2.031 3×10^-3	2.989 99	2.05417
	64	16	1.811	3.025 6×10^-4	3.596 1×10^-4	2.985 13	2.00616
	125	25	8.058	4.071 3×10^-5	9.419 0×10^-5	2.996 17	2.00127
算法3	8	4	0.062	1.364 4×10^-1	2.468 3×10^-2
	27	9	0.29	3.977 0×10^-3	2.031 3×10^-3	2.906 42	2.053 15
	64	16	1.62	3.025 5×10^-4	3.596 1×10^-4	2.984 79	2.006 16
	125	25	7.275	4.071 3×10^-5	9.419 0×10^-5	2.996 16	2.001 27

算法	ν	CPU/s	‖▽(u-u^h)‖₀	‖p-p^h‖₀
算法1	1	7.276	2.246 1×10^-6	9.419 0×10^-5
	10^-1	7.319	1.268 4×10^-5	9.419 0×10^-5
	10^-2	7.494	4.071 3×10^-5	9.419 0×10^-5
	10^-3	7.466	3.394 0×10^-4	9.419 0×10^-5
	10^-4	8.023	5.379 9×10^-3	9.419 4×10^-5
算法2	1	7.846	2.246 1×10^-6	9.419 0×10^-5
	10^-1	7.855	1.268 4×10^-5	9.419 0×10^-5
	10^-2	8.058	4.071 3×10^-5	9.419 0×10^-5
	10^-3	8.043	3.394 0×10^-4	9.419 0×10^-5
	10^-4	8.599	5.260 0×10^-3	9.419 4×10^-5
算法3	1	7.247	2.246 1×10^-6	9.419 0×10^-5
	10^-1	7.085	1.268 4×10^-5	9.419 0×10^-5
	10^-2	7.275	4.071 3×10^-5	9.419 0×10^-5
	10^-3	7.22	3.390 9×10^-4	9.419 0×10^-5
	10^-4	7.778	3.177 4×10^-3	9.419 0×10^-5

算法	ν	CPU/s	‖▽(u-u^h)‖₀	‖p-p^h‖₀
算法1	1	7.276	2.246 1×10^-6	9.419 0×10^-5
	10^-1	7.319	1.268 4×10^-5	9.419 0×10^-5
	10^-2	7.494	4.071 3×10^-5	9.419 0×10^-5
	10^-3	7.466	3.394 0×10^-4	9.419 0×10^-5
	10^-4	8.023	5.379 9×10^-3	9.419 4×10^-5
算法2	1	7.846	2.246 1×10^-6	9.419 0×10^-5
	10^-1	7.855	1.268 4×10^-5	9.419 0×10^-5
	10^-2	8.058	4.071 3×10^-5	9.419 0×10^-5
	10^-3	8.043	3.394 0×10^-4	9.419 0×10^-5
	10^-4	8.599	5.260 0×10^-3	9.419 4×10^-5
算法3	1	7.247	2.246 1×10^-6	9.419 0×10^-5
	10^-1	7.085	1.268 4×10^-5	9.419 0×10^-5
	10^-2	7.275	4.071 3×10^-5	9.419 0×10^-5
	10^-3	7.22	3.390 9×10^-4	9.419 0×10^-5
	10^-4	7.778	3.177 4×10^-3	9.419 0×10^-5

算法	α	CPU/s	‖▽(u-u^h)‖₀	‖p-p^h‖₀
算法2	0	7.258	2.465 5×10^-4	9.419 0×10^-5
	10^-2	7.296	1.268 4×10^-4	9.419 0×10^-5
	10^-1	7.275	4.071 3×10^-5	9.419 0×10^-5
	1	7.186	3.390 9×10^-5	9.419 0×10^-5
	10	7.247	3.177 4×10^-5	9.419 0×10^-5