Local and parallel finite element algorithms for the incompressible Navier-Stokes equations with damping

Eid Wassim; Yueqiang Shang

doi:10.3934/dcdsb.2022022

Article Contents

2022, Volume 27, Issue 11: 6823-6840. Doi: 10.3934/dcdsb.2022022

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Local and parallel finite element algorithms for the incompressible Navier-Stokes equations with damping

Eid Wassim and
Yueqiang Shang^,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

^* Corresponding author: Yueqiang Shang
^* Corresponding author: Yueqiang Shang

Received: April 2021

Revised: December 2021

Early access: February 2022

Published: November 2022

This work is supported by the Natural Science Foundation of Chongqing Municipality, China (No. cstc2021jcyj-msxmX1044).

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Abstract

Using two-grid discretizations strategy, we present some local and parallel finite element algorithms for simulating the steady incompressible Navier-Stokes equations with a nonlinear damping term. In these algorithms, we compute a solution of the Navier-Stokes system with a nonlinear damping term on a coarse grid, and then adjust the solution by some local and parallel procedures on overlapped fine grid subomains. With the use of theoretical tool of local a priori estimate of the finite element solution, we estimate the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we give some numerical results to verify the theoretical predictions and demonstrate the efficiency of the proposed algorithms.

Keywords:

Mathematics Subject Classification: Primary: 65N30, 65N15, 65N55; Secondary: 76D05, 76M10.

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Figure 1. A comparison of the computed velocities with different numbers of subdomains, different damping parameters $ \alpha $ and $ r $

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Figure 2. Comparison of $ u_1 $-velocity profile along the vertical centerline for the lid driven cavity flow at $ \nu = 0.01 $

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Figure 3. Comparison of $ u_2 $-velocity profile along the horizontal centerline for the lid driven cavity flow at $ \nu = 0.01 $

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Figure 4. A comparison of computed streamlines by Algorithm A1 with the reference data for lid-driven cavity flow: $ \nu = 0.01, \alpha = 1, r = 3 $

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Figure 5. A comparison of computed streamlines by Algorithm A1 with the reference data for lid-driven cavity flow: $ \nu = 0.01, \alpha = 10, r = 6 $

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Figure 6. Computed streamlines by Algorithm A1 with $ 4\times 4 $ sub-domains for lid-driven cavity flow at $ \nu = 0.01 $

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Figure 7. Computed streamlines by Algorithm A1 for lid-driven cavity flow with $ 2\times 2 $ subdomains, $ \nu = 1 $ and $ r = 3 $

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Figure 8. Computed streamlines by Algorithm A1 for lid-driven cavity flow with $ 4\times 4 $ subdomains, $ \nu = 1 $ and $ r = 3 $

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Table 1. Errors of the approximate solutions obtained by present parallel finite element algorithm (Algorithm A1)

$ h $	$ H $	$ it $	CPU(s)	$ \|\lVert \nabla u -\nabla u_h \|\rVert $$ _{0, \Omega} $	$ \|\lVert p - p_h \|\rVert $$ _{0, \Omega} $	Rate
1/27	1/18	3	0.923	0.000108646	0.000341526	-
1/64	1/32	3	2.634	2.06697e-05	6.35083e-05	1.94277
1/125	1/50	3	9.141	5.43566e-06	1.63504e-05	2.01912
1/216	1/72	3	22.593	1.97845e-06	5.53918e-06	1.94531

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Table 2. Errors of the approximate solutions by the standard two-level finite element method

$ h $	$ H $	$ it $	CPU(s)	$ \lVert \nabla u -\nabla u_h \rVert $$ _{0, \Omega} $	$ \lVert p - p_h \rVert $$ _{0, \Omega} $	Rate
1/27	1/18	2	0.801	0.000115269	0.000354182	-
1/64	1/32	2	4.255	2.05843e-05	6.30369e-05	1.99904
1/125	1/50	2	15.352	5.43494e-06	1.65252e-05	1.99732
1/216	1/72	2	36.999	1.92045e-06	5.53561e-06	1.9749

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Table 3. A comparision of the methods with different values of viscosity

Method	$ \nu $	$ it $	CPU(s)	$ \|\lVert\nabla u -\nabla u_h \|\rVert $$ _{0, \Omega} $	$ \|\lVert p - p_h \|\rVert $$ _{0, \Omega} $
Present	1	2	2.251	2.06697e-05	6.30688e-05
	0.1	3	2.959	2.06697e-05	6.35083e-05
	0.01	4	3.438	2.06698e-05	9.93789e-05
	0.001	8	5.852	2.06818e-05	0.000770386
Two-level method	1	2	4.025	2.05843e-05	6.30492e-05
	0.1	2	4.065	2.05843e-05	6.30369e-05
	0.01	3	4.681	2.05845e-05	6.30368e-05
	0.001	6	6.229	2.05969e-05	6.30368e-05

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