American Institute of Mathematical Sciences

Advanced
January  2019, 24(1): 351-361. doi: 10.3934/dcdsb.2018086

A locking free Reissner-Mindlin element with weak Galerkin rotations

Ruishu Wang 1, , Lin Mu 2,3,, and  Xiu Ye 2,3,

1. 

Department of Mathematics, Jilin University, Changchun, China
2. 

Computer Science and Mathematics Division Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA
3. 

Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA

* Corresponding author: Lin Mu

Received  October 2016 Revised  October 2017 Published  January 2019 Early access  March 2018

Fund Project: The research of second author was supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under award number ERKJE45; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC., for the U.S. Department of Energy under Contract DE-AC05-00OR22725. This research of third author was supported in part by National Science Foundation Grant DMS-1620016.

A locking free finite element method is developed for the Reissner-Mindlin equations in their primary form. In this method, the transverse displacement is approximated by continuous piecewise polynomials of degree $k+1$ and the rotation is approximated by weak Galerkin elements of degree $k$ for $k≥1$. A uniform convergence in thickness of the plate is established for this finite element approximation. The numerical examples demonstrate locking free of the method.

Keywords: Weak Galerkin, finite element methods, the Reissner-Mindlin plate.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: 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
References:
[1]

D. Arnold, F. Brezzi and D. Marini, A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate, J. Sci. Comput., 22 (2005), 25-45.  doi: 10.1007/s10915-004-4134-8.  Google Scholar

[2]

D. Arnold, F. Brezzi, R. Falk and D. Marini, Locking-free Reissner-Mindlin elements without reduced integration, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3660-3671.  doi: 10.1016/j.cma.2006.10.023.  Google Scholar

[3]

D. Arnold and R. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal., 26 (1989), 1276-1290.  doi: 10.1137/0726074.  Google Scholar

[4]

D. N. Arnold and X. Liu, Interior estimates for a low order finite element method for the Reissner-Mindlin plate model, Adv. in Comp. Math., 7 (1997), 337-360.  doi: 10.1023/A:1018907205385.  Google Scholar

[5]

S. Brenner, Korn's inequalities for piecewise $H^1$ vector fields, Math. Comput., 73 (2004), 1067-1087.   Google Scholar

[6]

F. Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plates, Math. Comp., 47 (1986), 151-158.  doi: 10.1090/S0025-5718-1986-0842127-7.  Google Scholar

[7]

F. Brezzi, K. Bathe and M. Fortin, Mixed interpolated elements for Reissner-Mindlin plates, Int. J. Numer. Methods Eng., 28 (1989), 1787-1801.  doi: 10.1002/nme.1620280806.  Google Scholar

[8]

G. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Mathematical Models and Methods in Applied Sciences, 1 (1991), 125-151.  doi: 10.1142/S0218202591000083.  Google Scholar

[9]

D. Chapelle and R. Stenberg, An optimal low-order locking-free finite element method for Reissner-Mindlin plates, Mathematical Models and Methods in Applied Sciences, 8 (1998), 407-430.  doi: 10.1142/S0218202598000172.  Google Scholar

[10]

R. Duran and E. Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp., 58 (1992), 561-573.  doi: 10.1090/S0025-5718-1992-1106965-0.  Google Scholar

[11]

R. Falk and T. Tu, Locking-free finite elements for the Reissner-Mindlin plate, Math. Comp., 69 (2000), 911-928.   Google Scholar

[12]

P. Hansbo, D. Heintz and M. Larson, A finite element method with discontinuous rotations for the Mindlin-Reissner plate model, Comput. Methods Appl. Mech. Engrg., 200 (2011), 638-648.  doi: 10.1016/j.cma.2010.09.009.  Google Scholar

[13]

C. Lovadina and D. Marini, Nonconforming locking-free finite elements for Reissner-Mindlin plates, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3448-3460.  doi: 10.1016/j.cma.2005.06.025.  Google Scholar

[14]

L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method with polynomial reduction, Journal of Computational and Applied Mathematics, 285 (2015), 45-58.  doi: 10.1016/j.cam.2015.02.001.  Google Scholar

[15]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element method for second-order elliptic problems on polytopal meshes, International Journal of Numerical Analysis and Modeling, 12 (2015), 31-53.   Google Scholar

[16]

R. Pierre, Convergence Properties and Numerical Approximation of the Solution of the Mindlin Plate Bending Problem, Math Comp., 51 (1988), 15-25.  doi: 10.1090/S0025-5718-1988-0942141-9.  Google Scholar

[17]

J. Wang and X. Ye, A superconvergent finite element scheme for the reissner-mindlin plate by projection methods, International Journal of Numeerical Analysis and Modeling, 1 (2004), 99-110.   Google Scholar

[18]

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

[19]

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

[20]

X. Ye, Stabilized finite element approximations for the Reissner-Mindlin plate, Advances in Computational Mathematics, 13 (2000), 375-386.  doi: 10.1023/A:1016693613626.  Google Scholar

[21]

X. Ye, A Rectangular Element for the Reissner-Mindlin Plate, Numer. Method for PDE, 16 (2000), 184-193.  doi: 10.1002/(SICI)1098-2426(200003)16:2<184::AID-NUM3>3.0.CO;2-B.  Google Scholar

show all references

References:
[1]

D. Arnold, F. Brezzi and D. Marini, A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate, J. Sci. Comput., 22 (2005), 25-45.  doi: 10.1007/s10915-004-4134-8.  Google Scholar

[2]

D. Arnold, F. Brezzi, R. Falk and D. Marini, Locking-free Reissner-Mindlin elements without reduced integration, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3660-3671.  doi: 10.1016/j.cma.2006.10.023.  Google Scholar

[3]

D. Arnold and R. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal., 26 (1989), 1276-1290.  doi: 10.1137/0726074.  Google Scholar

[4]

D. N. Arnold and X. Liu, Interior estimates for a low order finite element method for the Reissner-Mindlin plate model, Adv. in Comp. Math., 7 (1997), 337-360.  doi: 10.1023/A:1018907205385.  Google Scholar

[5]

S. Brenner, Korn's inequalities for piecewise $H^1$ vector fields, Math. Comput., 73 (2004), 1067-1087.   Google Scholar

[6]

F. Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plates, Math. Comp., 47 (1986), 151-158.  doi: 10.1090/S0025-5718-1986-0842127-7.  Google Scholar

[7]

F. Brezzi, K. Bathe and M. Fortin, Mixed interpolated elements for Reissner-Mindlin plates, Int. J. Numer. Methods Eng., 28 (1989), 1787-1801.  doi: 10.1002/nme.1620280806.  Google Scholar

[8]

G. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Mathematical Models and Methods in Applied Sciences, 1 (1991), 125-151.  doi: 10.1142/S0218202591000083.  Google Scholar

[9]

D. Chapelle and R. Stenberg, An optimal low-order locking-free finite element method for Reissner-Mindlin plates, Mathematical Models and Methods in Applied Sciences, 8 (1998), 407-430.  doi: 10.1142/S0218202598000172.  Google Scholar

[10]

R. Duran and E. Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp., 58 (1992), 561-573.  doi: 10.1090/S0025-5718-1992-1106965-0.  Google Scholar

[11]

R. Falk and T. Tu, Locking-free finite elements for the Reissner-Mindlin plate, Math. Comp., 69 (2000), 911-928.   Google Scholar

[12]

P. Hansbo, D. Heintz and M. Larson, A finite element method with discontinuous rotations for the Mindlin-Reissner plate model, Comput. Methods Appl. Mech. Engrg., 200 (2011), 638-648.  doi: 10.1016/j.cma.2010.09.009.  Google Scholar

[13]

C. Lovadina and D. Marini, Nonconforming locking-free finite elements for Reissner-Mindlin plates, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3448-3460.  doi: 10.1016/j.cma.2005.06.025.  Google Scholar

[14]

L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method with polynomial reduction, Journal of Computational and Applied Mathematics, 285 (2015), 45-58.  doi: 10.1016/j.cam.2015.02.001.  Google Scholar

[15]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element method for second-order elliptic problems on polytopal meshes, International Journal of Numerical Analysis and Modeling, 12 (2015), 31-53.   Google Scholar

[16]

R. Pierre, Convergence Properties and Numerical Approximation of the Solution of the Mindlin Plate Bending Problem, Math Comp., 51 (1988), 15-25.  doi: 10.1090/S0025-5718-1988-0942141-9.  Google Scholar

[17]

J. Wang and X. Ye, A superconvergent finite element scheme for the reissner-mindlin plate by projection methods, International Journal of Numeerical Analysis and Modeling, 1 (2004), 99-110.   Google Scholar

[18]

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

[19]

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

[20]

X. Ye, Stabilized finite element approximations for the Reissner-Mindlin plate, Advances in Computational Mathematics, 13 (2000), 375-386.  doi: 10.1023/A:1016693613626.  Google Scholar

[21]

X. Ye, A Rectangular Element for the Reissner-Mindlin Plate, Numer. Method for PDE, 16 (2000), 184-193.  doi: 10.1002/(SICI)1098-2426(200003)16:2<184::AID-NUM3>3.0.CO;2-B.  Google Scholar

Table 1.  Example 1. Error profile and convergence rate on triangular mesh.
h $\frac{\|{\boldsymbol{\rm{e}}}_0\|}{\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\|}$ Rate $\frac{|||{\boldsymbol{\rm{e}}}_h|||_{\theta}}{|||{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}|||}$ Rate $\frac{\| \xi_h\|}{\|w_I\|}$ Rate $\frac{||| \xi_h|||_w}{||| w_I|||_w}$ Rate
t=1
1/4 2.2928e-1 8.2826e-1 1.1156e-1 9.4025e-2
1/8 7.2951e-2 1.65 6.1481e-1 0.43 1.4546e-2 2.94 1.7212e-2 2.45
1/16 1.8952e-2 1.94 3.6061e-1 0.77 1.8584e-3 2.97 2.4738e-3 2.80
1/32 4.7756e-3 1.99 1.8925e-1 0.93 2.4951e-4 2.90 3.3612e-4 2.88
1/64 1.1961e-3 2.00 9.5844e-2 0.98 3.8498e-5 2.70 4.8047e-5 2.81
t = 1e − 3
1/4 2.3633e-1 2.3639e-1 2.7175e-1 2.9153e-1
1/8 6.8597e-2 1.78 6.8675e-2 1.78 8.7506e-2 1.63 1.1448e-1 1.35
1/16 1.9470e-2 1.82 1.9542e-2 1.81 2.4404e-2 1.84 3.4550e-2 1.73
1/32 5.0660e-3 1.94 5.1359e-3 1.93 6.2578e-3 1.96 9.0799e-3 1.93
1/64 1.2169e-3 2.06 1.2876e-3 2.00 1.5154e-3 2.05 2.2184e-3 2.03
t = 1e − 6
1/4 2.3636e-1 2.3636e-1 2.7183e-1 2.9161e-1
1/8 6.8703e-2 1.78 6.8703e-2 1.78 8.7716e-2 1.63 1.1467e-1 1.35
1/16 1.9610e-2 1.81 1.9610e-2 1.81 2.4634e-2 1.83 3.4772e-2 1.72
1/32 5.0329e-3 1.96 5.0329e-3 1.96 4.4698e-3 2.46 7.7775e-3 2.16
1/64 1.2582e-3 2.00 1.2582e-3 2.00 1.1175e-3 2.00 1.9444e-3 2.00
Table Options

Download as excel

h $\frac{\|{\boldsymbol{\rm{e}}}_0\|}{\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\|}$ Rate $\frac{|||{\boldsymbol{\rm{e}}}_h|||_{\theta}}{|||{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}|||}$ Rate $\frac{\| \xi_h\|}{\|w_I\|}$ Rate $\frac{||| \xi_h|||_w}{||| w_I|||_w}$ Rate
t=1
1/4 2.2928e-1 8.2826e-1 1.1156e-1 9.4025e-2
1/8 7.2951e-2 1.65 6.1481e-1 0.43 1.4546e-2 2.94 1.7212e-2 2.45
1/16 1.8952e-2 1.94 3.6061e-1 0.77 1.8584e-3 2.97 2.4738e-3 2.80
1/32 4.7756e-3 1.99 1.8925e-1 0.93 2.4951e-4 2.90 3.3612e-4 2.88
1/64 1.1961e-3 2.00 9.5844e-2 0.98 3.8498e-5 2.70 4.8047e-5 2.81
t = 1e − 3
1/4 2.3633e-1 2.3639e-1 2.7175e-1 2.9153e-1
1/8 6.8597e-2 1.78 6.8675e-2 1.78 8.7506e-2 1.63 1.1448e-1 1.35
1/16 1.9470e-2 1.82 1.9542e-2 1.81 2.4404e-2 1.84 3.4550e-2 1.73
1/32 5.0660e-3 1.94 5.1359e-3 1.93 6.2578e-3 1.96 9.0799e-3 1.93
1/64 1.2169e-3 2.06 1.2876e-3 2.00 1.5154e-3 2.05 2.2184e-3 2.03
t = 1e − 6
1/4 2.3636e-1 2.3636e-1 2.7183e-1 2.9161e-1
1/8 6.8703e-2 1.78 6.8703e-2 1.78 8.7716e-2 1.63 1.1467e-1 1.35
1/16 1.9610e-2 1.81 1.9610e-2 1.81 2.4634e-2 1.83 3.4772e-2 1.72
1/32 5.0329e-3 1.96 5.0329e-3 1.96 4.4698e-3 2.46 7.7775e-3 2.16
1/64 1.2582e-3 2.00 1.2582e-3 2.00 1.1175e-3 2.00 1.9444e-3 2.00
Table 2.  Example 2. Error profile and convergence rate on triangular mesh.
$h$ $\frac{\|{\boldsymbol{\rm{e}}}_0\|}{\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\|}$ Rate $\frac{|||{\boldsymbol{\rm{e}}}_h|||_{\theta}}{|||{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}|||}$ Rate $\frac{\| \xi_h\|}{\|w_I\|}$ Rate $\frac{||| \xi_h|||_w}{||| w_I|||_w}$ Rate
$t=1$
1/4 7.1054e-2 4.3252e-1 1.3468e-3 1.9340e-3
1/81.7700e-22.012.2528e-10.943.0173e-42.164.7669e-42.02
1/164.4178e-32.001.1386e-10.987.3026e-52.051.1882e-42.00
1/321.1038e-32.005.7085e-21.001.8108e-52.012.9695e-52.00
1/642.7591e-42.002.8562e-21.004.5181e-62.007.4241e-62.00
$t=1e-3$
1/47.8909e-27.8920e-21.5185e-22.1876e-2
1/81.9680e-22.001.9692e-22.003.4940e-32.125.3275e-32.04
1/164.9063e-32.004.9182e-32.008.6948e-42.011.3334e-32.00
1/321.2254e-32.001.2373e-31.992.1650e-42.013.3216e-42.01
1/643.0652e-42.003.1821e-41.965.2591e-52.048.0763e-52.04
$t=1e-6$
1/47.8904e-27.8904e-21.5196e-22.1889e-2
1/81.9658e-22.011.9658e-22.013.5363e-32.105.3726e-32.03
1/164.8250e-32.034.8250e-32.031.0325e-31.781.5119e-31.83
1/321.2063e-32.001.2063e-32.002.5812e-42.003.7798e-42.00
1/643.0156e-42.003.0156e-42.006.4531e-52.009.4494e-52.00
Table Options

Download as excel

$h$ $\frac{\|{\boldsymbol{\rm{e}}}_0\|}{\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\|}$ Rate $\frac{|||{\boldsymbol{\rm{e}}}_h|||_{\theta}}{|||{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}|||}$ Rate $\frac{\| \xi_h\|}{\|w_I\|}$ Rate $\frac{||| \xi_h|||_w}{||| w_I|||_w}$ Rate
$t=1$
1/4 7.1054e-2 4.3252e-1 1.3468e-3 1.9340e-3
1/81.7700e-22.012.2528e-10.943.0173e-42.164.7669e-42.02
1/164.4178e-32.001.1386e-10.987.3026e-52.051.1882e-42.00
1/321.1038e-32.005.7085e-21.001.8108e-52.012.9695e-52.00
1/642.7591e-42.002.8562e-21.004.5181e-62.007.4241e-62.00
$t=1e-3$
1/47.8909e-27.8920e-21.5185e-22.1876e-2
1/81.9680e-22.001.9692e-22.003.4940e-32.125.3275e-32.04
1/164.9063e-32.004.9182e-32.008.6948e-42.011.3334e-32.00
1/321.2254e-32.001.2373e-31.992.1650e-42.013.3216e-42.01
1/643.0652e-42.003.1821e-41.965.2591e-52.048.0763e-52.04
$t=1e-6$
1/47.8904e-27.8904e-21.5196e-22.1889e-2
1/81.9658e-22.011.9658e-22.013.5363e-32.105.3726e-32.03
1/164.8250e-32.034.8250e-32.031.0325e-31.781.5119e-31.83
1/321.2063e-32.001.2063e-32.002.5812e-42.003.7798e-42.00
1/643.0156e-42.003.0156e-42.006.4531e-52.009.4494e-52.00
Table 3.  Example 3. Error profile and convergence rate on triangular mesh.
$h$ $\frac{\|{\boldsymbol{\rm{e}}}_0\|}{\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\|}$Rate$\frac{|||{\boldsymbol{\rm{e}}}_h|||_{\theta}}{|||{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}|||}$Rate $\frac{\| \xi_h\|}{\|w_I\|}$Rate$\frac{||| \xi_h|||_w}{||| w_I|||_w}$Rate
$t=1$
1/42.8817e-18.7886e-13.4967e-22.8229e-2
1/86.4481e-22.165.8311e-10.594.9549e-32.824.7686e-32.57
1/161.5738e-22.033.2520e-10.847.6558e-42.697.2651e-42.71
1/323.9112e-32.011.6784e-10.951.4638e-42.391.1930e-42.60
1/649.7633e-42.008.4614e-20.993.3163e-52.142.3538e-52.34
$t=1e-3$
1/41.9791e-11.9802e-12.8790e-12.8340e-01
1/85.5770e-21.835.5851e-21.831.0222e-11.491.0240e-011.47
1/161.5321e-21.861.5393e-21.862.8291e-21.852.8723e-021.83
1/323.8890e-31.983.9578e-31.967.2000e-31.977.3429e-031.97
1/649.1919e-42.089.8893e-42.001.7371e-32.051.7727e-032.05
$t=1e-6$
1/41.9793e-11.9793e-12.8800e-12.8350e-01
1/85.5887e-21.825.5887e-21.821.0241e-11.491.0258e-011.47
1/161.5435e-21.861.5435e-21.862.8442e-21.852.8873e-021.83
1/323.3377e-32.213.3377e-32.214.4595e-32.674.9441e-032.55
1/648.3442e-42.008.3442e-42.001.1149e-32.001.2360e-032.00
Table Options

Download as excel

$h$ $\frac{\|{\boldsymbol{\rm{e}}}_0\|}{\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\|}$Rate$\frac{|||{\boldsymbol{\rm{e}}}_h|||_{\theta}}{|||{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}|||}$Rate $\frac{\| \xi_h\|}{\|w_I\|}$Rate$\frac{||| \xi_h|||_w}{||| w_I|||_w}$Rate
$t=1$
1/42.8817e-18.7886e-13.4967e-22.8229e-2
1/86.4481e-22.165.8311e-10.594.9549e-32.824.7686e-32.57
1/161.5738e-22.033.2520e-10.847.6558e-42.697.2651e-42.71
1/323.9112e-32.011.6784e-10.951.4638e-42.391.1930e-42.60
1/649.7633e-42.008.4614e-20.993.3163e-52.142.3538e-52.34
$t=1e-3$
1/41.9791e-11.9802e-12.8790e-12.8340e-01
1/85.5770e-21.835.5851e-21.831.0222e-11.491.0240e-011.47
1/161.5321e-21.861.5393e-21.862.8291e-21.852.8723e-021.83
1/323.8890e-31.983.9578e-31.967.2000e-31.977.3429e-031.97
1/649.1919e-42.089.8893e-42.001.7371e-32.051.7727e-032.05
$t=1e-6$
1/41.9793e-11.9793e-12.8800e-12.8350e-01
1/85.5887e-21.825.5887e-21.821.0241e-11.491.0258e-011.47
1/161.5435e-21.861.5435e-21.862.8442e-21.852.8873e-021.83
1/323.3377e-32.213.3377e-32.214.4595e-32.674.9441e-032.55
1/648.3442e-42.008.3442e-42.001.1149e-32.001.2360e-032.00
[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]

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
[5]

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
[6]

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
[7]

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
[8]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489
[15]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873
[16]

A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769
[17]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807
[18]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021074
[19]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677
[20]

Mohammad Asadzadeh, Anders Brahme, Jiping Xin. Galerkin methods for primary ion transport in inhomogeneous media. Kinetic & Related Models, 2010, 3 (3) : 373-394. doi: 10.3934/krm.2010.3.373

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (164)
  • HTML views (621)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences