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A locking free Reissner-Mindlin element with weak Galerkin rotations
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 |
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.
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. |
[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. |
[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. |
[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. |
[5] |
S. Brenner,
Korn's inequalities for piecewise $H^1$ vector fields, Math. Comput., 73 (2004), 1067-1087.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
R. Falk and T. Tu,
Locking-free finite elements for the Reissner-Mindlin plate, Math. Comp., 69 (2000), 911-928.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
S. Brenner,
Korn's inequalities for piecewise $H^1$ vector fields, Math. Comput., 73 (2004), 1067-1087.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
R. Falk and T. Tu,
Locking-free finite elements for the Reissner-Mindlin plate, Math. Comp., 69 (2000), 911-928.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
h | Rate | Rate | Rate | 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 |
h | Rate | Rate | Rate | 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 |
|
Rate | Rate | Rate | Rate | |||||
|
|||||||||
1/4 | 7.1054e-2 | 4.3252e-1 | 1.3468e-3 | 1.9340e-3 | |||||
1/8 | 1.7700e-2 | 2.01 | 2.2528e-1 | 0.94 | 3.0173e-4 | 2.16 | 4.7669e-4 | 2.02 | |
1/16 | 4.4178e-3 | 2.00 | 1.1386e-1 | 0.98 | 7.3026e-5 | 2.05 | 1.1882e-4 | 2.00 | |
1/32 | 1.1038e-3 | 2.00 | 5.7085e-2 | 1.00 | 1.8108e-5 | 2.01 | 2.9695e-5 | 2.00 | |
1/64 | 2.7591e-4 | 2.00 | 2.8562e-2 | 1.00 | 4.5181e-6 | 2.00 | 7.4241e-6 | 2.00 | |
1/4 | 7.8909e-2 | 7.8920e-2 | 1.5185e-2 | 2.1876e-2 | |||||
1/8 | 1.9680e-2 | 2.00 | 1.9692e-2 | 2.00 | 3.4940e-3 | 2.12 | 5.3275e-3 | 2.04 | |
1/16 | 4.9063e-3 | 2.00 | 4.9182e-3 | 2.00 | 8.6948e-4 | 2.01 | 1.3334e-3 | 2.00 | |
1/32 | 1.2254e-3 | 2.00 | 1.2373e-3 | 1.99 | 2.1650e-4 | 2.01 | 3.3216e-4 | 2.01 | |
1/64 | 3.0652e-4 | 2.00 | 3.1821e-4 | 1.96 | 5.2591e-5 | 2.04 | 8.0763e-5 | 2.04 | |
1/4 | 7.8904e-2 | 7.8904e-2 | 1.5196e-2 | 2.1889e-2 | |||||
1/8 | 1.9658e-2 | 2.01 | 1.9658e-2 | 2.01 | 3.5363e-3 | 2.10 | 5.3726e-3 | 2.03 | |
1/16 | 4.8250e-3 | 2.03 | 4.8250e-3 | 2.03 | 1.0325e-3 | 1.78 | 1.5119e-3 | 1.83 | |
1/32 | 1.2063e-3 | 2.00 | 1.2063e-3 | 2.00 | 2.5812e-4 | 2.00 | 3.7798e-4 | 2.00 | |
1/64 | 3.0156e-4 | 2.00 | 3.0156e-4 | 2.00 | 6.4531e-5 | 2.00 | 9.4494e-5 | 2.00 |
|
Rate | Rate | Rate | Rate | |||||
|
|||||||||
1/4 | 7.1054e-2 | 4.3252e-1 | 1.3468e-3 | 1.9340e-3 | |||||
1/8 | 1.7700e-2 | 2.01 | 2.2528e-1 | 0.94 | 3.0173e-4 | 2.16 | 4.7669e-4 | 2.02 | |
1/16 | 4.4178e-3 | 2.00 | 1.1386e-1 | 0.98 | 7.3026e-5 | 2.05 | 1.1882e-4 | 2.00 | |
1/32 | 1.1038e-3 | 2.00 | 5.7085e-2 | 1.00 | 1.8108e-5 | 2.01 | 2.9695e-5 | 2.00 | |
1/64 | 2.7591e-4 | 2.00 | 2.8562e-2 | 1.00 | 4.5181e-6 | 2.00 | 7.4241e-6 | 2.00 | |
1/4 | 7.8909e-2 | 7.8920e-2 | 1.5185e-2 | 2.1876e-2 | |||||
1/8 | 1.9680e-2 | 2.00 | 1.9692e-2 | 2.00 | 3.4940e-3 | 2.12 | 5.3275e-3 | 2.04 | |
1/16 | 4.9063e-3 | 2.00 | 4.9182e-3 | 2.00 | 8.6948e-4 | 2.01 | 1.3334e-3 | 2.00 | |
1/32 | 1.2254e-3 | 2.00 | 1.2373e-3 | 1.99 | 2.1650e-4 | 2.01 | 3.3216e-4 | 2.01 | |
1/64 | 3.0652e-4 | 2.00 | 3.1821e-4 | 1.96 | 5.2591e-5 | 2.04 | 8.0763e-5 | 2.04 | |
1/4 | 7.8904e-2 | 7.8904e-2 | 1.5196e-2 | 2.1889e-2 | |||||
1/8 | 1.9658e-2 | 2.01 | 1.9658e-2 | 2.01 | 3.5363e-3 | 2.10 | 5.3726e-3 | 2.03 | |
1/16 | 4.8250e-3 | 2.03 | 4.8250e-3 | 2.03 | 1.0325e-3 | 1.78 | 1.5119e-3 | 1.83 | |
1/32 | 1.2063e-3 | 2.00 | 1.2063e-3 | 2.00 | 2.5812e-4 | 2.00 | 3.7798e-4 | 2.00 | |
1/64 | 3.0156e-4 | 2.00 | 3.0156e-4 | 2.00 | 6.4531e-5 | 2.00 | 9.4494e-5 | 2.00 |
|
| Rate | Rate | | Rate | Rate | |||
| |||||||||
1/4 | 2.8817e-1 | 8.7886e-1 | 3.4967e-2 | 2.8229e-2 | |||||
1/8 | 6.4481e-2 | 2.16 | 5.8311e-1 | 0.59 | 4.9549e-3 | 2.82 | 4.7686e-3 | 2.57 | |
1/16 | 1.5738e-2 | 2.03 | 3.2520e-1 | 0.84 | 7.6558e-4 | 2.69 | 7.2651e-4 | 2.71 | |
1/32 | 3.9112e-3 | 2.01 | 1.6784e-1 | 0.95 | 1.4638e-4 | 2.39 | 1.1930e-4 | 2.60 | |
1/64 | 9.7633e-4 | 2.00 | 8.4614e-2 | 0.99 | 3.3163e-5 | 2.14 | 2.3538e-5 | 2.34 | |
| |||||||||
1/4 | 1.9791e-1 | 1.9802e-1 | 2.8790e-1 | 2.8340e-01 | |||||
1/8 | 5.5770e-2 | 1.83 | 5.5851e-2 | 1.83 | 1.0222e-1 | 1.49 | 1.0240e-01 | 1.47 | |
1/16 | 1.5321e-2 | 1.86 | 1.5393e-2 | 1.86 | 2.8291e-2 | 1.85 | 2.8723e-02 | 1.83 | |
1/32 | 3.8890e-3 | 1.98 | 3.9578e-3 | 1.96 | 7.2000e-3 | 1.97 | 7.3429e-03 | 1.97 | |
1/64 | 9.1919e-4 | 2.08 | 9.8893e-4 | 2.00 | 1.7371e-3 | 2.05 | 1.7727e-03 | 2.05 | |
| |||||||||
1/4 | 1.9793e-1 | 1.9793e-1 | 2.8800e-1 | 2.8350e-01 | |||||
1/8 | 5.5887e-2 | 1.82 | 5.5887e-2 | 1.82 | 1.0241e-1 | 1.49 | 1.0258e-01 | 1.47 | |
1/16 | 1.5435e-2 | 1.86 | 1.5435e-2 | 1.86 | 2.8442e-2 | 1.85 | 2.8873e-02 | 1.83 | |
1/32 | 3.3377e-3 | 2.21 | 3.3377e-3 | 2.21 | 4.4595e-3 | 2.67 | 4.9441e-03 | 2.55 | |
1/64 | 8.3442e-4 | 2.00 | 8.3442e-4 | 2.00 | 1.1149e-3 | 2.00 | 1.2360e-03 | 2.00 |
|
| Rate | Rate | | Rate | Rate | |||
| |||||||||
1/4 | 2.8817e-1 | 8.7886e-1 | 3.4967e-2 | 2.8229e-2 | |||||
1/8 | 6.4481e-2 | 2.16 | 5.8311e-1 | 0.59 | 4.9549e-3 | 2.82 | 4.7686e-3 | 2.57 | |
1/16 | 1.5738e-2 | 2.03 | 3.2520e-1 | 0.84 | 7.6558e-4 | 2.69 | 7.2651e-4 | 2.71 | |
1/32 | 3.9112e-3 | 2.01 | 1.6784e-1 | 0.95 | 1.4638e-4 | 2.39 | 1.1930e-4 | 2.60 | |
1/64 | 9.7633e-4 | 2.00 | 8.4614e-2 | 0.99 | 3.3163e-5 | 2.14 | 2.3538e-5 | 2.34 | |
| |||||||||
1/4 | 1.9791e-1 | 1.9802e-1 | 2.8790e-1 | 2.8340e-01 | |||||
1/8 | 5.5770e-2 | 1.83 | 5.5851e-2 | 1.83 | 1.0222e-1 | 1.49 | 1.0240e-01 | 1.47 | |
1/16 | 1.5321e-2 | 1.86 | 1.5393e-2 | 1.86 | 2.8291e-2 | 1.85 | 2.8723e-02 | 1.83 | |
1/32 | 3.8890e-3 | 1.98 | 3.9578e-3 | 1.96 | 7.2000e-3 | 1.97 | 7.3429e-03 | 1.97 | |
1/64 | 9.1919e-4 | 2.08 | 9.8893e-4 | 2.00 | 1.7371e-3 | 2.05 | 1.7727e-03 | 2.05 | |
| |||||||||
1/4 | 1.9793e-1 | 1.9793e-1 | 2.8800e-1 | 2.8350e-01 | |||||
1/8 | 5.5887e-2 | 1.82 | 5.5887e-2 | 1.82 | 1.0241e-1 | 1.49 | 1.0258e-01 | 1.47 | |
1/16 | 1.5435e-2 | 1.86 | 1.5435e-2 | 1.86 | 2.8442e-2 | 1.85 | 2.8873e-02 | 1.83 | |
1/32 | 3.3377e-3 | 2.21 | 3.3377e-3 | 2.21 | 4.4595e-3 | 2.67 | 4.9441e-03 | 2.55 | |
1/64 | 8.3442e-4 | 2.00 | 8.3442e-4 | 2.00 | 1.1149e-3 | 2.00 | 1.2360e-03 | 2.00 |
[1] |
Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873 |
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A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769 |
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Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete and 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 and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074 |
[19] |
Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete and 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 and Related Models, 2010, 3 (3) : 373-394. doi: 10.3934/krm.2010.3.373 |
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