  In this paper we investigate the approximation of a diffusion model problem
with contrasted diffusivity and the error analysis of various nonconforming
approximation methods. The essential difficulty is that the Sobolev smoothness
index of the exact solution may be just barely larger than one. The lack of
smoothness is handled by giving a weak meaning to the normal derivative of the
exact solution at the mesh faces. The error estimates are robust with respect
to the diffusivity contrast. We briefly show how the analysis can be extended
to the Maxwell's equations.
