  A unified study is presented in this paper for the design and analysis of
different finite element methods (FEMs), including conforming and nonconforming
FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid
discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG
and WG are shown to admit inf-sup conditions that hold uniformly with respect
to both mesh and penalization parameters. In addition, by taking the limit of
the stabilization parameters, a WG method is shown to converge to a mixed
method whereas an HDG method is shown to converge to a primal method.
Furthermore, a special class of DG methods, known as the mixed DG methods, is
presented to fill a gap revealed in the unified framework.
