  We study the weak finite element method solving convection-diffusion
equations. A weak finite element scheme is presented based on a spacial
variational form. We established a weak embedding inequality that is very
useful in the weak finite element analysis. The optimal order error estimates
are derived in the discrete $H^1$-norm, the $L_2$-norm and the $L_\infty$-norm,
respectively. In particular, the $H^1$-superconvergence of order $k+2$ is given
under certain condition. Finally, numerical examples are provided to illustrate
our theoretical analysis
