Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Summary

In an abstract framework we present a formalism which specifies 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 refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: −div (k(u)▽u) + c • ▽u = f in a two dimensional domain Ω with Dirichlet boundary conditions.