American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021112

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

Received  January 2021 Revised  March 2021 Published  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.

Keywords: Weak Galerkin finite element methods, weak gradient, nonlinear term.
Mathematics Subject Classification: Primary, 65N30, 65N15, 65N12; Secondary, 35B45, 35J50.
Citation: Hui Peng, Qilong Zhai. Weak Galerkin method for the Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
Table Options

Download as excel

(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
Table Options

Download as excel

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
Table Options

Download as excel

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
[1]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
[2]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096
[3]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126
[4]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097
[5]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[6]

Guanrong Li, Yanping Chen, Yunqing Huang. A hybridized weak Galerkin finite element scheme for general second-order elliptic problems. Electronic Research Archive, 2020, 28 (2) : 821-836. doi: 10.3934/era.2020042
[7]

Xiu Ye, Shangyou Zhang. A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4131-4145. doi: 10.3934/dcdsb.2020277
[8]

Ruishu Wang, Lin Mu, Xiu Ye. A locking free Reissner-Mindlin element with weak Galerkin rotations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 351-361. doi: 10.3934/dcdsb.2018086
[9]

Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295
[10]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024
[11]

Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131
[12]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29 (3) : 2375-2389. doi: 10.3934/era.2020120
[13]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116
[14]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53
[15]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
[16]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[17]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[18]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795
[19]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689
[20]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (41)
  • HTML views (71)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences