Chinese Journal of Computational Physics ›› 2024, Vol. 41 ›› Issue (4): 418-425.DOI: 10.19596/j.cnki.1001-246x.8731
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Yali WANG(
), Bo ZHENG, Yueqiang SHANG()
Received:2023-03-20
Online:2024-07-25
Published:2024-08-24
Contact:
Yueqiang SHANG
CLC Number:
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.
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URL: http://www.cjcp.org.cn/EN/10.19596/j.cnki.1001-246x.8731
| 算法 | 1/h | 1/H | CPU/s | ‖▽(u-uh)‖0 | ‖p-ph‖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 |
Table 1 Errors of approximate solutions by one-level algorithm, standard algorithm and algorithm 1 to algorithm 3
| 算法 | 1/h | 1/H | CPU/s | ‖▽(u-uh)‖0 | ‖p-ph‖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-uh)‖0 | ‖p-ph‖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 |
Table 2 Numerical results of algorithm 1 to algorithm 3 with different viscosity
| 算法 | ν | CPU/s | ‖▽(u-uh)‖0 | ‖p-ph‖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-uh)‖0 | ‖p-ph‖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 |
Table 3 Numerical results of algorithm 2 with different stabilization parameters
| 算法 | α | CPU/s | ‖▽(u-uh)‖0 | ‖p-ph‖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 |
Fig.1 Computed u1 by the standard two-level finite element algorithm (left) and algorithm 2 (right) with different values of ω (a) ω=10; (b) ω=102; (c) ω=103
Fig.2 Computed u1 by algorithm 2 with different values of ν (a)ν=1; (b)ν=10-1; (c)ν=10-2; (d)ν=10-3
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