Weak Galerkin method for the Stokes equations with damping

Hui Peng; Qilong Zhai

doi:10.3934/dcdsb.2021112

Article Contents

2022, Volume 27, Issue 4: 1853-1875. Doi: 10.3934/dcdsb.2021112

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Weak Galerkin method for the Stokes equations with damping

Hui Peng and
Qilong Zhai^,

Department of Mathematics, Jilin University, Changchun, 130012, China

^* Corresponding author: Qilong Zhai
^* Corresponding author: Qilong Zhai

Received: December 31, 2020

Revised: February 28, 2021

Early access: April 2021

Published: April 2022

The research is supported by China Natural National Science Foundation(12001232, 11901015)

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Abstract

In this paper, we introduce the weak Galerkin (WG) finite element method for the Stokes equations with damping. We establish the WG numerical scheme on general meshes and prove the well-posedness of the scheme. Optimal error estimates for the velocity and pressure are derived. Furthermore, in order to accelerate the WG algorithm, we present a two-level method and give the corresponding error estimates. Finally, some numerical examples are reported to validate the theoretical analysis.

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Mathematics Subject Classification: Primary, 65N30, 65N15, 65N12; Secondary, 35B45, 35J50.

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Figure 1. Left panel: The vectorgraph of velocity; Right panel: The streamlines of velocity

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Figure 2. Left panel: The vectorgraph of velocity; Right panel: The streamlines of velocity

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Table 1. The errors and the order of convergence by one-level method, for $ (7.1) $

h	$ {\|\!\|\!\|} Q_h \boldsymbol{u}- \boldsymbol{u}_{h} {\|\!\|\!\|} $	order	$ \\|Q_0 {{\mathbf{u}}}- {{\mathbf{u}}}_0\\|_0 $	order	$ \\|Q_h p-p_{h}\\|_0 $	order
$ \frac{1}{16} $	5.3818$ E- $01	-	1.8464$ E- $02	-	1.3715$ E- $01	-
$ \frac{1}{25} $	3.4671$ E- $01	0.99	7.6837$ E- $03	1.96	6.5631$ E- $02	1.65
$ \frac{1}{64} $	1.3613$ E- $01	0.99	1.1871$ E- $03	1.99	1.2856$ E- $02	1.73
$ \frac{1}{81} $	1.0761$ E- $01	1.0	7.4195$ E- $04	2.0	8.4724$ E- $03	1.77

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Table 2. The errors and the order of convergence by two-level method, for $ (7.1) $

(H, h)	$ {\|\!\|\!\|} Q_h \boldsymbol{u}- \boldsymbol{u}_{h} {\|\!\|\!\|} $	order	$ \\|Q_0 {{\mathbf{u}}}- {{\mathbf{u}}}_0\\|_0 $	order	$ \\|Q_h p-p_{h}\\|_0 $	order
($ \frac14 $, $ \frac{1}{16} $)	5.4020$ E- $01	-	1.8559$ E- $02	-	1.2727$ E- $01	-
($ \frac15 $, $ \frac{1}{25}) $	3.4732$ E- $01	0.99	7.7109$ E- $03	1.97	6.1089$ E- $02	1.64
($ \frac{1}{8} $, $ \frac{1}{64}) $	1.3618$ E- $01	1.0	1.1916$ E- $03	1.99	1.2175$ E- $02	1.72
($ \frac{1}{9} $, $ \frac{1}{81}) $	1.0763$ E- $01	1.0	7.4504$ E- $04	1.99	8.0646$ E- $03	1.75

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Table 3. The comparison of the CPU times (unit:s) for the one-level and two-level methods for $ (7.1) $

h	one-level	two level
($ \frac14 $, $ \frac{1}{16} $)	6.921	9.352
($ \frac{1}{5} $, $ \frac{1}{25}) $	19.923	24.005
($ \frac{1}{8} $, $ \frac{1}{64}) $	319.522	277.700
($ \frac{1}{9} $, $ \frac{1}{81}) $	974.475	691.726

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Table 4. The comparison of the CPU times (unit:s) for the one-level and two-level methods with different pairs of parameters, for $ (7.2) $

	one-level	two level
$ \alpha=1 $, $ r=3 $, $ \mu=1 $	308.325	278.230
$ \alpha=5 $, $ r=3 $, $ \mu=1 $	427.230	276.781
$ \alpha=10 $, $ r=3 $, $ \mu=1 $	425.516	277.020
$ \alpha=1 $, $ r=4 $, $ \mu=1 $	314.077	276.544
$ \alpha=1 $, $ r=5 $, $ \mu=1 $	371.219	277.493
$ \alpha=1 $, $ r=3 $, $ \mu=0.1 $	476.156	285.622

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