A Conforming Discontinuous Galerkin Finite Element Method: Part III

Abstract

The conforming discontinuous Galerkin (CDG) finite element methods were introduced in [12] on simplicial meshes and in [13] on polytopal meshes. The CDG method gets its name by combining the features of both conforming finite element method and discontinuous Galerkin (DG) finite element method. The goal of this paper is to continue our efforts on simplifying formulations for the finite element method with discontinuous approximation by constructing new spaces for the gradient approximation. Error estimates of optimal order are established for the corresponding CDG finite element approximation in both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented to confirm the theory.