  In this paper, we provide a number of new estimates on the stability and
convergence of both hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG)
methods. By using the standard Brezzi theory on mixed methods, we carefully
define appropriate norms for the various discretization variables and then
establish that the stability and error estimates hold uniformly with respect to
stabilization and discretization parameters. As a result, by taking appropriate
limit of the stabilization parameters, we show that the HDG method converges to
a primal conforming method and the WG method converge to a mixed conforming
method.
