April  2022, 27(4): 1853-1875. doi: 10.3934/dcdsb.2021112

Weak Galerkin method for the Stokes equations with damping

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

* Corresponding author: Qilong Zhai

Received  January 2021 Revised  March 2021 Published  April 2022 Early access  April 2021

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

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.

Citation: Hui Peng, Qilong Zhai. Weak Galerkin method for the Stokes equations with damping. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1853-1875. doi: 10.3934/dcdsb.2021112
References:
[1]

S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370.  doi: 10.1080/00029890.1980.11995034.

[2]

D. ArnoldF. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344.  doi: 10.1007/BF02576171.

[3]

S. AntontsevJ. $D\acute{l}az$ and H. de Oliveira, Stopping a viscous fluid by a feedback dissipative field. I. The stationary Stokes problem, J. Math. Fluid Mech., 6 (2004), 439-461.  doi: 10.1007/s00021-004-0106-x.

[4]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.

[5]

D. BreschB. Desjardins and C. Lin, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.

[6]

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Francaise Automat Informat. Recherche Op$\acute{e}$rationnelle S$\acute{e}$r. Rouge, 8 (1974), 129–151.

[7]

L. Chen and Y. Chen, Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods, J. Sci. Comput., 49 (2011), 383-401.  doi: 10.1007/s10915-011-9469-3.

[8]

W. ChenF. Wang and Y. Wang, Weak Galerkin method for the coupled Darcy-Stokes flow, IMA J. Numer. Anal., 36 (2016), 897-921.  doi: 10.1093/imanum/drv012.

[9]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[10]

X. HuL. Mu and X. Ye, A weak Galerkin finite element method for the Navier-Stokes equations, J. Comput. Appl. Math., 362 (2019), 614-625.  doi: 10.1016/j.cam.2018.08.022.

[11]

D. Liu and K. Li, Finite element analysis of the Stokes equations with damping, Math. Numer. Sin., 32 (2010), 433-448. 

[12]

M. LiD. Shi and Y. Dai, Stabilized low order finite elements for Stokes equations with damping, J. Math. Anal. Appl., 435 (2016), 646-660.  doi: 10.1016/j.jmaa.2015.10.040.

[13]

M. LiD. ShiZ. Li and H. Chen, Two-level mixed finite element methods for the Navier-Stokes equations with damping, J. Math. Anal. Appl., 470 (2019), 292-307.  doi: 10.1016/j.jmaa.2018.10.002.

[14]

L. MuJ. Wang and X. Ye, A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods, J. Comput. Phys., 273 (2014), 327-342.  doi: 10.1016/j.jcp.2014.04.017.

[15]

M. Mu and J. Xu, A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow, SIAM. J. Numer. Anal., 45 (2007), 1801-1813.  doi: 10.1137/050637820.

[16]

H. PengQ. ZhaiR. Zhang and S. Zhang, Weak Galerkin and continuous Galerkin coupled finite element methods for the Stokes-Darcy interface problem, Commun. Comput. Phys., 28 (2020), 1147-1175.  doi: 10.4208/cicp.oa-2019-0122.

[17]

D. Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations, Math. Comp., 79 (2010), 1303-1330.  doi: 10.1090/S0025-5718-10-02333-1.

[18]

Y. Shang and J. Qin, A finite element variational multiscale method based on two-grid discretization for the steady incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 300 (2016), 182-198.  doi: 10.1016/j.cma.2015.11.013.

[19]

D. Shi and Z. Yu, Superclose and superconvergence of finite element discretizations for the Stokes equations with damping, Appl. Math., 219 (2013), 7693-7698.  doi: 10.1016/j.amc.2013.01.057.

[20]

C. WangJ. WangR. Wang and R. Zhang, A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation, J. Comput. Appl. Math., 307 (2016), 346-366.  doi: 10.1016/j.cam.2015.12.015.

[21]

J. Wang and X. Ye, A weak Galerkin nite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.

[22]

J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.

[23]

J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155-174.  doi: 10.1007/s10444-015-9415-2.

[24]

R. Wang and R. Zhang, A weak Galerkin finite element method for the linear elasticity problem in mixed form, J. Comput. Math., 36 (2018), 469-491.  doi: 10.4208/jcm.1701-m2016-0733.

[25]

X. WangQ. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307 (2016), 13-24.  doi: 10.1016/j.cam.2016.04.031.

[26]

J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), 231-237.  doi: 10.1137/0915016.

[27]

Q. ZhaiH. XieR Zhang and Z Zhang, The weak Galerkin method for elliptic eigenvalue problems, Commun. Comput. Phys., 26 (2019), 160-191.  doi: 10.4208/cicp.OA-2018-0201.

[28]

T. Zhang and T. Lin, A stable weak Galerkin finite element method for Stokes problem, J. Comput. Appl. Math., 333 (2018), 235-246.  doi: 10.1016/j.cam.2017.10.042.

[29]

T. Zhang and T. Lin, An analysis of a weak Galerkin finite element method for stationary Navier-Stokes problems, J. Comput. Appl. Math., 362 (2019), 484-497.  doi: 10.1016/j.cam.2018.07.037.

[30]

Y. ZhangY. Qian and L. Mei, Discontinuous Galerkin methods for the Stokes equations with nonlinear damping term on general meshes, Comput. Math. Appl., 79 (2020), 2258-2275.  doi: 10.1016/j.camwa.2019.10.027.

[31]

Q. ZhangH XieR. Zhang and Z. Zhang, Acceleration of weak Galerkin methods for the Laplacian eigenvalue problem, J. Sci. Comput., 79 (2019), 914-934.  doi: 10.1007/s10915-018-0877-5.

show all references

References:
[1]

S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370.  doi: 10.1080/00029890.1980.11995034.

[2]

D. ArnoldF. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344.  doi: 10.1007/BF02576171.

[3]

S. AntontsevJ. $D\acute{l}az$ and H. de Oliveira, Stopping a viscous fluid by a feedback dissipative field. I. The stationary Stokes problem, J. Math. Fluid Mech., 6 (2004), 439-461.  doi: 10.1007/s00021-004-0106-x.

[4]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.

[5]

D. BreschB. Desjardins and C. Lin, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.

[6]

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Francaise Automat Informat. Recherche Op$\acute{e}$rationnelle S$\acute{e}$r. Rouge, 8 (1974), 129–151.

[7]

L. Chen and Y. Chen, Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods, J. Sci. Comput., 49 (2011), 383-401.  doi: 10.1007/s10915-011-9469-3.

[8]

W. ChenF. Wang and Y. Wang, Weak Galerkin method for the coupled Darcy-Stokes flow, IMA J. Numer. Anal., 36 (2016), 897-921.  doi: 10.1093/imanum/drv012.

[9]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[10]

X. HuL. Mu and X. Ye, A weak Galerkin finite element method for the Navier-Stokes equations, J. Comput. Appl. Math., 362 (2019), 614-625.  doi: 10.1016/j.cam.2018.08.022.

[11]

D. Liu and K. Li, Finite element analysis of the Stokes equations with damping, Math. Numer. Sin., 32 (2010), 433-448. 

[12]

M. LiD. Shi and Y. Dai, Stabilized low order finite elements for Stokes equations with damping, J. Math. Anal. Appl., 435 (2016), 646-660.  doi: 10.1016/j.jmaa.2015.10.040.

[13]

M. LiD. ShiZ. Li and H. Chen, Two-level mixed finite element methods for the Navier-Stokes equations with damping, J. Math. Anal. Appl., 470 (2019), 292-307.  doi: 10.1016/j.jmaa.2018.10.002.

[14]

L. MuJ. Wang and X. Ye, A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods, J. Comput. Phys., 273 (2014), 327-342.  doi: 10.1016/j.jcp.2014.04.017.

[15]

M. Mu and J. Xu, A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow, SIAM. J. Numer. Anal., 45 (2007), 1801-1813.  doi: 10.1137/050637820.

[16]

H. PengQ. ZhaiR. Zhang and S. Zhang, Weak Galerkin and continuous Galerkin coupled finite element methods for the Stokes-Darcy interface problem, Commun. Comput. Phys., 28 (2020), 1147-1175.  doi: 10.4208/cicp.oa-2019-0122.

[17]

D. Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations, Math. Comp., 79 (2010), 1303-1330.  doi: 10.1090/S0025-5718-10-02333-1.

[18]

Y. Shang and J. Qin, A finite element variational multiscale method based on two-grid discretization for the steady incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 300 (2016), 182-198.  doi: 10.1016/j.cma.2015.11.013.

[19]

D. Shi and Z. Yu, Superclose and superconvergence of finite element discretizations for the Stokes equations with damping, Appl. Math., 219 (2013), 7693-7698.  doi: 10.1016/j.amc.2013.01.057.

[20]

C. WangJ. WangR. Wang and R. Zhang, A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation, J. Comput. Appl. Math., 307 (2016), 346-366.  doi: 10.1016/j.cam.2015.12.015.

[21]

J. Wang and X. Ye, A weak Galerkin nite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.

[22]

J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.

[23]

J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155-174.  doi: 10.1007/s10444-015-9415-2.

[24]

R. Wang and R. Zhang, A weak Galerkin finite element method for the linear elasticity problem in mixed form, J. Comput. Math., 36 (2018), 469-491.  doi: 10.4208/jcm.1701-m2016-0733.

[25]

X. WangQ. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307 (2016), 13-24.  doi: 10.1016/j.cam.2016.04.031.

[26]

J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), 231-237.  doi: 10.1137/0915016.

[27]

Q. ZhaiH. XieR Zhang and Z Zhang, The weak Galerkin method for elliptic eigenvalue problems, Commun. Comput. Phys., 26 (2019), 160-191.  doi: 10.4208/cicp.OA-2018-0201.

[28]

T. Zhang and T. Lin, A stable weak Galerkin finite element method for Stokes problem, J. Comput. Appl. Math., 333 (2018), 235-246.  doi: 10.1016/j.cam.2017.10.042.

[29]

T. Zhang and T. Lin, An analysis of a weak Galerkin finite element method for stationary Navier-Stokes problems, J. Comput. Appl. Math., 362 (2019), 484-497.  doi: 10.1016/j.cam.2018.07.037.

[30]

Y. ZhangY. Qian and L. Mei, Discontinuous Galerkin methods for the Stokes equations with nonlinear damping term on general meshes, Comput. Math. Appl., 79 (2020), 2258-2275.  doi: 10.1016/j.camwa.2019.10.027.

[31]

Q. ZhangH XieR. Zhang and Z. Zhang, Acceleration of weak Galerkin methods for the Laplacian eigenvalue problem, J. Sci. Comput., 79 (2019), 914-934.  doi: 10.1007/s10915-018-0877-5.

Figure 1.  Left panel: The vectorgraph of velocity; Right panel: The streamlines of velocity
Figure 2.  Left panel: The vectorgraph of velocity; Right panel: The streamlines of velocity
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
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
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
(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
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
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
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
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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