TP or not TP, that is the question

We give here a comparative study of the mathematical analysis of two (classes of) discretisation schemes for the computation of approximate solutions to incompressible two phase flow problem in homogeneous porous media. The first scheme is the well-known finite volume scheme with a two-point flux approximation, classically used in industry. The second class contains the so-called approximate gradient schemes, which include finite elements with mass lumping, mixed finite elements, mimetic finite differences. Both (classes of) schemes are nonconforming and can be expressed using discrete function and gradient reconstructions within a variational formulation. Each class has its specific advantages and drawbacks: monotony properties are natural with the two point finite volume scheme, but meshes are restricted due to consistency issues; on the contrary, gradient schemes can be used on general meshes, but monotony properties are difficult to obtain.