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

In the paper $\mathit{\text{Preconditioning}}\mathit{\text{by}}\mathit{\text{inverting}}\mathit{\text{the}}\mathit{\text{Laplacian;}}\mathit{\text{an}}\mathit{\text{analysis}}\mathit{\text{of}}\mathit{\text{the}}\mathit{\text{eigenvalues.}}\mathit{\text{IMA}}\mathit{\text{Journal}}\mathit{\text{of}}\mathit{\text{Numerical}}\mathit{\text{Analysis}}\mathit{\text{29,}}\mathit{\text{1}}\mathit{\text{(2009),}}\mathit{\text{24–42}}$, Nielsen, Hackbusch and Tveito study the operator generated by using the inverse of the Laplacian as preconditioner for second order elliptic PDEs $\mathrm{\nabla }\cdot (k(x)\mathrm{\nabla }u)=f$. 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 ${\mathbf{L}}^{-\mathbf{1}}\mathbf{A}$, where $\mathbf{L}$ and $\mathbf{A}$ 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 ${\mathbf{L}}^{-\mathbf{1}}\mathbf{A}$ 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 ${\mathbf{L}}^{-\mathbf{1}}\mathbf{A}$. Our theoretical results are illuminated by several numerical experiments.