Sharp scale-resolution conditions for wavenumber-explicit hp finite element analysis of the time-harmonic Maxwell impedance problem

Nicaise and Tomezyk (2020) proved wavenumber-explicit quasi-optimality of a penalized, $H^1$-based hp finite element discretization of the time-harmonic Maxwell equations with impedance boundary conditions on domains with an analytic boundary. Their adjoint-approximation bound contains a penalty-induced algebraic term $kh/\sqrt{p}$ (mesh size $h$, polynomial degree $p$, wavenumber $k$), which they conjecture can be improved to $kh/p$, in agreement with their numerics showing that the minimal quasi-optimal degree satisfies $p^\star\sim k$ at fixed $h$. Later work of Melenk and Sauter (2024) proved quasi-optimality under exactly the conjectured $kh/p$ condition for the same boundary value problem discretized by conforming Nedelec edge elements, thereby resolving the question of penalty-free conforming enforcement and explaining the observed thresholds at the conforming-method level. The remaining restriction that survives this progress is specific to the penalized method: settle the sharpness of the $kh/\sqrt{p}$ term for the penalized $H^1$-based method, either by proving quasi-optimality under $kh/p$ together with $p\ge C\ln k$ or by proving a matching lower bound, yielding a sharp resolution guarantee for high-order discretizations of high-frequency electromagnetic scattering in the analytic-boundary setting of the source. Extending wavenumber-explicit hp quasi-optimality for the Maxwell impedance problem beyond analytic boundaries to non-smooth, piecewise-analytic polyhedral domains, incorporating corner and edge singularities into the regularity theory, is a related open direction of the same program rather than part of the requested result.

Background

The time-harmonic Maxwell equations describe monochromatic electromagnetic waves: given a wavenumber $k$, one seeks the electric and magnetic fields satisfying $\operatorname{curl} H - ikE = J$ and $\operatorname{curl} E + ikH = 0$. When a scattering problem is truncated to a bounded domain $\Omega$, an impedance (Leontovich) boundary condition $H\times n = \lambda_{\mathrm{imp}} E_t$ on $\partial\Omega$ (with $E_t$ the tangential trace and $\lambda_{\mathrm{imp}}>0$) absorbs outgoing waves to leading order. At high wavenumber, standard Galerkin discretizations suffer the pollution effect: the total error substantially exceeds the best-approximation error unless the mesh resolves the oscillations with a margin that grows with $k$. For the scalar Helmholtz equation, Melenk and Sauter (2011) developed the wavenumber-explicit theory showing that the hp finite element method (mesh size $h$, polynomial degree $p$) is quasi-optimal, meaning the Galerkin error is bounded by a constant multiple of the best-approximation error, provided the scale-resolution condition $kh/p$ is small together with the weak side constraint $p\ge C\ln k$; high order thus suppresses pollution. Two variational settings exist for Maxwell. In the standard $H(\operatorname{curl})$ formulation, discretized by conforming Nedelec edge elements (finite elements enforcing only tangential continuity across mesh faces), the impedance condition is a natural condition already encoded in the variational form. In the regularized $H^1$-based formulation introduced by Costabel, Dauge, and Nicaise, the field pair $(E,H)$ is sought in $H^1(\Omega)^6$ and the associated first-order system is elliptic, so continuous Lagrange elements apply and elliptic shift theorems give high regularity; the price is that the impedance condition becomes an essential constraint on the space. Nicaise and Tomezyk (2020) analyzed an hp discretization of this second formulation on a domain with analytic boundary. Since the impedance constraint cannot be imposed inside a Lagrange finite element space and no conforming interpolation operator preserving it could be constructed, they enforce it by penalty terms on the boundary, obtaining a nonconforming method. Their central estimate for the adjoint approximation property (the quantity controlling discrete stability via a duality argument; their Theorem 4.6 and formula (4.15)) contains, next to exponentially convergent terms, the algebraic term $kh/\sqrt{p}$, and quasi-optimality follows from the sufficient condition $kh/\sqrt{p}\le C_1$, $\ln k\le C_2 p$ (their condition (4.22)). The $\sqrt{p}$ degradation is caused specifically by the penalization. In Remark 4.7 the authors conjecture it is an artifact, the term being replaceable by $kh/p$, a belief confirmed by their experiments: the minimal degree $p^\star$ at which quasi-optimality sets in scales like $k$ for fixed $h$ (while condition (4.22) would force $p\gtrsim k^2$) and like $\ln k$ for fixed $kh$. Subsequently, Melenk and Sauter (2024) analyzed the identical boundary value problem (time-harmonic Maxwell, impedance condition, analytic boundary) by the conforming route: Nedelec elements of degree $p$ in the $H(\operatorname{curl})$ formulation, where the impedance condition is natural so no penalty arises, and proved quasi-optimality under $|k|h/p\le c_1$ and $p\ge c_2\ln|k|$ (their Theorem 9.7). This closes the source paper's conforming-enforcement question and proves the conjectured $kh/p$ threshold attainable, with exactly the observed $p^\star$ scalings, but for a different discretization. Tightening the analysis of the penalized $H^1$-based method itself, or proving any lower bound showing its $kh/\sqrt{p}$ term intrinsic, remains open; and both the 2020 and 2024 theories still require an analytic boundary, as do recent extensions to piecewise-smooth media with interior coefficient interfaces (Melenk and Worgutter 2026), while the $h$-version companion treatment of polyhedral domains by Nicaise and Tomezyk covers only convex-type polyhedra with partial stability results.

Problem Statement

The problem is to complete the wavenumber-explicit hp convergence theory for the time-harmonic Maxwell equations with impedance boundary conditions by settling the restriction that Nicaise and Tomezyk (2020) left open for their penalized method. The requested result serves a sharp quasi-optimality theory whose sufficient conditions match the numerically observed thresholds ($p^\star\sim k$ at fixed $h$; $p^\star\sim\ln k$ at fixed $kh$). Requested result (sharpness of the penalty-induced resolution condition): for the penalized $H^1$-based hp discretization of the Maxwell impedance problem on a domain with analytic boundary, prove that the adjoint-approximation bound (Theorem 4.6 / formula (4.15) of the source, following the preprint numbering) and the resulting quasi-optimality hold with the term $kh/\sqrt{p}$ replaced by $kh/p$; in particular, that the discrete problem is well posed and quasi-optimal under the scale-resolution condition $kh/p\le C$ together with $p\ge C\ln k$, as the source authors conjectured on the basis of their numerics; or, conversely, prove a matching lower bound establishing that the $kh/\sqrt{p}$ degradation is intrinsic to this penalized scheme, for instance by constructing a sequence of wavenumbers, meshes, and polynomial degrees with $kh/p$ bounded along which the quasi-optimality constant must degenerate. Either direction settles the problem. Extending $k$-explicit hp quasi-optimality for the Maxwell impedance problem to domains whose boundary is not analytic, for example polyhedral domains with piecewise-analytic faces going beyond the $h$-version and convex-type geometric assumptions of the existing polyhedral-domain treatment, with the corner and edge singularities of the solution incorporated $k$-explicitly into the regularity theory, is a related open direction of the same program; it is not a required component of the answer to this problem.

References

1. Serge Nicaise and Jerome Tomezyk. Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary. Numerical Methods for Partial Differential Equations, 36(6):1868-1903, 2020. DOI: 10.1002/num.22508. https://doi.org/10.1002/num.22508
2. Serge Nicaise and Jerome Tomezyk. The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains. In U. Langer, D. Pauly, and S. Repin (eds.), Maxwell's Equations: Analysis and Numerics, Radon Series on Computational and Applied Mathematics 24, pp. 285-340, De Gruyter, Berlin, 2019. DOI: 10.1515/9783110543612-009. https://doi.org/10.1515/9783110543612-009
3. J. M. Melenk and S. A. Sauter. Wavenumber-Explicit hp-FEM Analysis for Maxwell's Equations with Impedance Boundary Conditions. Foundations of Computational Mathematics, 24(6):1871-1939, 2024. DOI: 10.1007/s10208-023-09626-7. https://doi.org/10.1007/s10208-023-09626-7
4. Jens M. Melenk and Stefan A. Sauter. Wavenumber-Explicit hp-FEM Analysis for Maxwell's Equations with Transparent Boundary Conditions. Foundations of Computational Mathematics, 21(1):125-241, 2021. DOI: 10.1007/s10208-020-09452-1. https://doi.org/10.1007/s10208-020-09452-1
5. J. M. Melenk and S. Sauter. Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation. SIAM Journal on Numerical Analysis, 49(3):1210-1243, 2011. DOI: 10.1137/090776202. https://doi.org/10.1137/090776202
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7. Jerome Tomezyk. Resolution numerique de quelques problemes du type Helmholtz avec conditions au bord d'impedance ou des couches absorbantes (PML) [English title: Numerical resolution of some Helmholtz-type problems with impedance boundary condition or PML]. PhD thesis, 2019. HAL: tel-02304642. https://theses.hal.science/tel-02304642
8. Shuaishuai Lu and Haijun Wu. Preasymptotic Error Estimates of Linear EEM and CIP-EEM for the Time-Harmonic Maxwell Equations with Large Wave Number. SIAM Journal on Numerical Analysis, 64(2):326-349, 2026. DOI: 10.1137/24m1680362. https://doi.org/10.1137/24m1680362
9. Theophile Chaumont-Frelet, Jeffrey Galkowski, and Euan A. Spence. Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations. Mathematics of Computation, 2026 (online). DOI: 10.1090/mcom/4198. https://doi.org/10.1090/mcom/4198
10. Theophile Chaumont-Frelet, Andrea Moiola, and Euan A. Spence. Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media. Journal de Mathematiques Pures et Appliquees, 179:183-218, 2023. DOI: 10.1016/j.matpur.2023.09.004. https://doi.org/10.1016/j.matpur.2023.09.004 (preprint arXiv:2301.07092)
11. Luyu Cen, Shihua Gong, Euan A. Spence, and Yue Yu. Convergence of parallel overlapping domain decomposition methods with impedance boundary conditions for time-harmonic Maxwell equations in heterogeneous media. 2026. arXiv:2606.04982. https://arxiv.org/abs/2606.04982
12. Jens Markus Melenk and David Worgutter. Wavenumber-explicit hp-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media. 2026. arXiv:2603.17467. https://arxiv.org/abs/2603.17467