Discontinuous Galerkin derivative operators with applications to second-order elliptic problems and stability (Q2795278)

The authors develop the use of discontinuous Galerkin (DG) operators as a means for understanding and motivating the approximation of second-order elliptic problems. The objective of the paper is to complement the paper by \textit{T. Lewis} and \textit{M. Neilan} [J. Sci. Comput. 59, No. 3, 602--625 (2014; Zbl 1303.65092)] concerning the dual-wind discontinuous Galerkin (DWDG) method, which focuses primarily on the convergence analysis of the proposed method with respect to Dirichlet boundary conditions. The focus of the authors here is on how various DG methods can be constructed through DG derivative operators and then to use the common framework to analytically compare properties of the different methods. It is also observed that the DWDG method is related to the unified framework given by \textit{D. N. Arnold} et al. [SIAM J. Numer. Anal. 39, No. 5, 1749--1779 (2002; Zbl 1008.65080)].