The authors study a variational formulation of second-order elliptic equations in mixed form that is obtained by piecewise integrating one of the two equations in the system w.r.t. a partition of the domain into mesh cells. A Petrov-Galerkin discretization with optimal test functions is applied. These optimal test functions can be computed by solving local problems. Well-posedness and optimal error estimates are proved. In the second part of the paper, the application to convection-diffusion problems is studied. The available freedom in the variational formulation and in its optimal Petrov-Galerkin discretization is used to construct a method that allows a passing to a converging method in the convective limit, being a necessary condition to retain convergence and having a bound on the cost for a vanishing diffusion. The theoretical findings are supported by several numerical results.