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

Abstract

Summary. The study of the magnetohydrodynamic stability of a plasma leads to a problem of determination of the spectrum of a non-compact selfadjoint operator A. The spectrum of A will be approximated by the eigenvalues of A h, where A h is a linear operator approximating A in a finite dimensional space (finite element method) and h is a parameter which tends to zero.Generally the spectrum of A n "pollutes" spectrum of A, i.e. for each h there exists an eigenvalue 2 h of A h which, as h tends to zero, converges to 2 which is not in the spectrum of A.We present here a sufficient condition on the finite dimensional spaces used, in order to obtain good approximation properties of the spectrum of A and, especially, the "non-pollution" property. Definitions and NotationsLet U ~ V be two real Hilbert spaces with respect to inner products (., -)u, (-, .)

On spectral approximation. Part 1. The problem of convergence

“…Note that this operator can also be treated by a different method developped in [7] by J. Rappaz. For an one-dimensional example, see also [3].…”

On spectral approximation. Part 2. Error estimates for the Galerkin method

“…In order to illustrate this theorem, we consider the example developped in section 4 of part 1 of this paper [3] ; one can prove by Rappaz' method of élimination used in [6] the existence of an infinité number of isolated eigenvalues of finite multiplicities; by supposing the coefficients a, P, ... sufficiently smooth, one vérifies that the corresponding eigensubspaces are subsets of H 2 x (H 1 ) 2 ; consequently y h = O (h), y* = O (h) and the estimâtes of theorem 3 a, b are of order h 2 .…”

Mode coupling in two-dimensional magnetohydrodynamic flows

“…The radial direction is discretized using a finite-element method. In order to avoid poor resolution of the eigenvalues or extra spurious eigenvalues ('spectral pollution'),v 1 ,Ā 2 andĀ 3 are discretized using cubic elements, whereas the other variables are discretized using quadratic elements (Rappaz 1977). For the poloidal direction, a Fourier decomposition is made, where the different modes will couple owing to geometric effects.…”