Error analysis of piecewise constant approximations of Darcy's law

In this paper we present a short discussion on some finite element formulations for linear elliptic problems. For the sake of simplicity 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.