  A new finite element method with discontinuous approximation is introduced
for solving second order elliptic problem. Since this method combines the
features of both conforming finite element method and discontinuous Galerkin
(DG) method, we call it conforming DG method. While using DG finite element
space, this conforming DG method maintains the features of the conforming
finite element method such as simple formulation and strong enforcement of
boundary condition. Therefore, this finite element method has the flexibility
of using discontinuous approximation and simplicity in formulation of the
conforming finite element method. Error estimates of optimal order are
established for the corresponding discontinuous finite element approximation in
both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented
to confirm the theory.
