A unified analysis of elliptic problems with various boundary conditions and their approximation

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 boundeddomain. 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 unifiedconvergence analysis that simultaneously covers(i) all usual boundary conditions,(ii) several approximation methods.The considered approximations can be conforming (that is, the approximation functions can belong to theenergy space relative to the problem) or not, and include classical as well as recent numerical schemes.Convergence results, a priori and a posteriori error estimates are given. We finally briefly show how theabstract setting can also be applied to some models such as flows in fractured medium, elasticity equationsand diffusion equations on manifolds. A by-product of the analysis is a result on the equivalence betweengeneral Poincaré inequalities and the surjectivity of the divergence operator in appropriate spaces