  We study a non standard mixed formulation of the Poisson problem, sometimes
known as dual mixed formulation. For reasons related to the equilibration of
the flux, we use finite elements that are conforming in H(div) for the
approximation of the gradients, even if the formulation would allow for
discontinuous finite elements. The scheme is not uniformly inf-sup stable, but
we can show existence and uniqueness of the solution, as well as optimal error
estimates for the gradient variable when suitable regularity assumptions are
made. Several additional remarks complete the paper, shedding some light on the
sources of instability for mixed formulations.
