  We design an abstract setting for the approximation in Banach spaces of
operators acting in duality. A typical example are the gradient and divergence
operators in Lebesgue--Sobolev spaces on a bounded domain. We apply this
abstract setting to the numerical approximation of Leray-Lions type problems,
which include in particular linear diffusion. The main interest of the abstract
setting is to provide a unified convergence analysis that simultaneously covers
(i) all usual boundary conditions, (ii) several approximation methods. The
considered approximations can be conforming, or not (that is, the approximation
functions can belong to the energy space of the problem, or not), and include
classical as well as recent numerical schemes. Convergence results and error
estimates are given. We finally briefly show how the abstract setting can also
be applied to other models, including flows in fractured medium, elasticity
equations and diffusion equations on manifolds. A by-product of the analysis is
an apparently novel result on the equivalence between general Poincar{\'e}
inequalities and the surjectivity of the divergence operator in appropriate
spaces.
