Extending Eigenvalue Localization Results for Preconditioned PDE Discretizations

Extend the eigenvalue localization techniques developed for Laplacian-preconditioned second-order self-adjoint PDEs to broader classes of PDEs or to different preconditioners, establishing precise spectral localization results that explain iterative solver behavior.

Background

Recent work shows exact eigenvalue localization for Laplacian-preconditioned elliptic PDE discretizations, providing new insight into CG convergence beyond condition-number bounds.

The authors point to several specific open problems cataloged elsewhere and ask for generalization to other PDEs and preconditioners to broaden the explanatory power of these techniques.

References

Several specific open problems that exist in this context are mentioned inSection~9. More generally, try to extend the approach in to other types of PDEs or preconditioners.

— Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop  (2602.05394 - Amsel et al., 5 Feb 2026) in Subsection "Preconditioning and convergence of iterative solvers" (Section 2)

However, effective preconditioning for those iterative solvers applied to the time-harmonic wave equation remains an open challenge: the linear system resulting from the discretization of the equation is indefinite, rendering classical approaches such as shifted Laplacian, multigrid, and domain decomposition methods less effective.

— A discontinuous Petrov-Galerkin finite-element framework for the simulation of microwave-heated flows  (2609.03155 - Marquis et al., 2 Sep 2026) in Introduction