  Compatible schemes localize degrees of freedom according to the physical
nature of the underlying fields and operate a clear distinction between
topological laws and closure relations. For elliptic problems, the cornerstone
in the scheme design is the discrete Hodge operator linking gradients to fluxes
by means of a dual mesh, while a structure-preserving discretization is
employed for the gradient and divergence operators. The discrete Hodge operator
is sparse, symmetric positive definite and is assembled cellwise from local
operators. We analyze two schemes depending on whether the potential degrees of
freedom are attached to the vertices or to the cells of the primal mesh. We
derive new functional analysis results on the discrete gradient that are the
counterpart of the Sobolev embeddings. Then, we identify the two design
properties of the local discrete Hodge operators yielding optimal discrete
energy error estimates. Additionally, we show how these operators can be built
from local nonconforming gradient reconstructions using a dual barycentric
mesh. In this case, we also prove an optimal $L^2$-error estimate for the
potential for smooth solutions. Links with existing schemes (finite elements,
finite volumes, mimetic finite differences) are discussed. Numerical results
are presented on three-dimensional polyhedral meshes.
