# American Institute of Mathematical Sciences

June  2020, 28(2): 821-836. doi: 10.3934/era.2020042

## A hybridized weak Galerkin finite element scheme for general second-order elliptic problems

 1 School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, Guangdong, China 2 School of Mathematical Science, South China Normal University, Guangzhou 520631, Guangdong, China 3 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China

* Corresponding author: Yanping Chen

Received  February 2020 Revised  March 2020 Published  May 2020

Fund Project: The first author is supported by the Natural Science Foundation of Guangdong Province (No. 2016A030307017) and the Natural Science Foundation of Lingnan Normal University (No. ZL2038); The second author is supported by the State Key Program of National Natural Science Foundation of China (No.11931003) and the National Natural Science Foundation of China (No.41974133, No.11671157); The third author is supported by National Natural Science Foundation of China (No.11971410)

In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.

Citation: 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
##### References:
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Shi, Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition, Adv. Nonlinear Anal., 8 (2019), 1252-1285.   Google Scholar [26] R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the Biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), 559-585.  doi: 10.1007/s10915-014-9945-7.  Google Scholar

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##### References:
 [1] C. E. Baumann and J. T. Oden, A discontinuous $hp$ finite elelnent method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 175 (1999), 311-341.  doi: 10.1016/S0045-7825(98)00359-4.  Google Scholar [2] H. Berestycki and J. Wei, On least energy solutions to a semilinear elliptic equation in a strip, Discrete Contin. Dyn. Syst., 28 (2010), 1083-1099.  doi: 10.3934/dcds.2010.28.1083.  Google Scholar [3] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar [4] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers, RAIRO, 8 (1974), 129-151.   Google Scholar [5] F. Brezzi, J. Douglas and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235.  doi: 10.1007/BF01389710.  Google Scholar [6] B. Cockburn, B. Dong, J. Guzman, M. Restelli and R. Sacco, A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction, SIAM J. Sci. Comput., 31 (2019), 3827-3846.  doi: 10.1137/080728810.  Google Scholar [7] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), 1319-1365.  doi: 10.1137/070706616.  Google Scholar [8] B. Cockburn and K. Shi, Conditions for superconvergence of HDG methods for Stokes flow, Math. Comp., 82 (2013), 651-671.  doi: 10.1090/S0025-5718-2012-02644-5.  Google Scholar [9] J. I. Díaz, J. Hernández and Y. Sh. Ilyasov, On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets, Adv. Nonlinear Anal., 9 (2020), 1046-1065.   Google Scholar [10] J. Jr Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp., 44 (1985), 39-52.  doi: 10.1090/S0025-5718-1985-0771029-9.  Google Scholar [11] A. Erna, A. F. Stephansena and M. Vohralík, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems, J. Comput. Appl. Math., 234 (2010), 114-130.  doi: 10.1016/j.cam.2009.12.009.  Google Scholar [12] R. Lazarov and S. Tomov, A posteriori error estimates for finite volume element approximations of convection-diffusion-reaction equations, Computat. Geosci., 6 (2002), 483-503.  doi: 10.1023/A:1021247300362.  Google Scholar [13] Y. Li and J. Bao, Semilinear elliptic system with boundary singularity, Discrete Contin. Dyn. Syst., 40 (2020), 2189-2212.  doi: 10.3934/dcds.2020111.  Google Scholar [14] G. Li and Y. Chen, A new weak Galerkin finite element method for general second-order elliptic problems, J. Comput. Appl. Math., 344 (2018), 701-715.  doi: 10.1016/j.cam.2018.05.021.  Google Scholar [15] L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Meth. Part. D E, 30 (2014), 1003-1029.  doi: 10.1002/num.21855.  Google Scholar [16] L. Mu, J. Wang and X. Ye, A hybridized formulation for the weak Galerkin mixed finite element method, J. Comput. Appl. Math., 307 (2016), 335-345.  doi: 10.1016/j.cam.2016.01.004.  Google Scholar [17] V. Vougalter and V. Volpert, On the solvability conditions for the diffusion equation with convection terms, Commun. Pure Appl. Anal., 11 (2012), 365-373.  doi: 10.3934/cpaa.2012.11.365.  Google Scholar [18] C. Wang and J. Wang, A hybridized weak Galerkin finite element method for the Biharmonic equation, Int. J. Numer. Anal. Model., 12 (2015), 302-317.   Google Scholar [19] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.  Google Scholar [20] J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.  Google Scholar [21] 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 [22] L. Wang and D. Ye, Concentrating solutions for an anisotropic elliptic problem with large exponent, Discrete Contin. Dyn. Syst., 35 (2015), 3771-3797.  doi: 10.3934/dcds.2015.35.3771.  Google Scholar [23] P. Winkert, Multiplicity results for a class of elliptic problems with nonlinear boundary condition, Commun. Pure Appl. Anal., 12 (2013), 785-802.  doi: 10.3934/cpaa.2013.12.785.  Google Scholar [24] Q. Zhai, R. Zhang and X. Wang, A hybridized weak Galerkin finite element scheme for the Stokes equations, Sci. China Math., 58 (2015), 2455-2472.  doi: 10.1007/s11425-015-5030-4.  Google Scholar [25] Y. Zhang and L. Shi, Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition, Adv. Nonlinear Anal., 8 (2019), 1252-1285.   Google Scholar [26] R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the Biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), 559-585.  doi: 10.1007/s10915-014-9945-7.  Google Scholar
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