Consistency, Stability, a-Priori and a-Posteriori Errors for Nonlinear Problems

In an abstract framework similar to the one developed in Crouzeix-Rappaz [2], we present a formalism which makes precise the notions of consistency and stability for the finite element approximation of nonlinear elliptic problems. The consistency is connected to the approximation by interpolation, whereas the stability follows from discrete inf-sup conditions of the linearized problem. Moreover, contrary to the method developed in [2], this formalism does not require us to invert the principal part of the operator and allows us to obtain a priori and a posteriori error estimates for strongly nonlinear problems.