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Weak Galerkin method for the Stokes equations with damping
Department of Mathematics, Jilin University, Changchun, 130012, China |
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.
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. Arnold, F. Brezzi and M. Fortin,
A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344.
doi: 10.1007/BF02576171. |
[3] |
S. Antontsev, J. $D\acute{l}az$ 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. Bresch, B. 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. Chen, F. 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. Hu, L. 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. Li, D. 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. Li, D. Shi, Z. 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. Mu, J. 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. Peng, Q. Zhai, R. 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. Wang, J. Wang, R. 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. Wang, Q. 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. Zhai, H. Xie, R 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. Zhang, Y. 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. Zhang, H Xie, R. 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. Arnold, F. Brezzi and M. Fortin,
A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344.
doi: 10.1007/BF02576171. |
[3] |
S. Antontsev, J. $D\acute{l}az$ 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. Bresch, B. 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. Chen, F. 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. Hu, L. 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. Li, D. 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. Li, D. Shi, Z. 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. Mu, J. 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. Peng, Q. Zhai, R. 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. Wang, J. Wang, R. 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. Wang, Q. 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. Zhai, H. Xie, R 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. Zhang, Y. 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. Zhang, H Xie, R. 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. |


h | order | order | order | |||
5.3818 |
- | 1.8464 |
- | 1.3715 |
- | |
3.4671 |
0.99 | 7.6837 |
1.96 | 6.5631 |
1.65 | |
1.3613 |
0.99 | 1.1871 |
1.99 | 1.2856 |
1.73 | |
1.0761 |
1.0 | 7.4195 |
2.0 | 8.4724 |
1.77 |
h | order | order | order | |||
5.3818 |
- | 1.8464 |
- | 1.3715 |
- | |
3.4671 |
0.99 | 7.6837 |
1.96 | 6.5631 |
1.65 | |
1.3613 |
0.99 | 1.1871 |
1.99 | 1.2856 |
1.73 | |
1.0761 |
1.0 | 7.4195 |
2.0 | 8.4724 |
1.77 |
(H, h) | order | order | order | |||
( |
5.4020 |
- | 1.8559 |
- | 1.2727 |
- |
( |
3.4732 |
0.99 | 7.7109 |
1.97 | 6.1089 |
1.64 |
( |
1.3618 |
1.0 | 1.1916 |
1.99 | 1.2175 |
1.72 |
( |
1.0763 |
1.0 | 7.4504 |
1.99 | 8.0646 |
1.75 |
(H, h) | order | order | order | |||
( |
5.4020 |
- | 1.8559 |
- | 1.2727 |
- |
( |
3.4732 |
0.99 | 7.7109 |
1.97 | 6.1089 |
1.64 |
( |
1.3618 |
1.0 | 1.1916 |
1.99 | 1.2175 |
1.72 |
( |
1.0763 |
1.0 | 7.4504 |
1.99 | 8.0646 |
1.75 |
h | one-level | two level |
( |
6.921 | 9.352 |
( |
19.923 | 24.005 |
( |
319.522 | 277.700 |
( |
974.475 | 691.726 |
h | one-level | two level |
( |
6.921 | 9.352 |
( |
19.923 | 24.005 |
( |
319.522 | 277.700 |
( |
974.475 | 691.726 |
one-level | two level | |
308.325 | 278.230 | |
427.230 | 276.781 | |
425.516 | 277.020 | |
314.077 | 276.544 | |
371.219 | 277.493 | |
476.156 | 285.622 |
one-level | two level | |
308.325 | 278.230 | |
427.230 | 276.781 | |
425.516 | 277.020 | |
314.077 | 276.544 | |
371.219 | 277.493 | |
476.156 | 285.622 |
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