Consistency , stability , a priori and a posteriorierrors for Petrov - Galerkin methodsapplied

In an abstract framework we present a formalism which speciies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the reenement of the mesh in adaptive nite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: ?div (k(u)ru) + c ru = f in a two dimensional domain with Dirichlet boundary conditions.