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Title: A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods

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Journal Article: Publisher's Accepted Manuscript
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Journal of Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 273; Journal Issue: C; Related Information: CHORUS Timestamp: 2017-06-01 03:13:31; Journal ID: ISSN 0377-0427
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Citation Formats

Lin, Guang, Liu, Jiangguo, and Sadre-Marandi, Farrah. A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods. Belgium: N. p., 2015. Web. doi:10.1016/
Lin, Guang, Liu, Jiangguo, & Sadre-Marandi, Farrah. A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods. Belgium. doi:10.1016/
Lin, Guang, Liu, Jiangguo, and Sadre-Marandi, Farrah. 2015. "A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods". Belgium. doi:10.1016/
title = {A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods},
author = {Lin, Guang and Liu, Jiangguo and Sadre-Marandi, Farrah},
abstractNote = {},
doi = {10.1016/},
journal = {Journal of Computational and Applied Mathematics},
number = C,
volume = 273,
place = {Belgium},
year = 2015,
month = 1

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Publisher's Version of Record at 10.1016/

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Cited by: 7works
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  • We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form, then the formulation must reduce to that for conservative systems. Standard DGFEM formulations cannot be applied to nonconservative systems of partial differential equations. We therefore introduce the theory of weak solutions for nonconservative products into the DGFEM formulation leading to the new question how to define the path connecting left andmore » right states across a discontinuity. The effect of different paths on the numerical solution is investigated and found to be small. We also introduce a new numerical flux that is able to deal with nonconservative products. Our scheme is applied to two different systems of partial differential equations. First, we consider the shallow water equations, where topography leads to nonconservative products, in which the known, possibly discontinuous, topography is formally taken as an unknown in the system. Second, we consider a simplification of a depth-averaged two-phase flow model which contains more intrinsic nonconservative products.« less
  • Cited by 1
  • This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising frommore » the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.« less
  • This paper presents a family of weak Galerkin finite element methods (WGFEMs) for Darcy flow computation. The WGFEMs are new numerical methods that rely on the novel concept of discrete weak gradients. The WGFEMs solve for pressure unknowns both in element interiors and on the mesh skeleton. The numerical velocity is then obtained from the discrete weak gradient of the numerical pressure. The new methods are quite different than many existing numerical methods in that they are locally conservative by design, the resulting discrete linear systems are symmetric and positive-definite, and there is no need for tuning problem-dependent penalty factors.more » We test the WGFEMs on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in Darcy flow.« less
  • The weak Galerkin (WG) methods have been introduced in [11, 12, 17] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of theWG method and its Schur complement is established. The numerical results demonstrate themore » effectiveness of this new implementation technique.« less