Finite element approximation of some indefinite elliptic problems

We consider the finite element approximation of some indefinite Neumann problems in a domain of IR^N. From the Fredholm Alternative this kind of problem admits a solution if and only if the right hand term has zero mean value with respect to a measure whose density m is the solution of a homogeneous adjoint problem. The first step consists in the construction of piecewise linear finite element approximations m_h of m, showing their optimal rate of convergence both in energy and L^p norms. The functions m_h are then shown to be crucial in testing admissible data for the Neumann problem and also in its numerical resolution (actually, the standard Galerkin approximation may not be solvable without suitable perturbations of the data).