Ye2012]. The weak Galerkin method uses discontinuous piecewise polynomials inside each element and on the interfaces to approximate the variational solution. In [@WangYe2012], the authors have proved optimal convergence of the weak Galerkin method for the mixed formulation of second order elliptic equations on very general polygonal and polyhedral meshes. Most of the existing error analysis of finite element methods assume triangular, quadrilateral, or some commonly-seen three-dimensional meshes. To our knowledge, it is the first time that optimal convergence for the finite element solutions has been rigorously proved in [@WangYe2012] for general meshes of arbitrary polygons and polyhedrons.

The discontinuous Galerkin method imposes the interface continuity weakly, and is known to be able to handle non-conformal, hybrid meshes as well as a variety of basis functions. There have been many research works in this direction, for example, nodal discontinuous Galerkin methods [@Bergot10; @Cohen00; @Hesthaven00] for hyperbolic conservation laws. However, we would like to point out that so far there has been no theoretical analysis on the convergence rate of discontinuous Galerkin method, on very general polygonal or polyhedral meshes yet. Motivated by the work in [@WangYe2012], here we would like to fill the gap. The objective of this paper is to establish the theoretical analysis of the interior penalty discontinuous Galerkin method [@Arnold02] for elliptic equations on very general meshes and discrete spaces.

The paper is organized as follows. In Section 2, we briefly describe the interior penalty discontinuous Galerkin method in an abstract setting. In Section 3, several assumptions on the discrete spaces are listed, which form a minimum requirement for the well-posedness and the approximation property of the discrete formulation. Abstract error estimations are given. In Section 4, we discuss choices of meshes and discrete spaces that satisfy the assumptions given in Section 3. Finally, numerical results are presen