  We consider the problem of how to compute eigenvalues of a self-adjoint
operator when a direct application of the Galerkin (finite-section) method is
unreliable. The last two decades have seen the development of the so-called
quadratic methods for addressing this problem. Recently a new perturbation
approach has emerged, the idea being to perturb eigenvalues off the real line
and, consequently, away from regions where the Galerkin method fails. We
propose a simplified perturbation method which requires no \'a priori
information and for which we provide a rigorous convergence analysis. The
latter shows that, in general, our approach will significantly outperform the
quadratic methods. We also present a new spectral enclosure for operators of
the form $A+iB$ where $A$ is self-adjoint, $B$ is self-adjoint and bounded.
This enables us to control, very precisely, how eigenvalues are perturbed from
the real line. The main results are demonstrated with examples including
magnetohydrodynamics, Schr\"odinger and Dirac operators.
