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

Euan A. Spence

TL;DR

This work develops a rigorous framework for preconditioning high-frequency Helmholtz and time-harmonic Maxwell discretisations by perturbing coefficients or adding absorption. It proves that, under a discrete–continuous inf-sup comparability and small coefficient perturbations, the perturbed Galerkin inverse A_2^{-1} yields explicit, k-stable bounds on the preconditioned operators ||I - A_2^{-1} A_1|| and ||I - A_1 A_2^{-1}||, with a linear scaling in the absorption parameter in the special case μ_2=μ_1, ε_2=(1+iα)ε_1. The results generalise prior Helmholtz bounds to a broader class of Helmholtz and Maxwell problems, require assumptions only on the original problem a_1(·,·), and rely on a new discrete-to-continuous inf-sup bound in the preasymptotic regime. These findings underpin efficient, frequency-explicit preconditioning strategies and support two-level domain-decomposition analyses for Maxwell and Helmholtz, including connections to recent work on uncertainty quantification. Overall, the paper advances theoretical guarantees for absorption-based and coefficient-perturbation preconditioners in high-frequency wave simulations, with practical implications for robust iterative solvers in heterogeneous media.

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].

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

TL;DR

This work develops a rigorous framework for preconditioning high-frequency Helmholtz and time-harmonic Maxwell discretisations by perturbing coefficients or adding absorption. It proves that, under a discrete–continuous inf-sup comparability and small coefficient perturbations, the perturbed Galerkin inverse A_2^{-1} yields explicit, k-stable bounds on the preconditioned operators ||I - A_2^{-1} A_1|| and ||I - A_1 A_2^{-1}||, with a linear scaling in the absorption parameter in the special case μ_2=μ_1, ε_2=(1+iα)ε_1. The results generalise prior Helmholtz bounds to a broader class of Helmholtz and Maxwell problems, require assumptions only on the original problem a_1(·,·), and rely on a new discrete-to-continuous inf-sup bound in the preasymptotic regime. These findings underpin efficient, frequency-explicit preconditioning strategies and support two-level domain-decomposition analyses for Maxwell and Helmholtz, including connections to recent work on uncertainty quantification. Overall, the paper advances theoretical guarantees for absorption-based and coefficient-perturbation preconditioners in high-frequency wave simulations, with practical implications for robust iterative solvers in heterogeneous media.

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].

Paper Structure

This paper contains 31 sections, 12 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Preconditioning using absorption.
  4. "Nearby preconditioning".
  5. Content and motivation for the present paper.
  6. Informal statement of the main result
  7. Set-up at the continuous level.
  8. Set-up at the discrete level.
  9. Informal statement of the main result (Theorem \ref{['thm:main']} below).
  10. Discussion
  11. The main result specialised to preconditioning with absorption
  12. Recovering the Helmholtz results of GaGrSp:15 about preconditioning with absorption
  13. The condition that the discrete inf-sup constant is bounded by the continuous inf-sup constant
  14. How the results of this paper improve the Helmholtz results of GaGrSp:15GrPeSp:19
  15. Numerical experiments investigating the sharpness of the bounds \ref{['eq:main1a']}, \ref{['eq:main2a']}, \ref{['eq:abs1a']}, and \ref{['eq:abs2a']}
  16. ...and 16 more sections

Key Result

Lemma 2.2

Let $\Omega$ be a bounded Lipschitz open set, ${\cal H}_0 = L^2(\Omega)$ (scalar valued), ${\mathcal{D}}:= k^{-1} \nabla$, and For $\ell=1,2,$ let $\mu_\ell^{-1}$ be bounded, symmetric, matrix functions on $\Omega$ with ${\rm essinf}_\Omega\Re \mu_\ell^{-1}>0$ (in the sense of quadratic forms) and $\epsilon_\ell$ be bounded, scalar-valued functions on $\Omega$. If one of the following three point

Theorems & Definitions (32)

  • Lemma 2.2: Helmholtz problems satisfying Assumption \ref{['ass:abstract']}
  • Remark 2.3: The Helmholtz problems covered in Lemma \ref{['lem:Helmholtz']}
  • proof : Proof of Lemma \ref{['lem:Helmholtz']}
  • Lemma 2.4: Maxwell problems satisfying Assumption \ref{['ass:abstract']}
  • Remark 2.5: The Maxwell problems covered in Lemma \ref{['lem:Helmholtz']}
  • proof : Proof of Lemma \ref{['lem:Maxwell']}
  • Remark 2.6: The reason for the particular weighting with $k$
  • Theorem 3.1: Inf-sup condition equivalent to bound on inverse operator
  • proof : References for the proof
  • Lemma 3.2: $k$-independent equivalence of norms of ${\mathcal{A}}_\ell^{-1}$
  • ...and 22 more