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In the paper \textit{Preconditioning by inverting the {L}aplacian; ananalysis of the eigenvalues. IMA Journal of Numerical Analysis 29, 1 (2009),24--42}, Nielsen, Hackbusch and Tveito study the operator generated by usingthe inverse of the Laplacian as preconditioner for second order elliptic PDEs$\nabla \cdot (k(x) \nabla u) = f$. They prove that the range of $k(x)$ iscontained in the spectrum of the preconditioned operator, provided that $k$ iscontinuous. Their rigorous analysis only addresses mappings defined on infinitedimensional spaces, but the numerical experiments in the paper suggest that asimilar 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 matricesassociated with the Laplace operator and general second order ellipticoperators, 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 $k(x)$ ofthe supports of the FE nodal basis functions. As a consequence, we can showthat the nodal values of $k(x)$ yield accurate approximations of theeigenvalues of $\bf{L}^{-1}\bf{A}$. Our theoretical results are illuminated byseveral numerical experiments.

Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator