The finite element method approximates the spectrum of an operator S by computing the spectra of a sequence of operators S N defined in terms of the finite element spaces. For the case that S is compact, convergence of the approximate spectra follows from the convergence of S N to S in the operator norm. We consider the case that S is non-compact, in which case such operator norm convergence cannot take place, and the approximations may be polluted by spurious eigenvalues. Pollution-free convergence of the eigenvalues can, however, be guaranteed outside the essential numerical range of S , which is related to the essential spectrum of S . We present results for estimating this essential numerical range and apply them to an algorithm for the buckling of three-dimensional bodies (that gives rise to a non-compact S ). Our results show, for instance, that for the example of a circular disc, the algorithm will be free of spurious eigenvalues provided the body is thin enough. The case that singularities in the stresses can lead to non-physical spectral values being approximated is also investigated.