Optimal p-version error bounds for the hp interior penalty discontinuous Galerkin method for the quad-curl source and eigenvalue problems

The hp-version interior penalty discontinuous Galerkin (IPDG) method of Han and Zhang (BIT Numerical Mathematics 63, 2023) for the quad-curl equation $\mathrm{curl}^4 u$ and the associated quad-curl eigenvalue problem comes with a priori error bounds that are optimal in the mesh size $h$ but provably suboptimal in the polynomial degree $p$: the analysis loses 3 orders in $p$ on nonconforming simplicial meshes and 2 orders on conforming ones. The open problem is to complete the convergence theory of this method by determining the sharp polynomial-degree dependence of its a priori error bounds — the DG norm for the source problem, and the eigenvalue and eigenfunction errors — whether that sharp exponent is the optimal one or instead reflects a genuine residual loss. The paper itself also notes, within the same theory, that the $p$-independence on general simplicial meshes of the reconstruction-based constant behind the hp discrete compactness argument is currently verified only for $H(\mathrm{curl})$-conforming rectangular elements, and it indicates but does not carry out the extension of the hp-error analysis to general polygonal (2D) and polyhedral (3D) meshes; these are related context for the central $p$-optimality objective rather than separate completion requirements.

Background

For a vector field $u$ on a simply connected polyhedral domain $\Omega \subset \mathbb{R}^3$, the curl operator $\nabla \times u$ generalizes to the fourth-order quad-curl operator $\mathrm{curl}^4$. The quad-curl problem seeks $u$ with $\nabla \times \nabla \times \nabla \times \nabla \times u = f$ together with a divergence-free constraint $\nabla \cdot u = 0$ and suitable boundary conditions. It arises in the inverse electromagnetic scattering transmission eigenvalue problem and in the time discretization of resistive magnetohydrodynamics, and its eigenvalue problem $\mathrm{curl}^4 u = \lambda u$ governs so-called Maxwell transmission eigenvalues used to derive nondestructive testing criteria. Because the curl operator has a large kernel (gradient fields), the problem is not elliptic in the standard sense, and discretizations need special care: conforming methods require finite elements with continuous tangential curls across mesh faces ($H(\mathrm{curl}^2)$-conforming elements), while mixed and discontinuous formulations introduce auxiliary variables or penalty terms. Interior penalty discontinuous Galerkin (IPDG) methods use fully discontinuous piecewise-polynomial spaces and add mesh-face penalty terms, scaled by the polynomial degree $p$, to weakly enforce continuity of the field and its curls. The hp version refines both the mesh size $h$ and the polynomial degree $p$, and an analysis is called $p$-optimal when the polynomial-degree exponent in the error bound matches the best-approximation exponent of the underlying space. For second-order elliptic problems, hp-optimal DG bounds are known (e.g., Stamm and Wihler, 2010), but for fourth-order operators the hp-IPDG theory itself remains suboptimal in $p$ (Mozolevski, Süli, and Bösing proved analogous bounds for the biharmonic equation). For the quad-curl problem, Chen, Qiu, and Xu (2021) analyzed an $h$-version mixed IPDG method, and Sun (2016) gave the first mixed finite element method for the eigenvalue problem; $H(\mathrm{curl}^2)$-conforming spectral element methods (Wang, Shan, Li, and Zhang, 2021; Wang, Li, Q. Zhang, and Z. Zhang, 2025) provide high-order approximation spaces but without $p$-explicit error theory. Han and Zhang (2023) proposed the first hp-version IPDG method for the quad-curl source problem (with and without the divergence-free constraint) and its eigenvalue problem, with two interior penalty parameters. They proved well-posedness via a discrete Poincaré inequality in the discrete divergence-free space, established hp-version discrete compactness (the property, central to Maxwell-type eigenvalue approximation since Kikuchi and Boffi, that bounded discrete divergence-free fields are precompact in $L^2$, ruling out spurious eigenmodes), and then invoked the Babuška–Osborn spectral approximation framework (the classical perturbation theory relating operator convergence to eigenvalue/eigenvector convergence) to obtain eigenvalue convergence. Their bounds are optimal in $h$ but suboptimal in $p$: for the source problem, the DG-norm bound behaves like $h^{\min(p+1,s)-2} p^{3.5-s}$ on nonconforming meshes (meshes with hanging nodes) and $h^{\min(p+1,s)-2} p^{3-s}$ on conforming meshes for solutions in $H^s$, and the eigenvalue bounds behave like $h^{2\min(s,p+1)-4} p^{7-2s}$ and $p^{6-2s}$ respectively — by the authors' accounting, a loss of 3 orders of $p$ on nonconforming simplicial meshes and 2 orders on conforming ones. Their numerical experiments nevertheless show exponential $p$-convergence for smooth solutions, so the defect is in the theory (the algebraic $p$-exponents under finite regularity), where the losses trace back to $p$-dependent constants in trace inequalities, in the interpolation analysis, and in a reconstruction operator: the discrete compactness proof rests on an inequality whose hidden constant is $h$-independent by results of Houston, Perugia, Schneebeli, and Schötzau (2005) for time-harmonic Maxwell IPDG, but whose $p$-independence is verified by the authors only for $H(\mathrm{curl})$-conforming rectangular (Nédélec-type edge) elements. p-version discrete compactness is known for conforming discrete differential forms (Boffi, Costabel, Dauge, Demkowicz, and Hiptmair, 2011), but that result has not been transferred to the DG divergence-free reconstruction setting.

Problem Statement

For the hp-version interior penalty discontinuous Galerkin method of Han and Zhang (BIT Numerical Mathematics 63, 2023) for the quad-curl source problem $\mathrm{curl}^4 u = f$ (both with and without the divergence-free constraint) and the associated quad-curl eigenvalue problem, on simplicial (triangular/tetrahedral) meshes with polynomial degree $p$: determine the exact sharp dependence on the polynomial degree $p$ of each of the three a priori error bounds — the DG-norm error of the source problem, the eigenvalue errors, and the eigenfunction errors — on nonconforming and on conforming meshes. The three quantities are separately resolvable: fixing the sharp $p$-exponent of any one of them settles that quantity and leaves the other two open (the loss currently reported in the theory being 3 orders of $p$ on nonconforming meshes and 2 orders on conforming meshes).

The paper notes two related gaps in the same convergence theory, recorded here as context rather than as requirements on a solution: the hidden constant in the reconstruction-based inequality underlying the hp discrete compactness of the discrete divergence-free DG space is verified to be independent of $h$, but its independence of $p$ is established only for $H(\mathrm{curl})$-conforming rectangular elements (Remark 4.1); and the extension of the hp-error analysis to general polygonal (2D) and polyhedral (3D) meshes is indicated but not performed (Remarks 3.1 and 4.2), including the conditions on element edge/face counts under which the uniform convergence of the discrete solution operators can be guaranteed.

References

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