A conforming discontinuous Galerkin finite element method: Part II

Abstract: A conforming discontinuous Galerkin (DG) finite element method has been introduced in [21] on simplicial meshes, which has the flexibility of using discontinuous approximation and the simplicity in formulation of the classic continuous finite element method. The goal of this paper is to extend the conforming DG finite element method in \cite{cdg1} so that it can work on general polytopal meshes by designing weak gradient $\nabla_w$ appropriately. Two different conforming DG formulations on polytopal meshes are introduced which handle boundary conditions differently. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented to confirm the theory.