不可压缩流的并行两水平稳定有限元算法

doi:10.19596/j.cnki.1001-246x.8411

摘要/Abstract

摘要：

提出一种数值求解定常不可压缩Stokes方程的并行两水平Grad-div稳定有限元算法。首先在粗网格中求解Grad-div稳定化的全局解, 再在相互重叠的细网格子区域上并行纠正。通过对稳定化参数、粗细网格尺寸恰当的选取, 该方法可得到最优收敛率, 数值结果验证了算法的高效性。

关键词: Stokes方程, 有限元方法, 两水平方法, Grad-div稳定项, 并行算法

Abstract:

A parallel two-level Grad-div stabilized finite element algorithm for steady incompressible Stokes equations is proposed. Basic idea of the algorithm is to solve global Grad-div stabilized solution in a coarse mesh firstly, and then correct it in parallel on overlapping fine mesh subdomains. With reasonable selection of stabilization parameters and mesh sizes, an optimal convergence rate can be obtained. Numerical results verify efficiency of the algorithm.

Key words: Stokes equations, finite element algorithm, two-level method, Grad-div stabilization, parallel algorithm

朱家莉, 尚月强. 不可压缩流的并行两水平稳定有限元算法[J]. 计算物理, 2022, 39(3): 309-317.

Jiali ZHU, Yueqiang SHANG. A Parallel Two-level Stablized Finite Element Algorithm for Incompressible Flows[J]. Chinese Journal of Computational Physics, 2022, 39(3): 309-317.

图/表 11

参考文献 22

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h	H	CPU/s	$\frac{{{{\left\\| {\|\nabla \left( {{\mathit{\boldsymbol{u}}} - {{\mathit{\boldsymbol{u}}}^h}} \right)\|} \right\\|}_{0, \mathit{\Omega} }}}}{{{{\left\\| {\nabla {\mathit{\boldsymbol{u}}}} \right\\|}_{0, \mathit{\Omega} }}}}$	$\frac{\left\\|p-p^h\right\\|_{0, \mathit{\Omega}}}{\\|p\\|_{0, \mathit{\Omega}}}$	收敛阶
1/27	1/18	0.3	0.002 231 99	0.002 671 32
1/64	1/32	1.26	0.000 445 662	0.000 423 465	1.895 44
1/125	1/50	4.19	0.000 118 793	0.000 105 77	1.984 19

h	H	CPU/s	$\frac{{{{\left\\| {\|\nabla \left( {{\mathit{\boldsymbol{u}}} - {{\mathit{\boldsymbol{u}}}^h}} \right)\|} \right\\|}_{0, \mathit{\Omega} }}}}{{{{\left\\| {\nabla {\mathit{\boldsymbol{u}}}} \right\\|}_{0, \mathit{\Omega} }}}}$	$\frac{\left\\|p-p^h\right\\|_{0, \mathit{\Omega}}}{\\|p\\|_{0, \mathit{\Omega}}}$	收敛阶
1/27	1/18	0.3	0.002 231 99	0.002 671 32
1/64	1/32	1.26	0.000 445 662	0.000 423 465	1.895 44
1/125	1/50	4.19	0.000 118 793	0.000 105 77	1.984 19

h	H	CPU/s	$\frac{{{{\left\\| {\|\nabla \left({\mathit{\boldsymbol{u}} - {\mathit{\boldsymbol{u}}^h}} \right)\|} \right\\|}_{0, \mathit{\Omega} }}}}{{{{\left\\| {\nabla \mathit{\boldsymbol{u}}} \right\\|}_{0, \mathit{\Omega} }}}}$	$\frac{\left\\|p-p^h\right\\|_{0, \mathit{\Omega}}}{\\|p\\|_{0, \mathit{\Omega}}}$	收敛阶
1/27	1/18	0.33	0.008 306 41	0.002 654 18
1/64	1/32	1.24	0.001 451 32	0.000 407 09	2.026 32
1/125	1/50	4.214	0.000 307 177	0.000 127 241	2.297 95

h	H	CPU/s	$\frac{{{{\left\\| {\|\nabla \left({\mathit{\boldsymbol{u}} - {\mathit{\boldsymbol{u}}^h}} \right)\|} \right\\|}_{0, \mathit{\Omega} }}}}{{{{\left\\| {\nabla \mathit{\boldsymbol{u}}} \right\\|}_{0, \mathit{\Omega} }}}}$	$\frac{\left\\|p-p^h\right\\|_{0, \mathit{\Omega}}}{\\|p\\|_{0, \mathit{\Omega}}}$	收敛阶
1/27	1/18	0.33	0.008 306 41	0.002 654 18
1/64	1/32	1.24	0.001 451 32	0.000 407 09	2.026 32
1/125	1/50	4.214	0.000 307 177	0.000 127 241	2.297 95

h	H	CPU/s	$\frac{\left\\|\nabla\left(\boldsymbol{u}-\boldsymbol{u}^h\right)\right\\|_{0, \mathit{\Omega}}}{\\|\nabla \boldsymbol{u}\\|_{0, \mathit{\Omega}}}$	$\frac{\left\\|p-p^h\right\\|_{0, \mathit{\Omega}}}{\\|p\\|_{0, \mathit{\Omega}}}$	收敛阶
1/27	1/18	0.66	0.002 244 77	0.002 268 77
1/64	1/32	3.45	0.000 433 208	0.000 401 801	1.915 99
1/125	1/50	14.652	0.000 117 007	0.000 106 459	1.958 06