  In this paper, we develop global superconvergence estimates for the lowest
order Raviart--Thomas mixed finite element method for second order elliptic
equations with general boundary conditions on triangular meshes, where most
pairs of adjacent triangles form approximate parallelograms. In particular, we
prove the $L^{2}$-distance between the numerical solution and canonical
interpolant for the vector variable is of order $1+\rho$, where $\rho\in(0,1]$
is dependent on the mesh structure. By a cheap local postprocessing operator
$G_{h}$, we prove the $L^{2}$-distance between the exact solution and the
postprocessed numerical solution for the vector variable is of order $1+\rho$.
As a byproduct, we also obtain the superconvergence estimate for
Crouzeix--Raviart nonconforming finite elements on triangular meshes of the
same type.
