### A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM

Yuping Zeng1, Feng Wang2, Zhifeng Weng3, Hanzhang Hu1

1. 1. School of Mathematics, Jiaying University, Meizhou 514015, China;
2. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China;
3. Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
• Received:2019-01-21 Revised:2019-06-03 Online:2021-09-15 Published:2021-10-15
• Supported by:
The authors thank the anonymous referees for their valuable comments and suggestions which helped to improve the quality of this article. The first author was supported by Guangdong Basic and Applied Basic Research Foundation (Grant Nos. 2018A030307024 and 2020A1515011032), and by National Natural Science Foundation of China (Grant No. 11526097). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11871272 and 11871281). The third author was supported by National Natural Science Foundation of China (Grant No. 11701197). The fourth author was supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2018A0303100016).

Yuping Zeng, Feng Wang, Zhifeng Weng, Hanzhang Hu. A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM[J]. Journal of Computational Mathematics, 2021, 39(5): 755-776.

In this paper, we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems. We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method, though they have essentially different bilinear forms. More precisely, we prove its reliability and efficiency for the actual error measured in the standard DG norm. We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution. Numerical results are presented to verify the theoretical analysis.

CLC Number:

 [1] M. Ainsworth, A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. Numer. Anal., 45(2007), 1777-1798.[2] M. Ainsworth and G. Fu, Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations, J. Sci. Comput., 77(2018), 443-466.[3] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39(2002), 1749-1779.[4] R. Becker, P. Hansbo and M. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods, Comput. Methods Appl. Mech. Engrg., 192(2003), 723-733.[5] D. Braess, T. Fraunholz and R.H.W. Hoppe, An equilibrated a posteriori error estimator for the interior penalty discontinuous Galerkin method, SIAM J. Numer. Anal., 52(2014), 2121-2136.[6] S.C. Brenner, T. Gudi and L.Y. Sung, A posteriori error control for a weakly over-penalized symmetric interior penalty method, J. Sci. Comput., 40(2009), 37-50.[7] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods (3rd edn), Springer-Verlag, New York, 2008.[8] C. Carstensen, T. Gudi and M. Jensen, A unifying theory of a posteriori error control for discontinuous Galerkin FEM, Numer. Math., 112(2009), 363-379.[9] C. Carstensen and J. Hu, A unifying theory of a posteriori error control for nonconforming finite element methods, Numer. Math., 107(2007), 473-502.[10] G. Chen, M. Feng and X. Xie, A robust WG finite element method for convection-diffusion-reaction equations, J. Comput. Appl. Math., 315(2017), 107-125.[11] G. Chen, M. Feng and X. Xie, Robust globally divergence-free weak Galerkin methods for Stokes equations, J. Comput. Math., 34(2016), 549-570.[12] G. Chen and X. Xie, A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses, Comput. Methods Appl. Math., 16(2016), 389-408.[13] L. Chen, iFEM:an integrated finite element methods package in MATLAB, University of California at Irvine, 2009.[14] L. Chen, J. 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.[15] W. Chen, F. Wang and Y. Wang, Weak Galerkin method for the coupled Darcy-Stokes flow, IMA J. Numer. Anal., 2(2016), 897-921.[16] Y. Chen, G. Chen and X. Xie, Weak Galerkin finite element method for Biot's consolidation problem, J. Comput. Appl. Math., 330(2018), 398-416.[17] Z. Chen and H. Wu, Selected Topics in Finite Element Methods, Science Press, Beijing, 2010.[18] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.[19] P. Clément, Approximation by finite element functions using local regularization, RAIRO Modél. Math. Anal. Numér., 9(1975), 77-84.[20] B. Cockburn and W. Zhang, A posteriori error estimates for HDG methods, J. Sci. Comput., 51(2012), 582-607.[21] B. Cockburn and W. Zhang, A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 51(2013), 676-693.[22] V. Dolejší, I. Šebestová and M. Vohralík, Algebraic and discretization error estimation by equilibrated fluxes for discontinuous Galerkin methods on nonmatching grids, J. Sci. Comput., 64(2015), 1-34.[23] W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33(1996), 1106-1124.[24] Y. Du and Z. Zhang, A numerical analysis of the weak Galerkin method for the Helmholtz equation with high wave number, Commun. Comput. Phys., 22(2017), 133-156.[25] A. Ern and M. Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal., 53(2015), 1058-1081.[26] F. Gao and X. Wang, A modified weak Galerkin finite element method for a class of parabolic problems, J. Comput. Appl. Math., 271(2014), 1-19.[27] F. Gao and X. Wang, A modified weak Galerkin finite element method for Sobolev equation, J. Comput. Math., 33(2015), 307-322[28] T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comput., 79(2010), 2169-2189.[29] Q. Hong, F. Wang, S. Wu and J. Xu, A unified study of continuous and discontinuous Galerkin methods, Sci. China Math., 62(2019), 1-32.[30] P. Houston, D. Schötzau and T. P. Wihler, Energy norm a posteriori error estimation of hpadaptive discontinuous Galerkin methods for elliptic problems, Math. Models Methods Appl. Sci., 17(2007), 33-62.[31] X. Hu, L. Mu and X. Ye, Weak Galerkin method for the Biot's consolidation model, Comput. Math. Appl., 75(2018), 2017-2030.[32] Y. Huang, J. Li and D. Li, Developing weak Galerkin finite element methods for the wave equation, Numer. Methods PDEs., 33(2017), 868-884.[33] O.A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal., 41(2003), 2374-2399.[34] O. A. Karakashian and F. Pascal, Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems, SIAM J. Numer. Anal., 45(2007), 641-665.[35] T. Lewis and M. Neilan, Convergence analysis of a symmetric dual-wind discontinuous Galerkin method, J. Sci. Comput., 59(2014), 602-625.[36] Q. Li and J. Wang, Weak Galerkin finite element methods for parabolic equations, Numer. Methods PDEs., 29(2013), 2004-2024.[37] R. Li, Y. Gao, J. Li and Z. Chen, A weak Galerkin finite element method for a coupled StokesDarcy problem on general meshes, J. Comput. Appl. Math., 334(2018), 111-127.[38] G. Lin, J. Liu, L. Mu and X. Ye, Weak Galerkin finite element methods for Darcy flow:Anisotropy and heterogeneity, J. Comput. Phys., 276(2014), 422-437.[39] R. Lin, X. Ye, S. Zhang and P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction Problems, SIAM J. Numer. Anal., 56(2018), 1482-1497.[40] C. Lovadina and L. D. Marini, A posteriori error estimates for discontinuous Galerkin approximations of second order elliptic problems, J. Sci. Comput., 40(2009), 340-359.[41] L. Mu, J. Wang, G. Wei, X. Ye and S. Zhao, Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250(2013), 106-125.[42] L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Methods PDEs., 30(2014), 1003-1029.[43] L. Mu, J. Wang and X. Ye, A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods, J. Comput. Phys., 273(2014), 327-342.[44] L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12(2015), 31-53.[45] L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method with polynomial reduction, J. Comput. Appl. Math., 285(2015), 45-58.[46] L. Mu, J. Wang and X. Ye, A new weak Galerkin finite element method for the Helmholtz equation, IMA J. Numer. Anal., 35(2015), 1228-1255.[47] L. Mu, J. Wang and X. Ye, A weak Galerkin method for the Reissner-Mindlin plate in primary form, J. Sci. Comput., 75(2018), 782-802.[48] L. Mu, J. Wang, X. Ye and S. Zhang, A C0-weak Galerkin finite element method for the biharmonic equation, J. Sci. Comput., 59(2014), 473-495.[49] L. Mu, J. Wang, X. Ye and S. Zhang, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65(2015), 363-386.[50] L. Mu, X. Wang and X. Ye, A modified weak Galerkin finite element method for the Stokes equations, J. Comput. Appl. Math., 275(2015) 79-90.[51] C. Wang, New discretization schemes for time-harmonic Maxwell equations by weak Galerkin finite element methods, J. Comput. Appl. Math., 341(2018), 127-143.[52] C. Wang and J. Wang, An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes, Comput. Math. Appl., 68(2014), 2314-2330.[53] C. Wang and J. Wang, A primal-dual weak Galerkin finite element method for Fokker-Planck type equations, arXiv preprint arXiv:1704.05606, 2017.[54] C. Wang and J. Wang, A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form, Math. Comput., 87(2018), 515-545.[55] C. Wang, J. Wang, R. Wang and R. Zhang, A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation, J. Comput. Appl. Math., 307(2016), 346-366.[56] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241(2013), 103-115.[57] J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83(2014), 2101-2126.[58] J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42(2016), 155-174.[59] J. Wang and X. Ye, The basics of weak Galerkin finite element methods, arXiv preprint arXiv:1901.10035, 2019.[60] J. Wang, Q. Zhai, R. Zhang and S. Zhang, A weak Galerkin finite element scheme for the CahnHilliard equation, Math. Comput., 88(2019), 211-235.[61] J. Wang and Z. Zhang, A hybridizable weak Galerkin method for the Helmholtz equation with large wave number hp analysis, Int. J. Numer. Anal. Model., 14(2017), 744-761.[62] R. Wang, X. Wang and R. Zhang, A modified weak Galerkin finite element method for the poroelasticity problems, Numer. Math. Theory Methods Appl., 11(2018), 518-539.[63] X. Wang, N.S. Malluwawadu, F. Gao and T.C. McMillan, A modified weak Galerkin finite element method, J. Comput. Appl. Math., 271(2014), 319-327.[64] X. Wang, Q. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307(2016), 13-24.[65] R. Schneider, Y. Xu and A. Zhou, An analysis of discontinuous Galerkin methods for elliptic problems, Adv. Comput. Math., 25(2006), 259-286.[66] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54(1994), 483-493.[67] S. Shields, J. Li and E.A. Machorro, Weak Galerkin methods for time-dependent Maxwell's equations, Comput. Math. Appl., 74(2017), 2106-2124.[68] L. Song, S. Zhao and K. Liu, A relaxed weak Galerkin method for elliptic interface problems with low regularity, Appl. Numer. Math., 128(2018), 65-80.[69] M. Sun and H. Rui, A coupling of weak Galerkin and mixed finite element methods for poroelasticity, Comput. Math. Appl., 73(2017), 804-823.[70] R. Verfürth, A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.[71] J. Yang and Y. Chen, A unified a posteriori error analysis for discontinuous Galerkin approximations of reactive transport equations, J. Comput. Math., 24(2006), 425-434.[72] Y. Zeng, J. Chen and F. Wang, Convergence analysis of a modified weak Galerkin finite element method for Signorini and obstacle problems, Numer. Methods PDEs., 33(2017), 1459-1474.[73] Q. Zhai, H. Xie, R. Zhang and Z. Zhang, The weak Galerkin method for elliptic eigenvalue problems, Commun. Comput. Phys., 26(2019), 160-191.[74] Q. Zhai, H. Xie, R. Zhang and Z. Zhang, Acceleration of weak Galerkin methods for the Laplacian eigenvalue problem, J. Sci. Comput., 79(2019), 914-934.[75] Q. Zhai and R. Zhang, Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes, Discret. Contin. Dyn. Syst. -B, 24(2019), 403-413.[76] 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.[77] 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.[78] T. Zhang and T. Lin, A posteriori error estimate for a modified weak Galerkin method solving elliptic problems, Numer. Methods PDEs., 33(2017), 381-398.
 [1] Ram Manohar, Rajen Kumar Sinha. ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR FULLY DISCRETE SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(2): 147-176. [2] Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski. CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK [J]. Journal of Computational Mathematics, 2020, 38(5): 748-767. [3] Tianliang Hou, Yanping Chen. MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS [J]. Journal of Computational Mathematics, 2015, 33(2): 158-178. [4] Yuping Zeng, Jinru Chen, Feng Wang, Yanxia Meng. A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A WEAKLY OVER-PENALIZED INTERIOR PENALTY METHOD FOR NON-SELF-ADJOINT AND INDEFINITE PROBLEMS [J]. Journal of Computational Mathematics, 2014, 32(3): 332-347. [5] Yanzhen Chang, Danping Yang. A POSTERIORI ERROR ESTIMATE OF FINITE ELEMENT METHOD FOR THE OPTIMAL CONTROL WITH THE STATIONARY BÉNARD PROBLEM [J]. Journal of Computational Mathematics, 2013, 31(1): 68-87. [6] Karl Kunisch, Wenbin Liu, Yanzhen Chang, Ningning Yan, Ruo Li. ADAPTIVE FINITE ELEMENT APPROXIMATION FOR A CLASS OF PARAMETER ESTIMATION PROBLEMS [J]. Journal of Computational Mathematics, 2010, 28(5): 645-675. [7] Tang Liu, Ningning Yan and Shuhua Zhang. RICHARDSON EXTRAPOLATION AND DEFECT CORRECTION OF FINITE ELEMENTMETHODS FOR OPTIMAL CONTROL PROBLEMS [J]. Journal of Computational Mathematics, 2010, 28(1): 55-71. [8] R.H.W. Hoppe, J. Sch\. CONVERGENCE OF ADAPTIVE EDGE ELEMENT METHODS FOR THE 3D EDDY CURRENTS EQUATIONS [J]. Journal of Computational Mathematics, 2009, 27(5): 657-676. [9] Lei Yuan, Danping Yang. A POSTERIORI ERROR ESTIMATE OF OPTIMAL CONTROL PROBLEM OF PDE WITH INTEGRAL CONSTRAINT FOR STATE [J]. Journal of Computational Mathematics, 2009, 27(4): 525-542. [10] Michael Hinze, Ningning Yan, Zhaojie Zhou. Variational Discretization for Optimal Control Governed by Convection Dominated Diffusion Equations [J]. Journal of Computational Mathematics, 2009, 27(2-3): 237-253. [11] Xiaobing Feng Haijun Wu. A Posteriori Error Estimates for Finite Element Approximations of the Cahn-Hilliard Equation and the Hele-Shaw Flow [J]. Journal of Computational Mathematics, 2008, 26(6): 767-796. [12] Ji-ming Yang,Yan-ping Chen. A UNIFIED A POSTERIORI ERROR ANALYSIS FOR DISCONTINUOUS GALERKINAPPROXIMATIONS OF REACTIVE TRANSPORT EQUATIONS [J]. Journal of Computational Mathematics, 2006, 24(3): 425-434. [13] Hui-po Liu,Ning-ning Yan. SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES FOR BOUNDARYCONTROL GOVERNED BY STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2006, 24(3): 343-356. [14] Zhi Min ZHANG. POLYNOMIAL PRESERVING RECOVERY FOR ANISOTROPIC AND IRREGULAR GRIDS [J]. Journal of Computational Mathematics, 2004, 22(2): 331-340.
Viewed
Full text

Abstract