  Under some regularity assumptions, we report an a priori error analysis of a
dG scheme for the Poisson and Stokes flow problem in their dual mixed
formulation. Both formulations satisfy a Babu\v{s}ka-Brezzi type condition
within the space H(div) x L2. It is well known that the lowest order
Crouzeix-Raviart element paired with piecewise constants satisfies such a
condition on (broken) H1 x L2 spaces. In the present article, we use this pair.
The continuity of the normal component is weakly imposed by penalizing jumps of
the broken H(div) component. For the resulting methods, we prove well-posedness
and convergence with constants independent of data and mesh size. We report
error estimates in the methods natural norms and optimal local error estimates
for the divergence error. In fact, our finite element solution shares for each
triangle one DOF with the CR interpolant and the divergence is locally the
best-approximation for any regularity. Numerical experiments support the
findings and suggest that the other errors converge optimally even for the
lowest regularity solutions and a crack-problem, as long as the crack is
resolved by the mesh.
