For the time-harmonic Maxwell equations at wavenumber kk, the conforming hp-finite-element method is known to be quasi-optimal and free of the pollution effect under the scale-resolution conditions kh/p≤c1kh/p\le c_1 and p≥c2log⁡kp\ge c_2\log k, but for the exterior scattering setting this was established (Melenk and Sauter, 2021) only on the unit ball with an exactly implemented transparent boundary condition, exploiting the spherical-harmonic representation of the capacity (Dirichlet-to-Neumann) operator. The problem is the extension left open by that paper: k-explicit quasi-optimality of H(curl)H(\mathrm{curl})-conforming Nédélec hp-discretizations on general domains (non-spherical surfaces, Lipschitz and polyhedral boundaries with edge and corner singularities), for practical approximations of the outgoing radiation condition (exact or discrete Dirichlet-to-Neumann maps on general surfaces, FEM-BEM coupling), together with the sharp kk-dependence of the intermediate constants (an O(k3)O(k^3) continuity constant and a non-sharp h-version condition kp+shp≲1k^{p+s}h^p\lesssim 1 with s=7s=7). Impedance boundary conditions on analytic boundaries (2023), piecewise analytic media (2026), and sharp h-FEM bounds with perfectly matched layers on smooth-partition domains (2026) have since been settled; the open core is the general-geometry theory, the non-spherical/approximate-DtN theory, and the sharp constants.

Background

The time-harmonic Maxwell equations model monochromatic electromagnetic waves with wavenumber kk (proportional to the number of wavelengths per unit length), so the solution oscillates at frequency kk and any grid-based numerics must at least resolve these oscillations. Beyond this natural resolution requirement, Galerkin discretizations at high frequency suffer the pollution effect: accumulated dispersion error forces meshes substantially finer than a fixed number of points per wavelength as kk grows. For the scalar Helmholtz equation, the hp-finite element method (FEM) — combining mesh width hh with polynomial degree pp — overcomes pollution: it is quasi-optimal (the Galerkin error is bounded by the best-approximation error times a constant) under the scale-resolution conditions kh/p≤c1kh/p\le c_1 and p≥c2log⁡kp\ge c_2\log k with c1c_1 small.

For Maxwell's equations the FEM spaces must be subspaces of H(curl)H(\mathrm{curl}), the space of square-integrable vector fields with square-integrable curl, discretized by Nédélec edge elements. The vector case is structurally harder because the embedding H(curl)⊂L2H(\mathrm{curl})\subset L^2 is not compact, so the classical Schatz duality argument (the standard route to quasi-optimality for elliptic problems via solving an adjoint problem) fails; a k-explicit theory needs (i) a regularity-by-decomposition result splitting the exact solution into a low-regularity part with kk-uniform bounds, a gradient field, and an analytic part, and (ii) interpolation operators with a commuting-diagram property (they commute with the de Rham gradient/curl sequence, the exact sequence linking gradient, curl, and divergence, so that discrete solenoidal, i.e. divergence-free, parts are handled correctly) whose approximation errors are explicit in hh, pp, and kk.

The source paper (Melenk and Sauter, Foundations of Computational Mathematics 21:125-241, 2021) proves the Maxwell analogue: for the full-space time-harmonic Maxwell problem reformulated on the unit ball Ω=B1(0)\Omega=B_1(0) with an exact transparent boundary condition on Γ=∂Ω\Gamma=\partial\Omega, the conforming hp-Galerkin method with Nédélec type-I elements of degree pp is quasi-optimal under kh/p≤c1kh/p\le c_1 and p≥c2log⁡kp\ge c_2\log k. The transparent boundary condition is expressed through the capacity operator TkT_k, the Maxwell Dirichlet-to-Neumann (DtN) map that takes the tangential electric trace on Γ\Gamma to the tangential magnetic trace of the outgoing solution of the exterior problem, thereby representing the unbounded exterior exactly. Three ingredients of the proof are tied to the spherical geometry: the frequency splitting of TkT_k into low- and high-frequency parts via its spherical-harmonic eigenfunction expansion with modewise symbol bounds; a kk-explicit continuity constant Ccont,k=O(k3)C_{\mathrm{cont},k}=O(k^3) for the sesquilinear form; and the adjoint regularity decompositions built on the analyticity of Γ\Gamma. The paper also derives, for fixed polynomial degree pp, an h-version scale-resolution condition kp+shp≲1k^{p+s}h^p\lesssim 1 with s=7s=7, explicitly stating that this ss is not expected to be sharp, and it announces a companion treatment of impedance (Leontovich) boundary conditions curl E×n−ikET=gT\mathrm{curl}\,E\times n - ikE_T=g_T, which model imperfectly conducting surfaces.

Problem Statement

As left open by Melenk and Sauter (2021), the problem is to extend the k-explicit convergence theory of the hp-finite-element method for the time-harmonic Maxwell equations beyond the model situation of the unit ball with an exactly implemented transparent boundary condition, in the following interlinked directions:

  1. General geometries. Establish k-explicit quasi-optimality — in particular pollution-free quasi-optimality under scale-resolution conditions of the form kh/p≤c1kh/p\le c_1, p≥c2log⁡kp\ge c_2\log k — for H(curl)H(\mathrm{curl})-conforming Nédélec hp-discretizations on general domains: non-spherical truncation surfaces and, most importantly, Lipschitz or polyhedral boundaries on which Maxwell solutions develop edge and corner singularities. This requires k-explicit analogues, in the presence of such singularities, of the regularity-by-decomposition of the solution and of the dual problem, of the boundary-operator (capacity/DtN) frequency-splitting and symbol bounds, and of the commuting-projector approximation bounds η6alg≲kh/p\eta_6^{\mathrm{alg}}\lesssim kh/p.

  2. Practical truncations of the radiation condition. Analyze how the k-explicit estimates and the scale-resolution conditions change when the exact transparent boundary condition is replaced by an implementable approximation, for example at least one of: discrete or local DtN approximations, boundary-element (FEM-BEM) coupling, in which the exterior problem is discretized by a boundary integral equation on the truncation surface, or perfectly matched layers — absorbing artificial layers appended around the computational domain that damp outgoing waves and thereby truncate the unbounded exterior — in the hp regime with pp growing like log⁡k\log k, so that the theory certifies realistic scattering computations rather than the idealized ball model.

  3. Sharpness. Determine the optimal kk-dependence of the analysis: the sharp h-version scale-resolution condition for fixed polynomial degree (the source derives kp+shp≲1k^{p+s}h^p\lesssim 1 with s=7s=7 and states that ss is not expected to be sharp, so the minimal ss is open), the sharpness of intermediate constants such as the O(k3)O(k^3) continuity constant Ccont,kC_{\mathrm{cont},k} and the adjoint analyticity exponents, and consequently the minimal polynomial degree p(k)p(k) needed for pollution-free approximation.

The three directions are jointly posed: the general-geometry theory (direction 1), the radiation-condition truncation theory (direction 2), and the sharp constants (direction 3), with the sharpness exponents in direction 3 established by lower bounds or rigorous counterexamples. Settings already settled in the literature — impedance boundary conditions on analytic boundaries, piecewise analytic media with analytic boundary, and sharp h-FEM bounds with PML on smooth-partition domains with fixed order — lie outside the open core of the problem, which is the general-geometry and non-spherical/approximate-DtN theory together with the sharp constants.

References

  1. J. M. Melenk and S. A. Sauter, Wavenumber-Explicit hp-FEM Analysis for Maxwell's Equations with Transparent Boundary Conditions, Foundations of Computational Mathematics 21 (2021) 125-241. DOI: 10.1007/s10208-020-09452-1, https://doi.org/10.1007/s10208-020-09452-1 (arXiv:1803.01619, https://arxiv.org/abs/1803.01619)
  2. 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 (published online 2023) 1871-1939. DOI: 10.1007/s10208-023-09626-7, https://doi.org/10.1007/s10208-023-09626-7 (arXiv:2201.02602, https://arxiv.org/abs/2201.02602)
  3. J. M. Melenk and D. Wörgötter, Wavenumber-explicit hp-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media, preprint (2026), arXiv:2603.17467, https://arxiv.org/abs/2603.17467
  4. T. Chaumont-Frelet, J. Galkowski, and E. A. Spence, Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations, Mathematics of Computation (published online 2026). DOI: 10.1090/mcom/4198, https://doi.org/10.1090/mcom/4198 (arXiv:2408.04507, https://arxiv.org/abs/2408.04507)
  5. M. Bernkopf, T. Chaumont-Frelet, and J. M. Melenk, Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media, Mathematics of Computation 94 (2025) 73-122. DOI: 10.1090/mcom/3958, https://doi.org/10.1090/mcom/3958
  6. T. Chaumont-Frelet and A. Ern, Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations, Comptes Rendus. Mathématique 363 (2025) 1083-1101. DOI: 10.5802/crmath.757, https://doi.org/10.5802/crmath.757 (arXiv:2309.14189, https://arxiv.org/abs/2309.14189)
  7. T. Chaumont-Frelet and P. Vega, Frequency-explicit approximability estimates for time-harmonic Maxwell's equations, Calcolo 59 (2022), article 22. DOI: 10.1007/s10092-022-00464-7, https://doi.org/10.1007/s10092-022-00464-7
  8. T. Chaumont-Frelet, A. Moiola, and E. A. Spence, Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media, Journal de Mathématiques Pures et Appliquées 179 (2023) 183-218. DOI: 10.1016/j.matpur.2023.09.004, https://doi.org/10.1016/j.matpur.2023.09.004
  9. S. Nicaise and J. 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 (2020) 1868-1903. DOI: 10.1002/num.22508, https://doi.org/10.1002/num.22508
  10. S. Nicaise and J. Tomezyk, The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains, in Maxwell's Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, Berlin (2019) 285-340. DOI: 10.1515/9783110543612-009, https://doi.org/10.1515/9783110543612-009
  11. J. M. Melenk and C. Rojik, On commuting p-version projection-based interpolation on tetrahedra, Mathematics of Computation 89 (2020) 45-87. DOI: 10.1090/mcom/3454, https://doi.org/10.1090/mcom/3454