  The gradient scheme framework is based on a small number of properties and
encompasses a large number of numerical methods for diffusion models. We recall
these properties and develop some new generic tools associated with the
gradient scheme framework. These tools enable us to prove that classical
schemes are indeed gradient schemes, and allow us to perform a complete and
generic study of the well-known (but rarely well-studied) mass lumping process.
They also allow an easy check of the mathematical properties of new schemes, by
developing a generic process for eliminating unknowns via barycentric
condensation, and by designing a concept of discrete functional analysis
toolbox for schemes based on polytopal meshes.
