|
|
- 2017
Navier-Stokes方程的回溯两水平有限元变分多尺度方法
|
Abstract:
基于两重网格离散和回溯两水平方法，提出了一种求解大雷诺数不可压缩流定常Navier-Stokes方程的回溯两水平有限元变分多尺度方法.其基本思想是：首先在一粗网格上求解带有亚格子模型稳定项的Navier-Stokes方程，然后在细网格上求解一个亚格子模型稳定化的线性Oseen问题，最后又回到粗网格上求解全线性化校正问题.通过适当的稳定化参数和粗细网格尺寸的选取，这些算法能取得最优渐近收敛阶.我们通过数值模拟，验证了其高效性.
Based on the two-grid discretization method and the two-level method with backtracking, a two-level variational multiscale algorithm with backtracking finite element for the stationary Navier-Stokes equations at high Reynolds numbers is proposed in this paper. The key idea of our algorithm is as follows:first, to solve a fully nonlinear Navier-Stokes equation with a subgrid stabilization term on a coarse grid; next, to solve a subgrid stabilized linear fine grid problem based on one step of Oseen iteration; and finally, to correct on the coarse grid with full linearization. The theoretical and numerical results show that with suitable scalings of algorithmic parameters, this algorithm of ours can yield an optimal convergence rate. Numerical tests are made to verify the efficiency of the method
| [1] | XU J C. A Novel Two-Grid Method for Semilinear Elliptic Equations[J]. SIAM J Sci Comput, 1994, 15(1): 231-237. DOI:10.1137/0915016 |
| [2] | LAYTON W, LENFERINKW. A Multilevel Mesh Independence Principle for the Navier-Stokes Equations[J]. SIAM J Numer Anal, 1996, 33(1): 17-30. DOI:10.1137/0733002 |
| [3] | HE Y N, LI K T. Two-Level Stabilized Finite Element Method for the Steady Navier-Stokes Problem[J]. Computing, 2005, 74(4): 337-351. DOI:10.1007/s00607-004-0118-7 |
| [4] | LAYTON W, TOBISKA L. A Two-Level Method with Backtracking for the Navier-Stokes Equations[J]. SIAMJ Numer Anal, 1998, 35(5): 2035-2054. DOI:10.1137/S003614299630230X |
| [5] | SHANG Y Q. A Parallel Two-Level Finite Element Variational Multiscale Method for the Navier-Stokes Equations[J]. Nonlinear Anal, 2013, 84: 103-116. DOI:10.1016/j.na.2013.02.009 |
| [6] | SHANG Y Q, QIN J. A Two-Parameter Stabilized Finite Element Method for Incompressible Flows[J]. NumericalMethods for PDEs, 2017, 33(2): 1-20. |
| [7] | GHIA U, GHIA K, SHIN C. High-Re Solutions for Incompressibleflow Using the Navier-Stokes Equationsand a Multigird Method[J]. J Comput Phys, 1982, 48(3): 387-411. DOI:10.1016/0021-9991(82)90058-4 |
| [8] | ZHENG H B, HOU Y R, SHI F, et al. A Finite Element Variational Multiscale Method for Incompressible Flows Based on Two Local Gauss Integrations[J]. J Comput Phys, 2009, 228(16): 5961-5977. DOI:10.1016/j.jcp.2009.05.006 |
| [9] | ZHANG Y, HE Y N. Assessment of Subgrid-Scale Models for the Incompressible Navier-Stokes Equations[J]. J Comput Appl Math, 2010, 234(2): 593-604. DOI:10.1016/j.cam.2009.12.051 |
| [10] | SHANG Y Q. A Two-Level Subgrid Stabilized Oseen Iterative Method for the Steady Navier-Stokes Equations[J]. J Comput Phys, 2013, 233(1): 210-226. |
| [11] | HECHT F. New Development in Freefem++[J]. J Numer Math, 2012, 20(3-4): 251-265. |
