A Parallel Two-level Stablized Finite Element Algorithm for Incompressible Flows

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

Abstract

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

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.

Figures/Tables 11

References 22

1	OLSHANSKII M , LUBE G , HEISTER T , et al. Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198 (49-52): 3975- 3988.
2	LIN M . Pressure robust weak Galerkin finite element methods for Stokes problems[J]. SIAM Journal on Scientific Computing, 2020, 42 (3): 608- 629. DOI
3	曹念铮, 游仁然. 求解振荡及非定常Stokes流动的边界元方法[J]. 计算物理, 1990, 7 (3): 257- 267.
4	ZHANG S . Divergence-free finite elements on tetrahedral grids for k≥6[J]. Mathematics of Computation, 2011, 80 (274): 669- 695. DOI
5	JESUS C , BERNARDO C , DOMINIK S . Hybridized globally divergence-free LDG methods Part I: The Stokes problem[J]. Mathematics of Computation, 2005, 75 (254): 533- 563. DOI
6	FRUTOS J , GARCIA-ARCHILLA B , JOHN V , et al. Analysis of the Grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements[J]. Advances in Computational Mathematics, 2018, 44 (1): 195- 225.
7	OLSHANSKII M , LUBE G , HEISTER T , et al. Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198 (49-52): 3975- 3988.
8	LINKE A , REBHOLZ L . On a reduced sparsity stabilization of Grad-div type for incompressible flow problems[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 261 (15): 142- 153.
9	AHMED N . On the Grad-div stabilization for the steady Oseen and Navier-Stokes equations[J]. Calcolo, 2016, 54 (1): 1- 31.
10	SHANG Y . A parallel two-level finite element method for the Navier-Stokes equations[J]. Applied Mathematics and Mechanics, 2010, 84 (11): 103- 116.
11	周春华. 流动数值模拟中一种并行自适应有限元算法[J]. 计算物理, 2006, 23 (4): 412- 418.
12	LAWRENCE C E. Partial differential equations[M]. 2nd ed. American Mathmatical Society, 2010.
13	HE Y , XU J , ZHOU A H . Local and parallel finite element algorithms for the Navier-Stokes problem[J]. Journal of Computational Mathematics, 2006, 24 (3): 227- 238.
14	JENKINS E W , JOHN V , LINKE A , et al. On the parameter choice in Grad-div stabilization for the Stokes equations[J]. Advances in Computation Mathematics, 2014, 40 (2): 491- 516.
15	XU J , ZHOU A . Local and parallel finite element algorithms based on two-grid discretizations[J]. Mathematics of Computation, 2000, 69 (231): 881- 909.
16	YUAN L , RONG A . Penalty finite element method for Navier-Stokes equations with nonlinear slip boundary conditions[J]. International Journal for Numerical Methods in Fluids, 2012, 69 (3): 550- 566.
17	SHANG Y , JIN Q . Parallel finite element variational multiscale algorithms for incompressible flow at high Reynolds numbers[J]. Applied Numerical Mathematics, 2017, 117 (7): 1- 21.
18	XU J , ZHOU A . Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems[J]. Advances in Computational Mathematics, 2001, 14 (4): 293- 327.
19	HECHT F . New development in freefem++[J]. Journal of Numerical Mathematics, 2012, 20 (3-4): 251- 265.
20	OLSHANSKⅡ M , REUSKEN A . Grad-div stablilization for Stokes equations[J]. Mathematics of Computation, 2004, 73 (248): 1699- 1718.
21	HE Y , XU J , ZHOU A , et al. Local and parallel finite element algorithms for the Stokes problem[J]. Numerische Mathematik, 2008, 109 (3): 415- 434.
22	ZHENG B , ZHOU K , SHANG Y . Parallel pressure projection stabilized finite element algorithms based on two-grid discretizations for incompressible flows[J]. International Journal of Computer Mathematics, 2020, (8): 1563- 1585.

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