  A general analysis framework is presented in this paper for many different
types of finite element methods (including various discontinuous Galerkin
methods). For second order elliptic equation, this framework employs $4$
different discretization variables, $u_h, \bm{p}_h, \check u_h$ and $\check
p_h$, where $u_h$ and $\bm{p}_h$ are for approximation of $u$ and
$\bm{p}=-\alpha \nabla u$ inside each element, and $ \check u_h$ and $\check
p_h$ are for approximation of residual of $u$ and $\bm{p} \cdot \bm{n}$ on the
boundary of each element. The resulting 4-field discretization is proved to
satisfy inf-sup conditions that are uniform with respect to all discretization
and penalization parameters. As a result, most existing finite element and
discontinuous Galerkin methods can be analyzed using this general framework by
making appropriate choices of discretization spaces and penalization
parameters.
