Laplacian Preconditioning of Elliptic PDEs: Localization of the Eigenvalues of the Discretized Operator

Abstract

In [IMA J. Numer. Anal., 29 (2009), pp. 24--42], Nielsen, Tveito, and Hackbusch study the operator generated by using the inverse of the Laplacian as the 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 $k(x)$ 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 ${L}^{-1}{A}$, where ${L}$ and ${{A}}$ are the stiffness matrices associated with the Laplace operator and second order elliptic operators with a scalar coefficient function, respectively. Using only technical assumptions on $k(x)$, we prove the existence of a one-to-one pairing between the eigenvalues of ${L}^{-1}{A}$ and the intervals determined by the images under $k(x)$ of the supports of the finite element nodal basis functions. As a consequence, we can show that the nodal values of $k(x)$ yield accurate approximations of the eigenvalues of ${L}^{-1}{A}$. Our theoretical results, including their relevance for understanding how the convergence of the conjugate gradient method may depend on the whole spectrum of the preconditioned matrix, are illuminated by several numerical experiments.