Influence of numerical integration on convergence of eigenvalues in magnetohydrodynamics stability analysis

When the eigenvalue problem of the linearized magnetohydrodynamics equation is solved by finite-element methods: whether energy integrals are exactly carried out or not affects the convergence properties of eigenvalues. If th e energy integrals are exactly carried out, the eigenvalue of the most viol ent instability (the lowest eigenvalue) is approximated from "above," that is, the approximated eigenvalue decreases towards the true eigenvalue as th e number of elements increases. If the energy integrals are estimated by Ga ussian quadrature formulas in which errors are of the same order as those b y the finite-element method, the lowest eigenvalue is approximated from "be low."