weak form of an equation by continuous discontinuous galerkin method

Question

If we have the weak form of an equation like Poisson equation with discontinuous Galerkin method, to get the weak form of continuous discontinuous Galerkin method what terms are changed? Is it true if just remove the term contained $[u][v]$ in the following: $$a(u,v)= (f,v)$$ where$$a(u,v)=\sum_{T \in \cal T}(\nabla u, \nabla v)_T-\sum_{E \in \cal \partial T}(<n.\nabla u>, [v])_E-(n.\nabla u,v)_{\partial \Omega}+\alpha \sum_{E \in \cal \partial T}([u], <n.\nabla v>)_E+\alpha (u-g,n.\nabla v)_{\partial \Omega}+\sum _{E \in \cal \partial T}\beta h^{-1}([u],[v])_E+ \beta h^{-1}([u-g],[v])_{\partial \Omega}$$

Answer 1

Work backwards (disintegrate by parts) until the point where you introduce the elements.Take the elements out. This will turn all the jump terms back into continuous ones. Integrate by parts forwards without elements, eliminate things that are zero (by BCs or whatever), then introduce the elements. There's no magic steps here. You might get lucky with some operators (linear, symmetric, positive definite ones seem likely) that dropping the terms you want to drop reduces things to what you want, but it's unlikely this process will work in more general or non-linear cases.