The automatic hp-adaptive finite element strategy of Bürg (Appl. Numer. Math. 72, 2013) marks each element for either subdivision (h-refinement) or polynomial-degree enrichment (p-refinement) using a posteriori error estimators, and was proven convergent for a class of Maxwell-type problems — but only under uniform positive definiteness of both coefficient matrices, an assumption that excludes the frequency-domain (indefinite) case of main physical interest, where the zeroth-order coefficient is negative. The source leaves open: (i) extending the convergence proof to coefficients β\beta spectrally bounded below by a negative constant, with contraction in the H(curl)H(\mathrm{curl})-norm or a modified energy norm; (ii) weakening the data-approximation restriction whose required accuracy tightens with polynomial degree; and (iii) proving optimal convergence rates of the adaptive hp strategy. Later literature resolves the h-only indefinite case (Zhong et al. 2012 and successors) and hp-optimality for coercive problems (Canuto et al. 2017), but no work proves convergence or optimality of an hp or automatic h-vs-p strategy for indefinite Maxwell: the problem remains open.

Background

Time-harmonic (frequency-domain) electromagnetic scattering and cavity problems reduce, after elimination of the magnetic field, to a second-order boundary-value problem for the electric field uu: find uu with ∇×(α∇×u)+βu=f\nabla\times(\alpha\nabla\times u)+\beta u=f on a polyhedral domain Ω⊂R3\Omega\subset\mathbb{R}^3 with perfectly conducting boundary condition n×u=0n\times u=0, where α\alpha (a matrix-valued permeability-type coefficient) is uniformly positive definite and β\beta collects the permittivity and the angular frequency ω\omega. In the genuinely time-harmonic case β=−ω2ε\beta=-\omega^2\varepsilon is negative definite, so the differential operator is indefinite — it is not associated with a coercive (energy-minimizing) bilinear form. The natural variational space is H(curl;Ω)H(\mathrm{curl};\Omega), the space of square-integrable vector fields with square-integrable curl, and the conforming finite element discretizations of this space are the Nédélec edge elements, whose degrees of freedom live on mesh edges and which reproduce the curl operator exactly. In an hp-adaptive finite element method (hp-FEM), each element of the mesh is refined either by subdivision into smaller elements (h-refinement) or by raising the local polynomial degree (p-refinement); hp-refinement can achieve exponential convergence for piecewise-analytic solutions with corner/edge singularities, but the best local choice between h and p is not known in advance. This motivates automatic strategies that use a posteriori error estimators — computable quantities built from the discrete solution and the data that bound the true error — to decide, element by element, whether to refine h or p. The adaptive loop iterates: solve, estimate, mark (here: for each marked element decide h vs. p), refine. For h-adaptive methods a mature theory establishes contraction (each loop step reduces the error by a fixed factor once data oscillation — the unresolved part of the data — is controlled) and quasi-optimality (the error decays at the rate afforded by the underlying approximation class). Bürg (2013) analyzed such a fully automatic hp strategy for the Maxwell-type problem above and proved convergence, and a quasi-convergence variant, of the sequence of discrete solutions — but only under uniform positive definiteness of both α\alpha and β\beta, an assumption that holds for coercive model problems yet fails exactly in the frequency-domain case, where β\beta is negative. The proof also relies on a data-saturation assumption requiring the data approximation to be sufficiently accurate, with the required accuracy tightening as the polynomial degree grows; the paper suggests higher-order quadrature, local refinement of the data approximation, or explicit data-error control as remedies. Finally, convergence alone does not imply optimality, and the paper explicitly names the proof of optimal convergence rates as an open direction. The neighboring theory is: Zhong, Chen, Shu, Wittum and Xu (Math. Comp. 81, 2012) proved convergence and quasi-optimal rates of the standard adaptive h-FEM loop with Nédélec elements for the indefinite time-harmonic Maxwell equation (β=−k2\beta=-k^2), assuming a sufficiently fine initial mesh; Xie, Zhong and Liu (2020) and He, Yang and Wang (2020) extended such h-adaptive convergence results to variable coefficients; and Canuto, Nochetto, Stevenson and Verani (Numer. Math. 135, 2017) proved convergence and instance-optimality of an hp-AFEM — but for coercive problems in one and two space dimensions. On the computational side, automatic hp codes for electromagnetics (the Demkowicz HP3D line, begun with Demkowicz 2005; Jiang, Zhang and Zheng 2013) are used routinely without convergence proofs, and Daniel, Ern, Smears and Vohralík (2018) obtained a computable guaranteed error-reduction bound only for a coercive hp strategy. What no available result covers is the combination the source asks for: an automatic (h-vs-p deciding) hp strategy for the indefinite Maxwell problem, with contraction-type convergence, p-robust data handling, and optimal rates.

Problem Statement

For the automatic hp-adaptive finite element strategy of Bürg (2013) — the adaptive loop that solves the Nédélec edge-element discretization, estimates the error elementwise, and decides per marked element between h-subdivision and p-enrichment — prove contraction-type convergence for the Maxwell-type problem ∇×(α∇×u)+βu=f\nabla\times(\alpha\nabla\times u)+\beta u=f in Ω⊂R3\Omega\subset\mathbb{R}^3 with n×u=0n\times u=0 beyond the paper's verified setting, in which both α\alpha and β\beta are assumed uniformly positive definite: keep α\alpha uniformly positive definite and admit a spectrally bounded-below, possibly indefinite β\beta, that is, β⩾−β0I\beta\geqslant-\beta_0 I almost everywhere for some β0>0\beta_0>0 (the smallest eigenvalue of β(x)\beta(x) is at least −β0-\beta_0 for almost every x∈Ωx\in\Omega), where β0\beta_0 may depend on the problem data and where the usual restrictions ensuring well-posedness of the indefinite continuous problem — for example a sufficiently fine initial mesh or a wavenumber condition, as in the h-FEM theory of Zhong et al. (2012) — are admitted, so that the frequency-domain case β=−ω2ε\beta=-\omega^2\varepsilon (negative definite) falls within the target class, with contraction measured in the H(curl)H(\mathrm{curl})-norm or in a modified energy norm suited to the indefinite problem. The source requires uniform positive definiteness of β\beta load-bearingly in its convergence proof, does not treat indefinite coefficients, and in its numerical Example 2 can accommodate a physically realistic non-positive-definite β\beta only by a regularizing replacement χδ(β)\chi_\delta(\beta) — extending the convergence theory to the indefinite case is precisely what remains open. The proof must handle the data-saturation restriction whose required accuracy tightens as the polynomial degree grows rather than assuming it away. In addition, establish optimal convergence rates (quasi-optimality with respect to the relevant hp approximation classes) for the automatic hp-adaptive strategy for Maxwell's equations, which the source explicitly identifies as open — it states that its analysis does not guarantee optimality of the algorithm and that its convergence result might be a first step towards proving it — and which its convergence result does not imply. These are three linked aspects of one objective — a provably reliable and provably efficient automatic hp solver for frequency-domain electromagnetics. The target throughout is the automatic (h-vs-p deciding) hp strategy for the indefinite Maxwell problem: convergence results for h-only adaptive loops (Zhong et al. 2012) and hp results for coercive problems (Canuto et al. 2017) address different settings and do not settle it.

References

  1. M. Bürg, "Convergence of an automatic hp-adaptive finite element strategy for Maxwell's equations", Applied Numerical Mathematics 72 (2013) 188–204, DOI 10.1016/j.apnum.2013.04.008, https://doi.org/10.1016/j.apnum.2013.04.008 (open-access precursor: IWRMM Preprint Nr. 12/04, Karlsruhe Institute of Technology, 2012, DOI 10.5445/IR/1000028335, https://publikationen.bibliothek.kit.edu/1000028335)
  2. M. Bürg, "A Fully Automatic hp-Adaptive Refinement Strategy", Dissertation, Karlsruhe Institute of Technology (2012), DOI 10.5445/IR/1000029034, https://publikationen.bibliothek.kit.edu/1000029034 (Chapter 4 is based on IWRMM Preprint 12-04)
  3. Liuqiang Zhong, Long Chen, Shi Shu, Gabriel Wittum, Jinchao Xu, "Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations", Mathematics of Computation 81 (2012) 623–642, DOI 10.1090/S0025-5718-2011-02544-5, https://doi.org/10.1090/S0025-5718-2011-02544-5
  4. Yingying Xie, Liuqiang Zhong, Chunmei Liu, "Convergence of an AEFEM for time-harmonic Maxwell equations with variable coefficients", Journal of Computational and Applied Mathematics 372 (2020) 112712, DOI 10.1016/j.cam.2020.112712, https://doi.org/10.1016/j.cam.2020.112712
  5. Bin He, Wei Yang, Hao Wang, "Convergence analysis of adaptive edge finite element method for variable coefficient time-harmonic Maxwell's equations", Journal of Computational and Applied Mathematics 376 (2020) 112860, DOI 10.1016/j.cam.2020.112860, https://doi.org/10.1016/j.cam.2020.112860
  6. Claudio Canuto, Ricardo H. Nochetto, Rob Stevenson, Marco Verani, "Convergence and optimality of hp-AFEM", Numerische Mathematik 135 (2017) 1073–1119, DOI 10.1007/s00211-016-0826-x, https://doi.org/10.1007/s00211-016-0826-x (preprint arXiv:1503.03996)
  7. Patrik Daniel, Alexandre Ern, Iain Smears, Martin Vohralík, "An adaptive hp-refinement strategy with computable guaranteed bound on the error reduction factor", Computers & Mathematics with Applications 76 (2018) 967–983, DOI 10.1016/j.camwa.2018.05.034, https://doi.org/10.1016/j.camwa.2018.05.034
  8. Xue Jiang, Linbo Zhang, Weiying Zheng, "Adaptive hp-Finite Element Computations for Time-Harmonic Maxwell's Equations", Communications in Computational Physics 13 (2013) 559–582, DOI 10.4208/cicp.231111.090312a, https://doi.org/10.4208/cicp.231111.090312a
  9. L. Demkowicz, "Fully automatic hp-adaptivity for Maxwell's equations", Computer Methods in Applied Mechanics and Engineering 194 (2005) 605–624, DOI 10.1016/j.cma.2004.05.023, https://doi.org/10.1016/j.cma.2004.05.023