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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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