Uniform elliptic regularity pickup for the Neumann–Poisson problem on vertex patches of shape-regular simplicial meshes (the Dong–Ern conjecture, Assumption 3.5)

Dong and Ern (2026) constructed an H(curl)-conforming reconstruction operator for piecewise polynomial fields on shape-regular simplicial meshes and used it to derive hp a posteriori error estimates for interior-penalty discontinuous Galerkin approximations of Maxwell's equations. Their p-improved L^2 reconstruction bound, the local Sobolev embedding behind it, and the reliability theorem for the dG Maxwell estimator all rest on an explicitly conjectured assumption: that the homogeneous Neumann–Poisson problem on every mesh vertex patch exhibits an elliptic regularity pickup — an H^{1/2+δ} bound on the gradient of the potential — with a constant C and an exponent δ>0 depending only on the mesh shape-regularity parameter. The authors state that they found no precise result in the literature and formulate the statement as a conjecture to be proved or disproved. This record states that conjecture as the problem: prove that a uniform regularity pickup holds over all vertex patches of all shape-regular simplicial meshes (including patches truncated by the domain boundary), or refute it by exhibiting a mesh family with uniformly bounded shape regularity for which no uniform constant and exponent exist. The estimate (Assumption 3.5) is unresolved in the literature available up to 2026.

Background

Nonconforming and discontinuous Galerkin (dG) discretizations of Maxwell's equations approximate the electric field by piecewise polynomial fields that may jump across mesh faces and therefore fail to lie in $H(\mathrm{curl};D)$ , the space of square-integrable vector fields on the domain $D$ with square-integrable curl, where the Maxwell boundary value problem is posed. A posteriori error estimation for such methods (bounds on the discretization error in terms of computable quantities such as solution jumps and residuals) is classically built on reconstructing, from the discrete field, a conforming field in $H_0(\mathrm{curl};D)$ (the subspace of $H(\mathrm{curl};D)$ with vanishing tangential component on the boundary) and bounding the reconstruction error in terms of the jumps of the discrete field. In the $hp$ setting, where both the mesh size $h$ and the polynomial degree $p$ vary, the quality of such bounds is judged by their scaling in $h$ and $p$ : a bound is $hp$ -optimal when it decays at the best-approximation rate in both parameters.

Dong and Ern (2026) devised such a reconstruction on shape-regular simplicial meshes (meshes whose elements have uniformly bounded aspect ratio, quantified by a shape-regularity parameter $\kappa_{\mathcal{T}_h}$ ). The construction is local: using the partition of unity given by the piecewise-linear hat basis functions attached to the mesh vertices, combined with a local Helmholtz decomposition (a splitting of a vector field into a curl-free potential part and a divergence-free part) on each vertex patch $D_{\mathbf{a}}$ — the union of all mesh cells containing the vertex $\mathbf{a}$ , a simply connected Lipschitz polytope of diameter $h_{\mathbf{a}}$ — the broken field is corrected patch by patch. The proven bounds on the difference between the reconstructed field and the original one involve the tangential jumps of the field and the normal jumps of its curl; they are always $h$ -optimal, but $p$ -suboptimal by $1/2$ order in the broken curl norm (the curl evaluated cellwise) and by up to $3/2$ orders in the $L^2$ -norm.

The $L^2$ bound is obtained by an Aubin–Nitsche duality argument (a standard technique that bounds an $L^2$ error through the solution of an auxiliary dual problem) on each patch, and its improvement to $1/2$ -order $p$ -suboptimality requires that the homogeneous Neumann–Poisson problem on the patch pick up elliptic regularity: the potential $\zeta_{\boldsymbol{g}}$ solving $\Delta\zeta_{\boldsymbol{g}}=\nabla\cdot\boldsymbol{g}$ in $D_{\mathbf{a}}$ with $\boldsymbol{n}\cdot\nabla\zeta_{\boldsymbol{g}}=\boldsymbol{n}\cdot\boldsymbol{g}$ on $\partial D_{\mathbf{a}}$ must have $\nabla\zeta_{\boldsymbol{g}}$ in the Sobolev space $H^{1/2+\delta}(D_{\mathbf{a}})$ for some $\delta>0$ , with constants controlled uniformly over all patches of a mesh family. On nonsmooth polyhedral domains such regularity is governed by edge and corner singular exponents, which depend on the domain geometry in a non-explicit way; the open question is whether mesh shape regularity alone forces these exponents and constants to remain uniform across a mesh family. The authors examined the neighboring literature — explicit Poincaré constants for finite element stars (Veeser and Verfürth, 2012), continuity constants of Bogovskii-type and regularized Poincaré integral operators (Guzmán and Salgado, 2021), and computable Poincaré–Friedrichs constants over domains with shellable triangulations, a class that includes vertex stars (Chaumont-Frelet, Licht, and Vohralík, 2025) — all of which control $L^2$ -type potential or compactness quantities rather than this Sobolev-index pickup, and found no precise result. They therefore formulated the needed statement as their Assumption 3.5 and explicitly invited the community to prove or disprove it. The assumption is load-bearing: under it, the local $L^2$ reconstruction bound, a local Sobolev embedding used in its proof, and the reliability theorem for the dG Maxwell estimator all hold with constants depending only on $\kappa_{\mathcal{T}_h}$ ; on convex domains the analogous globally posed reconstruction achieves full $hp$ -optimality via the classical full elliptic regularity pickup of the global domain, which isolates the uniform local, patch-level validity as the sole gap.

Problem Statement

The problem is Assumption 3.5 ("Elliptic regularity pickup for Neumann problem") of Dong and Ern (2026), which its authors formulate as a conjecture to be proved or disproved.

Let $D\subset\mathbb{R}^d$ (the source fixes $d=3$ and states that everything readily extends to $d=2$ ) be an open, bounded, connected Lipschitz polyhedron covered exactly by a simplicial mesh $\mathcal{T}_h$ with shape-regularity parameter $\kappa_{\mathcal{T}_h}$ . For a mesh vertex $\mathbf{a}$ , let $D_{\mathbf{a}}$ denote the vertex patch (the union of all mesh cells containing $\mathbf{a}$ ), let $h_{\mathbf{a}}$ be its diameter, and let $H^1(D_{\mathbf{a}})/\mathbb{R}$ be the Sobolev space $H^1$ modulo additive constants. For $\boldsymbol{g}\in[H^1(D_{\mathbf{a}})/\mathbb{R}]^d$ , let $\zeta_{\boldsymbol{g}}\in H^1(D_{\mathbf{a}})/\mathbb{R}$ be the unique zero-mean solution of the homogeneous Neumann problem $(\nabla\zeta_{\boldsymbol{g}},\nabla\rho)_{D_{\mathbf{a}}}=(\boldsymbol{g},\nabla\rho)_{D_{\mathbf{a}}}\quad\forall\rho\in H^1(D_{\mathbf{a}})/\mathbb{R},$ that is, $\Delta\zeta_{\boldsymbol{g}}=\nabla\cdot\boldsymbol{g}$ in $D_{\mathbf{a}}$ and $\boldsymbol{n}_{D_{\mathbf{a}}}\cdot\nabla\zeta_{\boldsymbol{g}}=\boldsymbol{n}_{D_{\mathbf{a}}}\cdot\boldsymbol{g}$ on $\partial D_{\mathbf{a}}$ .

Question: do there exist a constant $C(\kappa_{\mathcal{T}_h})$ and an exponent $\delta(\kappa_{\mathcal{T}_h})>0$ , both depending only on the mesh shape-regularity parameter, such that for every mesh vertex $\mathbf{a}$ (including vertices on the domain boundary, whose patches are truncated by $\partial D$ ) and every $\boldsymbol{g}\in[H^1(D_{\mathbf{a}})/\mathbb{R}]^d$ , $\|\zeta_{\boldsymbol{g}}\|_{D_{\mathbf{a}}}+h_{\mathbf{a}}^{\frac{3}{2}+\delta}\,|\nabla\zeta_{\boldsymbol{g}}|_{H^{\frac{1}{2}+\delta}(D_{\mathbf{a}})}\leq C(\kappa_{\mathcal{T}_h})\,h_{\mathbf{a}}^{2}\,\|\nabla\boldsymbol{g}\|_{D_{\mathbf{a}}}?$

Here $\|\cdot\|_{D_{\mathbf{a}}}$ denotes the $L^2(D_{\mathbf{a}})$ -norm and $|\cdot|_{H^{s}(D_{\mathbf{a}})}$ the seminorm of the Sobolev space $H^{s}(D_{\mathbf{a}})$ ; this estimate is Assumption 3.5 of the source, reproduced with its numbering of spaces, norms, and scaling exponents.

Resolve this question in one of two admissible ways: (i) prove the estimate, with $C$ and $\delta$ depending only on shape regularity, uniformly over all vertex patches of all meshes in the family (patches are simply connected Lipschitz polytopes; boundary patches must be included); or (ii) refute it by exhibiting a family of simplicial meshes with uniformly bounded $\kappa_{\mathcal{T}_h}$ , together with vertex patches and data $\boldsymbol{g}$ , for which no such uniform pair $(C,\delta)$ can exist. No restriction on dimension beyond the source's setting, on patch type, or on the method of proof or refutation is admissible.

References

Z. Dong and A. Ern, "H(curl)-reconstruction of piecewise polynomial fields with application to hp-a posteriori nonconforming error analysis for Maxwell's equations", SIAM Journal on Numerical Analysis (published 2026-07-20), DOI 10.1137/25M1789433, https://doi.org/10.1137/25M1789433; preprint arXiv:2508.12904 (v1 2025-08-18, v3 2026-03-20), https://arxiv.org/abs/2508.12904 (source of Assumption 3.5)
A. Veeser and R. Verfürth, "Poincaré constants for finite element stars", IMA Journal of Numerical Analysis 32 (2012), no. 1, 30–47, DOI 10.1093/imanum/drr011, https://doi.org/10.1093/imanum/drr011
J. Guzmán and A. J. Salgado, "Estimation of the continuity constants for Bogovskiĭ and regularized Poincaré integral operators", Journal of Mathematical Analysis and Applications 502 (2021), no. 1, Paper No. 125246, DOI 10.1016/j.jmaa.2021.125246, https://doi.org/10.1016/j.jmaa.2021.125246
T. Chaumont-Frelet, M. W. Licht, and M. Vohralík, "Computable Poincaré–Friedrichs constants for the L^p de Rham complex over convex domains and domains with shellable triangulations", preprint (2025), arXiv:2508.06741, https://arxiv.org/abs/2508.06741
Z. Dong, A. Ern, and T. Wadhawan, "hp-a posteriori error estimates for hybrid high-order methods applied to biharmonic problems", preprint (2026), arXiv:2602.06872, https://arxiv.org/abs/2602.06872
Z. Dong and T. Wadhawan, "A note on the constants in inverse trace inequalities for polynomials orthogonal to lower-order subspaces", preprint (2025), arXiv:2512.14570, https://arxiv.org/abs/2512.14570
T. Chaumont-Frelet and M. Vohralík, "A stable local commuting projector and optimal hp approximation estimates in H(curl)", Numerische Mathematik 156 (2024), no. 6, 2293–2342, DOI 10.1007/s00211-024-01431-w, https://doi.org/10.1007/s00211-024-01431-w
T. Chaumont-Frelet, A. Ern, and M. Vohralík, "Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem", Mathematics of Computation 91 (2022), no. 333, 37–74 (published online 2021-09-28), DOI 10.1090/mcom/3673, https://doi.org/10.1090/mcom/3673