Preconditioning FEM discretisations of the high-frequency Helmholtz and Maxwell equations by either perturbing the coefficients or adding absorption

Abstract

This paper investigates the following question: given a Galerkin matrix corresponding to a finite-element discretisation of either the Helmholtz or time-harmonic Maxwell equations with variable coefficients, suppose that the coefficients of the underlying PDE are perturbed; how good an approximate inverse (i.e., preconditioner) is the resulting Galerkin matrix to the original Galerkin matrix? An important special case is when the perturbation consists of adding absorption (in the spirit of "shifted Laplacian preconditioning"). The results of this paper improve the Helmholtz results in [Gander, Graham, Spence, 2015] and [Graham, Pembery, Spence, 2021], and extend these results to the time-harmonic Maxwell equations, confirming a conjecture in the recent preprint [Li, Hu, arXiv 2501.18305].