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A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form

Authors: Chunmei Wang and Junping Wang
Journal: Math. Comp. 87 (2018), 515-545
MSC (2010): Primary 65N30, 65N12, 35J15, 35D35
Published electronically: June 27, 2017
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Abstract: This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The numerical solution is characterized as a minimization of a non-negative quadratic functional with constraints that mimic the second order elliptic equation by using the discrete weak Hessian. The resulting Euler-Lagrange equation offers a symmetric finite element scheme involving both the primal and a dual variable known as the Lagrange multiplier, and thus the name of primal-dual weak Galerkin finite element method. Error estimates of optimal order are derived for the corresponding finite element approximations in a discrete $ H^2$-norm, as well as the usual $ H^1$- and $ L^2$-norms. The convergence theory is based on the assumption that the solution of the model problem is $ H^2$-regular, and that the coefficient tensor in the PDE is piecewise continuous and uniformly positive definite in the domain. Some numerical results are presented for smooth and non-smooth coefficients on convex and non-convex domains, which not only confirm the developed convergence theory but also a superconvergence result.

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  • [1] Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179-192. MR 0359352
  • [2] Susanne C. Brenner, Thirupathi Gudi, Michael Neilan, and L. Sung, $ \mathcal {C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp. 80 (2011), no. 276, 1979-1995. MR 2813346,
  • [3] Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954
  • [4] 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. MR 0365287
  • [5] Maurizio Chicco, Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients, Ann. Mat. Pura Appl. (4) 92 (1972), 13-22. MR 0328316
  • [6] Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [7] M. Crouzeix and V. Thomée, The stability in $ L_p$ and $ W^1_p$ of the $ L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521-532. MR 878688,
  • [8] A. Dedner and T. Pryer, Discontinuous Galerkin methods for nonvariational problems, arXiv:1304.2265v1.
  • [9] X. Feng, L. Hennings, and M. Neilan, $ C0$ discontinuous Galerkin finite element methods for second order linear elliptic partial differential equations in non-divergence form, arXiv:1505.02842.
  • [10] Wendell H. Fleming and H. Mete Soner, Controlled Markov Processes and Viscosity Solutions, 2nd ed., Stochastic Modelling and Applied Probability, vol. 25, Springer, New York, 2006. MR 2179357
  • [11] David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [12] Vivette Girault and Pierre-Arnaud Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR 851383
  • [13] Pierre Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original [MR0775683]; With a foreword by Susanne C. Brenner. MR 3396210
  • [14] Omar Lakkis and Tristan Pryer, A finite element method for second order nonvariational elliptic problems, SIAM J. Sci. Comput. 33 (2011), no. 2, 786-801. MR 2801189,
  • [15] Antonino Maugeri, Dian K. Palagachev, and Lubomira G. Softova, Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research, vol. 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. MR 2260015
  • [16] Lin Mu, Junping Wang, and Xiu Ye, Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model. 12 (2015), no. 1, 31-53. MR 3286455
  • [17] Lin Mu, Junping Wang, Xiu Ye, and Shangyou Zhang, A $ C^0$-weak Galerkin finite element method for the biharmonic equation, J. Sci. Comput. 59 (2014), no. 2, 473-495. MR 3188449,
  • [18] M. Neilan, Convergence analysis of a finite element method for second order non-variational elliptic problems, J. Numer. Math., DOI: 10.1515/jnma-2016-1017.
  • [19] Michael Neilan, Quadratic finite element approximations of the Monge-Ampère equation, J. Sci. Comput. 54 (2013), no. 1, 200-226. MR 3002620,
  • [20] R. H. Nochetto and W. Zhang, Discrete ABP estimate and convergence rates for linear elliptic equations in non-divergence form, arXiv:1411.6036.
  • [21] Iain Smears and Endre Süli, Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordès coefficients, SIAM J. Numer. Anal. 51 (2013), no. 4, 2088-2106. MR 3077903,
  • [22] Iain Smears and Endre Süli, Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cordes coefficients, SIAM J. Numer. Anal. 52 (2014), no. 2, 993-1016. MR 3196952,
  • [23] Chunmei Wang and Junping 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), no. 12, 2314-2330. MR 3284276,
  • [24] 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,
  • [25] Junping Wang and Xiu Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comp. 83 (2014), no. 289, 2101-2126. MR 3223326,
  • [26] Junping Wang and Xiu Ye, A weak Galerkin finite element method for the stokes equations, Adv. Comput. Math. 42 (2016), no. 1, 155-174. MR 3452926,

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Additional Information

Chunmei Wang
Affiliation: Department of Mathematics, Texas State University, San Marcos, Texas 78666-4684

Junping Wang
Affiliation: Division of Mathematical Sciences, National Science Foundation, Arlington, Virginia 22230

Keywords: Weak Galerkin, finite element methods, non-divergence form, weak Hessian operator, discontinuous coefficients, Cord\`es condition, polyhedral meshes
Received by editor(s): October 21, 2015
Received by editor(s) in revised form: September 16, 2016
Published electronically: June 27, 2017
Additional Notes: The research of the first author was partially supported by National Science Foundation Awards #DMS-1522586 and #DMS-1648171
The research of the second author was supported by the NSF IR/D program, while working at National Science 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
Article copyright: © Copyright 2017 American Mathematical Society

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