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

TL;DR

This paper analyzes the eigenvalues of discretized elliptic PDE operators preconditioned by the inverse Laplacian, showing that nodal values of the coefficient function approximate eigenvalues well, supported by theoretical proofs and numerical experiments.

Contribution

It proves a new relationship between eigenvalues of the preconditioned operator and the coefficient function's nodal values without requiring continuity of k(x).

Findings

Eigenvalues of the preconditioned matrix are paired with intervals from k(x)

Nodal values of k(x) approximate eigenvalues accurately

Theoretical results are confirmed by numerical experiments

Abstract

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.