Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids

2021 ◽  
Author(s):  
Mahboub Baccouch
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Elliptic Problems ◽  
Local Discontinuous Galerkin Method ◽  
Second Order ◽  
Cartesian Grids ◽  
Local Discontinuous Galerkin ◽  
Convergence And Superconvergence ◽  
Second Order Elliptic Problems

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

2019 ◽  
Vol 19 (4) ◽  
pp. 849-861 ◽  
Author(s):  
Xiao Zhang ◽  
Xiaoping Xie ◽  
Shiquan Zhang
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Convergence Rates ◽  
Elliptic Problems ◽  
Second Order ◽  
Lagrangian Multiplier ◽  
Piecewise Polynomials ◽  
Numer Anal ◽  
Second Order Elliptic Problems

AbstractThe 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 ({k\geq 0}) 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.

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