Numerical approximation of the spectra of non-compact operators arising in buckling problems

Abstract

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.