  We propose employing the extension of the Lehmann-Maehly-Goerisch method
developed by Zimmermann and Mertins, as a highly effective tool for the
pollution-free finite element computation of the eigenfrequencies of the
resonant cavity problem on a bounded region. This method gives complementary
bounds for the eigenfrequencies which are adjacent to a given real parameter.
We present a concrete numerical scheme which provides certified enclosures in a
suitable asymptotic regime. We illustrate the applicability of this scheme by
means of some numerical experiments on benchmark data using Lagrange elements
and unstructured meshes.
