October  2021, 26(10): 5217-5226. doi: 10.3934/dcdsb.2020340

A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition

1. 

Nanhu College, Jiaxing University, Jiaxing, 314001, China

2. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China, Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

3. 

College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

* Corresponding author: P. Zhu

Received  September 2018 Revised  September 2019 Published  October 2021 Early access  November 2020

Fund Project: P. Zhu is supported by Zhejiang Provincial Natural Science Foundation of China (Grant No.LY19A010008)

A reliable and efficient a posteriori error estimator is presented for a weak Galerkin finite element method without stabilizer for the second order elliptic equation with mixed boundary conditions. The upper bound of the estimator is proved by Helmholtz decomposition technique and lower bound is hold naturally. The performance of the estimator is illustrated by numerical experiments.

Citation: Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340
References:
[1]

M. Ainsworth, Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal., 42 (2005), 2320-2341.  doi: 10.1137/S0036142903425112.

[2]

R. BeckerP. Hansbo and M. G. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods, Computer Methods in Applied Mechanics & Engineering, 192 (2003), 723-733.  doi: 10.1016/S0045-7825(02)00593-5.

[3]

L. ChenJ. Wang and X. Ye, A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput., 59 (2014), 496-511.  doi: 10.1007/s10915-013-9771-3.

[4]

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.

[5]

W. Dörlfer, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054.

[6]

V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[7]

H. LiL. Mu and X. Ye, A posteriori error estimates for the weak Galerkin finite element methods on polytopal meshes, Commun. Comput. Phys., 26 (2019), 558-578.  doi: 10.4208/cicp.OA-2018-0058.

[8]

L. MuJ. WangY. WangX. Ye and S. Zhang, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386.  doi: 10.1007/s10915-014-9964-4.

[9]

L. MuJ. Wang and X. Ye, A weak Galerkin method for the Reissner-Mindlin plate in primary form, J. Sci. Comput., 75 (2018), 782-802.  doi: 10.1007/s10915-017-0564-y.

[10]

R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, John Wiley, Chichester, 1996.

[11]

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

[12]

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.

[13]

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.

[14]

T. Zhang and T. Lin, A posteriori error estimate for a modified weak Galerkin method solving elliptic problems, Numer. Methods Partial Differential Eq., 33 (2017), 381-398.  doi: 10.1002/num.22114.

show all references

References:
[1]

M. Ainsworth, Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal., 42 (2005), 2320-2341.  doi: 10.1137/S0036142903425112.

[2]

R. BeckerP. Hansbo and M. G. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods, Computer Methods in Applied Mechanics & Engineering, 192 (2003), 723-733.  doi: 10.1016/S0045-7825(02)00593-5.

[3]

L. ChenJ. Wang and X. Ye, A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput., 59 (2014), 496-511.  doi: 10.1007/s10915-013-9771-3.

[4]

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.

[5]

W. Dörlfer, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054.

[6]

V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[7]

H. LiL. Mu and X. Ye, A posteriori error estimates for the weak Galerkin finite element methods on polytopal meshes, Commun. Comput. Phys., 26 (2019), 558-578.  doi: 10.4208/cicp.OA-2018-0058.

[8]

L. MuJ. WangY. WangX. Ye and S. Zhang, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386.  doi: 10.1007/s10915-014-9964-4.

[9]

L. MuJ. Wang and X. Ye, A weak Galerkin method for the Reissner-Mindlin plate in primary form, J. Sci. Comput., 75 (2018), 782-802.  doi: 10.1007/s10915-017-0564-y.

[10]

R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, John Wiley, Chichester, 1996.

[11]

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

[12]

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.

[13]

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.

[14]

T. Zhang and T. Lin, A posteriori error estimate for a modified weak Galerkin method solving elliptic problems, Numer. Methods Partial Differential Eq., 33 (2017), 381-398.  doi: 10.1002/num.22114.

Figure 1.  Initial mesh for adaptive refinement
Figure 2.  Effectivity index for Example 1
Figure 3.  Example 1. Final adaptive refinement mesh and WG solution
Figure 4.  Effectivity index for Example 2
Figure 5.  Example 2. Final adaptive refinement mesh and WG solution
Figure 6.  Effectivity index for Example 3
Figure 7.  Example 3. Final adaptive refinement mesh and WG solution
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