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A weak Galerkin mixed finite element method for second order elliptic problems


Authors: Junping Wang and Xiu Ye
Journal: Math. Comp. 83 (2014), 2101-2126
MSC (2010): Primary 65N15, 65N30, 76D07; Secondary 35B45, 35J50
DOI: https://doi.org/10.1090/S0025-5718-2014-02852-4
Published electronically: May 5, 2014
MathSciNet review: 3223326
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Abstract: A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete $ H^1$ and $ L^2$ norms are established for the corresponding weak Galerkin mixed finite element solutions.


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  • [1] D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7-32 (English, with French summary). MR 813687 (87g:65126)
  • [2] Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749-1779. MR 1885715 (2002k:65183), https://doi.org/10.1137/S0036142901384162
  • [3] Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179-192. MR 0359352 (50 #11806)
  • [4] L. Beirão da Veiga, K. Lipnikov, and G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes, SIAM J. Numer. Anal. 49 (2011), no. 5, 1737-1760. MR 2837482 (2012i:65229), https://doi.org/10.1137/100807764
  • [5] L. Beirão da Veiga, K. Lipnikov, and G. Manzini, Convergence analysis of the high-order mimetic finite difference method, Numer. Math. 113 (2009), no. 3, 325-356. MR 2534128 (2010g:65180), https://doi.org/10.1007/s00211-009-0234-6
  • [6] M. Berndt, K. Lipnikov, D. Moulton, and M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation, East-West J. Numer. Math. 9 (2001), no. 4, 265-284. MR 1879474 (2003a:65085)
  • [7] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258 (95f:65001)
  • [8] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129-151 (English, with loose French summary). MR 0365287 (51 #1540)
  • [9] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • [10] Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237-250. MR 890035 (88f:65190), https://doi.org/10.1007/BF01396752
  • [11] Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217-235. MR 799685 (87g:65133), https://doi.org/10.1007/BF01389710
  • [12] Franco Brezzi, Konstantin Lipnikov, and Mikhail Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal. 43 (2005), no. 5, 1872-1896 (electronic). MR 2192322 (2006j:65311), https://doi.org/10.1137/040613950
  • [13] K. Chand and B. Henshaw, Overture Demo Introduction, 9th Overset Grid Symposium, Penn State University.
  • [14] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58 #25001)
  • [15] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319-1365. MR 2485455 (2010b:65251), https://doi.org/10.1137/070706616
  • [16] Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148
  • [17] Jérôme Droniou and Robert Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numer. Math. 105 (2006), no. 1, 35-71. MR 2257385 (2008d:65121), https://doi.org/10.1007/s00211-006-0034-1
  • [18] Paola Causin and Riccardo Sacco, A discontinuous Petrov-Galerkin method with Lagrangian multipliers for second order elliptic problems, SIAM J. Numer. Anal. 43 (2005), no. 1, 280-302 (electronic). MR 2177145 (2006h:65185), https://doi.org/10.1137/S0036142903427871
  • [19] Jaroslav Fořt, Jiří Fürst, Jan Halama, Raphaèle Herbin, and Florence Hubert (eds.), Finite volumes for complex applications. VI. Problems & perspectives. Volume 1, 2, Springer Proceedings in Mathematics, vol. 4, Springer, Heidelberg, 2011. MR 2882736 (2012h:00040)
  • [20] Raphaèle Herbin and Florence Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, Finite volumes for complex applications V, ISTE, London, 2008, pp. 659-692. MR 2451465 (2010a:65211)
  • [21] A. Khamayseh1 and V. Almeida, Adaptive hybrid mesh refinement for multiphysics applications, Journal of Physics: Conference Series 78 (2007) 012039.
  • [22] L. Mu, J. Wang, and X. Ye, A weak Galerkin finite element methods with polynomial reduction, arXiv:1304.6481.
  • [23] L. Mu, J. Wang, and X. Ye, Weak Galerkin finite element methods on polytopal meshes, arXiv:1204.3655v2, submitted to International J. of Numerical Analysis and Modeling.
  • [24] L. Mu, J. Wang, Y. Wang, and X. Ye, A computational study of the weak Galerkin method for second order elliptic equations, Numerical Algorithms 63 (2013), 753-777. DOI:10.1007/s11075-012-9651-1.
  • [25] L. Mu, J. Wang, X. Ye, and S. Zhao, A numerical study on the weak Galerkin method for the Helmholtz equation with large wave numbers, Communication in Computational Physics, accepted.
  • [26] L. Mu, J. Wang, G. Wei, X. Ye, and S. Zhao, Weak Galerkin methods for second order elliptic interface problems, Journal of Computational Physics 250 (2013), 106-125. doi:10.1016/j.jcp.2013.04.042.
  • [27] P. Raviart and J. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method, I. Galligani, E. Magenes, eds., Lectures Notes in Math. 606, Springer-Verlag, New York, 1977.
  • [28] J. Wang, Mixed finite element methods, Numerical Methods in Scientific and Engineering Computing, Eds: W. Cai, Z. Shi, C-W. Shu, and J. Xu, Academic Press.
  • [29] Junping Wang and Xiu Ye, A weak Galerkin finite element method for second order elliptic problems, J. Comput. Appl. Math. 241 (2013), 103-115. MR 2994424, https://doi.org/10.1016/j.cam.2012.10.003
  • [30] Junping Wang and Xiu Ye, A weak Galerkin finite element method for the Stokes equation, arXiv:1302.2707, 2013.

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Additional Information

Junping Wang
Affiliation: Division of Mathematical Sciences, National Science Foundation, Arlington, Virginia 22230
Email: jwang@nsf.gov

Xiu Ye
Affiliation: Department of Mathematics, University of Arkansas at Little Rock, Little Rock, Arkansas 72204
Email: xxye@ualr.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02852-4
Keywords: Weak Galerkin, finite element methods, discrete weak divergence, second order elliptic problems, mixed finite element methods
Received by editor(s): March 20, 2012
Received by editor(s) in revised form: November 23, 2012, and December 11, 2012
Published electronically: May 5, 2014
Additional Notes: The research of the first author was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation
This research was supported in part by National Science Foundation Grant DMS-1115097
Article copyright: © Copyright 2014 American Mathematical Society

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