  In the paper \textit{Preconditioning by inverting the {L}aplacian; an
analysis of the eigenvalues. IMA Journal of Numerical Analysis 29, 1 (2009),
24--42}, Nielsen, Hackbusch and Tveito study the operator generated by using
the inverse of the Laplacian as preconditioner for second order elliptic PDEs
$\nabla \cdot (k(x) \nabla u) = f$. They prove that the range of $k(x)$ is
contained in the spectrum of the preconditioned operator, provided that $k$ is
continuous. Their rigorous analysis only addresses mappings defined on infinite
dimensional spaces, but the numerical experiments in the paper suggest that a
similar property holds in the discrete case.
  % Motivated by this investigation, we analyze the eigenvalues of the matrix
$\bf{L}^{-1}\bf{A}$, where $\bf{L}$ and ${\bf{A}}$ are the stiffness matrices
associated with the Laplace operator and general second order elliptic
operators, respectively. Without any assumption about the continuity of $k(x)$,
we prove the existence of a one-to-one pairing between the eigenvalues of
$\bf{L}^{-1}\bf{A}$ and the intervals determined by the images under $k(x)$ of
the supports of the FE nodal basis functions. As a consequence, we can show
that the nodal values of $k(x)$ yield accurate approximations of the
eigenvalues of $\bf{L}^{-1}\bf{A}$. Our theoretical results are illuminated by
several numerical experiments.
