Piecewise constant pressure for Darcy law

We present a short discussion on some finite element formulations for linear elliptic problems. For the sake of symplicity we consider the Poisson equation, taking the notation from Darcy's law. Among the zillions of such methods, we shall concentrate our attention on FEM leading to a final system of linear algebraic equations "M P=F" where each unknown represents the constant value of the approximated pressure in a single element. It is indeed well known that for some applications there is a certain demand for these types of schemes