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Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations

Théophile Chaumont-Frelet, Jeffrey Galkowski, Euan A. Spence

TL;DR

This work develops sharp, wavenumber-explicit error bounds for the $h$-version of edge-element discretisations (Nédélec elements) of the time-harmonic Maxwell equations in a bounded domain with PEC boundary conditions. The authors build a general abstract framework combining a Gårding inequality, regularity shifts, and a divergence-conformity analysis via projections to handle the nontrivial curl-kernel, enabling preasymptotic error control for fixed polynomial degree $p$ and arbitrary order. A central innovation is the introduction of a smoothing-based operator $S$ and the auxiliary self-adjoint operator $P^{\#}$, together with a refined duality argument that yields $k$-explicit quasi-optimality and relative-error bounds in the preasymptotic regime, even when the scatterer is nonempty. The theory accommodates complex-valued, piecewise regular coefficients and radiating conditions via PML, and is supported by Weber-type regularity results extended to complex coefficients. Overall, the results provide rigorous guidelines for mesh design and polynomial degree choices in high-frequency Maxwell simulations, with practical implications for scattering and PML-absorbed configurations.

Abstract

We prove sharp wavenumber-explicit error bounds for first- or second-family-Nédélec-element (a.k.a. edge-element) conforming discretisations, of arbitrary (fixed) order, of the variable-coefficient time-harmonic Maxwell equations posed in a bounded domain with perfect electric conductor (PEC) boundary conditions. The PDE coefficients are allowed to be piecewise regular and complex-valued; this set-up therefore includes scattering from a PEC obstacle and/or variable real-valued coefficients, with the radiation condition approximated by a perfectly matched layer (PML). In the analysis of the $h$-version of the finite-element method, with fixed polynomial degree $p$, applied to the time-harmonic Maxwell equations, the $\textit{asymptotic regime}$ is when the meshwidth, $h$, is small enough (in a wavenumber-dependent way) that the Galerkin solution is quasioptimal independently of the wavenumber, while the $\textit{preasymptotic regime}$ is the complement of the asymptotic regime. The results of this paper are the first preasymptotic error bounds for the time-harmonic Maxwell equations using first-family Nédélec elements or higher-than-lowest-order second-family Nédélec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.

Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations

TL;DR

This work develops sharp, wavenumber-explicit error bounds for the -version of edge-element discretisations (Nédélec elements) of the time-harmonic Maxwell equations in a bounded domain with PEC boundary conditions. The authors build a general abstract framework combining a Gårding inequality, regularity shifts, and a divergence-conformity analysis via projections to handle the nontrivial curl-kernel, enabling preasymptotic error control for fixed polynomial degree and arbitrary order. A central innovation is the introduction of a smoothing-based operator and the auxiliary self-adjoint operator , together with a refined duality argument that yields -explicit quasi-optimality and relative-error bounds in the preasymptotic regime, even when the scatterer is nonempty. The theory accommodates complex-valued, piecewise regular coefficients and radiating conditions via PML, and is supported by Weber-type regularity results extended to complex coefficients. Overall, the results provide rigorous guidelines for mesh design and polynomial degree choices in high-frequency Maxwell simulations, with practical implications for scattering and PML-absorbed configurations.

Abstract

We prove sharp wavenumber-explicit error bounds for first- or second-family-Nédélec-element (a.k.a. edge-element) conforming discretisations, of arbitrary (fixed) order, of the variable-coefficient time-harmonic Maxwell equations posed in a bounded domain with perfect electric conductor (PEC) boundary conditions. The PDE coefficients are allowed to be piecewise regular and complex-valued; this set-up therefore includes scattering from a PEC obstacle and/or variable real-valued coefficients, with the radiation condition approximated by a perfectly matched layer (PML). In the analysis of the -version of the finite-element method, with fixed polynomial degree , applied to the time-harmonic Maxwell equations, the is when the meshwidth, , is small enough (in a wavenumber-dependent way) that the Galerkin solution is quasioptimal independently of the wavenumber, while the is the complement of the asymptotic regime. The results of this paper are the first preasymptotic error bounds for the time-harmonic Maxwell equations using first-family Nédélec elements or higher-than-lowest-order second-family Nédélec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.
Paper Structure (50 sections, 51 theorems, 287 equations, 1 figure)

This paper contains 50 sections, 51 theorems, 287 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. The context and novelty of the main result
  4. The asymptotic and preasymptotic regimes.
  5. State of the art in the asymptotic regime for the Helmholtz $h$-FEM.
  6. State of the art in the preasymptotic regime for the Helmholtz $h$-FEM.
  7. Duality-argument analysis of the Maxwell $h$-FEM using Nédélec finite elements.
  8. Current state of the art for wavenumber-explicit bounds on the Maxwell $h$-FEM using Nédélec finite elements.
  9. Summary of the ideas behind the proof of Theorem \ref{['thm:intro']}.
  10. Outline
  11. The main result in abstract form
  12. Abstract framework and assumptions
  13. The Galerkin method
  14. The quantity $\gamma_{\rm dv}(P)$
  15. The main abstract theorem
  16. ...and 35 more sections

Key Result

Theorem 1.3

Suppose that Assumption ass:regularity holds for an integer $m\geq1$. Given $1\leq p\leq m$ and $k_0, C_{\rm osc}>0$ there exist $C_1, C_2, C_3>0$ such that the following holds. Let ${\cal H}_h\subset H_0({\rm curl}\,,\Omega)$ be the space of first- or second-family Nédélec finite-elements of degree the Galerkin solution $E_h$ exists, is unique, and satisfies Furthermore, if the data $f$ is $k$-o

Figures (1)

  • Figure 1.1: An example of $\Omega$ (shaded) with $\overline{\Omega}= \cup_{j=1}^n \overline{\Omega_j}$ satisfying Definition \ref{['def:Crpartition']}.

Theorems & Definitions (105)

  • Definition 1.1: $C^\ell$ with respect to a partition
  • Theorem 1.3: The main result
  • Remark 1.4: The origin of the assumptions of Theorem \ref{['thm:intro']}
  • Remark 1.5: The main result applied to differential $r$-forms
  • Remark 1.6: The norm of the solution operator $C_{\rm sol}$
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['lem:kernel_closed']}
  • Lemma 2.4: Application to Maxwell
  • Remark 2.5: The regularity assumptions on $\epsilon,\mu$, and $\partial\Omega$
  • Remark 2.6: $\Pi_1$ projects to functions that are $\epsilon$-divergence free
  • ...and 95 more