SIGEST

Abstract

SIAM Review, Volume 66, Issue 1, Page 123-123, February 2024.
The SIGEST article in this issue is “A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators,” by Bjørn Fredrik Nielsen and Zdeněk Strakoš. This paper studies the eigenvalues of second-order self-adjoint differential operators in the continuum and discrete settings. In particular, they investigate second-order diffusion with a diffusion tensor preconditioned by the inverse Laplacian. They prove that there is a one-to-one correspondence between the spectrum of the preconditioned system and the eigenvalues of the diffusion tensor. Moreover, they investigate the relationship between the spectrum of the preconditioned operator and the generalized eigenvalue problem for its discretized counterpart and show that the latter asymptotically approximates the former. The results presented in the paper are fundamental to anyone wanting to solve elliptic PDEs. Understanding the distribution of eigenvalues is crucial for solving associated linear systems via, e.g., conjugate gradient descent whose convergence rate depends on the spread of the spectrum of the system matrix. The approach of operator preconditioning as done here with the inverse Laplacian turns the unbounded spectrum of a second-order diffusion operator into one that is completely characterized by the diffusion tensor itself. This carries over to the discrete setting, where the support of the spectrum without preconditioning is increasing as one over the squared mesh size, while in the operator preconditioned case mesh independent bounds for the eigenvalues, completely determined by the diffusion tensor, can be obtained. The original version of this article appeared in the SIAM Journal on Numerical Analysis in 2020 and has been recognized as an outstanding and well-presented result in the community. In preparing this SIGEST version, the authors have added new material to sections 1 and 2 in order to increase accessibility, added clarifications to sections 6 and 7, and added the new section 8, which contains a description of more recent results concerning the numerical approximation of the continuous spectrum. It also comments on the related differences between the (generalized) PDE eigenvalue problems for compact and noncompact operators and provides several new references.