  In this work we present a generic framework for non-conforming finite
elements on polytopal meshes, characterised by elements that can be generic
polygons/polyhedra. We first present the functional framework on the example of
a linear elliptic problem representing a single-phase flow in porous medium.
This framework gathers a wide variety of possible non-conforming methods, and
an error estimate is provided for this simple model. We then turn to the
application of the functional framework to the case of a steady degenerate
elliptic equation, for which a mass-lumping technique is required; here, this
technique simply consists in using a different --piecewise constant-- function
reconstruction from the chosen degrees of freedom. A convergence result is
stated for this degenerate model. Then, we introduce a novel specific
non-conforming method, dubbed Locally Enriched Polytopal Non-Conforming
(LEPNC). These basis functions comprise functions dedicated to each face of the
mesh (and associated with average values on these faces), together with
functions spanning the local $\mathbb{P}^1$ space in each polytopal element.
The analysis of the interpolation properties of these basis functions is
provided, and mass-lumping techniques are presented. Numerical tests are
presented to assess the efficiency and the accuracy of this method on various
examples. Finally, we show that generic polytopal non-conforming methods,
including the LEPNC, can be plugged into the gradient discretization method
framework, which makes them amenable to all the error estimates and convergence
results that were established in this framework for a variety of models.
