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 & 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.  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
[1]

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

[2]

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

[3]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089

[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]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[6]

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

[7]

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

[8]

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

[9]

Hsueh-Chen Lee, Hyesuk Lee. An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows. Electronic Research Archive, 2021, 29 (4) : 2755-2770. doi: 10.3934/era.2021012

[10]

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

[11]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

[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]

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

[14]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216

[15]

Hatim Tayeq, Amal Bergam, Anouar El Harrak, Kenza Khomsi. Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2557-2570. doi: 10.3934/dcdss.2020400

[16]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095

[17]

Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583

[18]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185

[19]

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

[20]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (118)
  • HTML views (347)
  • Cited by (0)

Other articles
by authors

[Back to Top]