Approximation of the spectrum of a non-compact operator given by the magnetohydrodynamic stability of a plasma

Abstract

The study of the magnetohydrodynamic stability of a plasma leads to a problem of determination of the spectrum of a non-compact selfadjoint operatorA. The spectrum ofA will be approximated by the eigenvalues ofA h , whereA h is a linear operator approximatingA in a finite dimensional space (finite element method) andh is a parameter which tends to zero. Generally the spectrum ofA h "pollutes" spectrum ofA, i.e. for eachh there exists an eigenvalue ?h ofA h which, ash tends to zero, converges to ? which is not in the spectrum ofA. We present here a sufficient condition on the finite dimensional spaces used, in order to obtain good approximation properties of the spectrum ofA and, especially, the "non-pollution" property.