If we use CG elements (continuous Galerkin), the boundary integration in FEM can be easily converted to sum over quadrature points using node basis functions of the edges. However, in DG elements (discontinuous Galerkin), there is no shared node basis and each elements have its own node basis.

- How the boundary integration can be done in DG case?
- There is a concept of topology used to describe elements. By assigning DOF to node, edge, element. This topology is very different in CG vs DG. Is it possible to generalize, this transformation of integration to finite sum, using this topology?