  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.
