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An Optimal Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems

Xiao Zhang, Xiaoping Xie and Shiquan Zhang

Abstract

The embedded discontinuous Galerkin (EDG) method by Cockburn, Gopalakrishnan and Lazarov [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, 2, 1319–1365] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second-order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees k+1, k+1, k (k0) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11771312

Award Identifier / Grant number: 11401407

Award Identifier / Grant number: 91430105

Funding statement: This work was supported in part by National Natural Science Foundation of China (11771312, 11401407) and Major Research Plan of National Natural Science Foundation of China (91430105).

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Received: 2017-11-09
Revised: 2018-02-05
Accepted: 2018-02-27
Published Online: 2018-03-20
Published in Print: 2019-10-01

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