Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Matteo Giacomini; Ruben Sevilla

REVIEW 3 major objections 4 minor 126 references

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that degree-adaptive error decreases monotonically to the tolerance only when the curved boundary is represented exactly (NURBS-enhanced HDG); a frozen polynomial geometry lets the loop stop on the wrong solution.

desk verdict Useful consolidation of the authors' HDG variants, but the key new p-adaptivity claim rests on an undefined 'exact error' reference and needs a reproducibility pass before I'd trust the strong version. read the letter →

arxiv 1908.06604 v1 pith:N3NHTPB4 submitted 2019-08-19 math.NA cs.CEcs.NA

Matteo Giacomini , Ruben Sevilla This is my paper · ORCID

classification math.NAcs.CEcs.NA MSC 65N3065N5074S05

keywords hybridizablediscontinuousGalerkindegreeadaptivityNURBS-enhancedfiniteelementmethodexactgeometryHDG-Voigtformulationsuperconvergenterrorindicatorface-centeredvolumenear-incompressibleelasticity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper surveys three strands of hybridizable discontinuous Galerkin (HDG) methods for computational mechanics and makes a central argument about degree adaptivity: when a curved boundary is approximated once with fixed polynomials and only the solution degree is adapted, the error estimator can declare success while the true error stays far from the tolerance, because the iteration is converging to the solution of a different problem. The authors show that representing the geometry exactly with NURBS-enhanced finite elements (HDG-NEFEM) removes this failure, giving monotone decrease of the error to a user-specified tolerance on a coarse, fixed mesh. The same superconvergence machinery that builds the cheap error indicator also underlies an HDG-Voigt formulation for linear elasticity with pointwise symmetric stresses, optimal convergence at low polynomial degree, and no volumetric locking near the incompressible limit. The paper closes with the face-centered finite volume (FCFV) method, the lowest-order member of the HDG family, applied to a Stokes flow around 39 red-blood-cell-shaped particles in a mesh of nearly nine million tetrahedra.

What carries the argument

The engine of the paper is the pairing of three existing constructions. First, the hybridizable discontinuous Galerkin (HDG) formulation rewrites a second-order PDE into local element problems whose only globally coupled unknowns live on mesh faces, which makes the degrees of freedom cheap and enables a local postprocessing step, Eqs. (5)–(6), that produces a superconvergent approximation $u^\star$ of degree $k+1$ and accuracy order $k+2$. Second, the NURBS-enhanced finite element method (NEFEM) represents the domain boundary by the exact CAD NURBS curves and adapts the functional space near the boundary to that geometry, so changing the polynomial degree never requires re-approximating or regenerating the curved boundary. Third, the superconvergent $u^\star$ feeds the elementwise error indicator $E_u^e$ of Eq. (7), and the a priori estimate of Eq. (8), $\varepsilon_e \le C h_e^{k_e+1+n_{\mathrm{sd}}/2}$, combined with Richardson extrapolation of $C$, produces the degree-update rule $\Delta k_e = \left\lceil \log(\varepsilon/E_u^e)/\log(h_e) \right\rceil$ of Eq. (9). The HDG-Voigt formulation replaces the stress tensor by its independent entries in Voigt notation, enforcing symmetry of the mixed variable pointwise and yielding optimal convergence with nodal equal-order spaces; FCFV then appears as the constant-degree limit of the same HDG machinery, where all local unknowns admit closed-form expressions in terms of the face unknowns.

What would settle it

Run the degree-adaptive loop of Section 3 on a smooth domain with a known analytic solution (for example, a harmonic function on a circular-arc domain representable both by NURBS and by polynomials) and compare, at every iteration, the estimated error from Eq. (7) with the true error against the manufactured solution, under both HDG-NEFEM and fixed-cubic subparametric geometry. If at any iteration the NEFEM path shows a true error far above the estimate, or fails to decrease monotonically to the prescribed tolerance, the central claim is refuted; if the fixed-cubic path again stops with a large true error while its estimate claims success, the claimed failure mode of polynomial-frozen adaptivity is reproduced.

Watch

Extended reading notes

Core claim

The central claim is that in a degree-adaptive HDG process the geometry of the computational domain must be represented exactly for the iteration to converge to the intended solution. Using the electrostatic problem of a square inclusion with rounded corners, the paper shows that an isoparametric approach with a fixed cubic polynomial description of the rounded fillet stops after eight iterations because the estimated error reaches the tolerance, while the exact error remains far above it—the method is solving a problem with a slightly different, non-smooth geometry. With HDG-NEFEM, in which the NURBS description of the fillet is kept regardless of the polynomial degree of the functional approximation, the exact and estimated errors both decrease monotonically until the desired tolerance of $0.5\times10^{-3}$ is achieved. The mechanism is the superconvergent postprocessed solution $u^\star$, which yields a cheap elementwise error indicator; an a priori bound of the form $\varepsilon_e \le C h_e^{k_e+1+n_{\mathrm{sd}}/2}$, with $C$ fixed by Richardson extrapolation, then dictates how much to raise the local polynomial degree. Supporting results establish an HDG-Voigt formulation for linear elasticity whose pointwise symmetric strain-rate variable converges optimally (order $k+1$) even for $k=1,2,3$ and drives error indicators based on either displacement or strain rate, as well as a lowest-order face-centered finite volume method that is LBB-stable, reconstruction-free, and insensitive to mesh distortion and cell stretching.

Load-bearing premise

The whole degree-adaptive loop rests on the assumption that the error bound $\varepsilon_e \le C h_e^{k_e+1+n_{\mathrm{sd}}/2}$, with the constant $C$ obtained by Richardson extrapolation from two computed polynomial degrees, correctly predicts how the error will fall when the degree is raised; the paper itself shows this prediction failing badly in Figure 3 (right) whenever the boundary is frozen as cubic polynomials, and it never independently verifies the reliability of the indicator for the NEFEM case.

Editorial extensions

If this is right

Degree-adaptive HDG on a fixed coarse mesh can reach a user-prescribed tolerance in electrostatics without mesh regeneration, provided the curved boundary is the exact NURBS geometry; with a frozen cubic boundary it stops prematurely.
The superconvergent postprocessed displacement provides an inexpensive elementwise error indicator that drives p-adaptivity in linear elasticity for both displacement- and stress-accuracy goals, the strain-rate indicator localizing stress concentrations.
HDG-Voigt supplies a pointwise symmetric stress and strain-rate approximation that is optimally convergent for $k=1,2,3$ and free of volumetric locking for nearly incompressible materials, as demonstrated on Cook's membrane with $\nu=0.499999975$.
The same HDG rationale, taken to constant degree, yields FCFV: a first-order, LBB-stable, reconstruction-free finite volume method insensitive to distorted and stretched cells, exercised here on a 61-million-unknown Stokes problem in a serial Matlab implementation.
Because the NEFEM boundary is degree-independent, the degree-adaptive strategy composes directly with h-refinement, yielding hp-adaptive processes that keep the exact CAD geometry throughout.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical diagnostic suggested by Figure 3 (right) but not stated in the paper: any existing p-adaptive code with a subparametric boundary could monitor the gap between its estimated and exact error on a representatively curved test case, and a persistent gap is a cheap signal that geometric error, not functional error, is dominating the iteration.
The clean separation NEFEM provides between geometric and functional approximation suggests a parameter-free way to size curved meshes: let the CAD model alone determine the boundary representation, so the geometry error is exactly zero rather than balanced against the functional error.
The HDG-Voigt strain-rate indicator could be carried into shape-optimization or fracture-preconditioning loops, where the quantity of interest is the stress concentration at a rounded notch—exactly the region the displacement-based indicator misses; this is an extension the paper motivates but does not perform.
If FCFV's closed-form local elimination is the reason for its cheap assembly, the same decoupling should make the global face system amenable to algebraic multigrid or domain-decomposition preconditioning at scales far beyond the reported serial experiment, a direct but untested consequence of the block structure in Eq. (26).

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this as a consolidated review of the authors' own HDG variants: HDG-NEFEM for exact geometry, HDG-Voigt for linear elasticity, and FCFV for large-scale Stokes flow. The genuinely new material is the degree-adaptivity comparison in Section 3 and the 39-RBC FCFV demonstration in Section 5. The formulations are standard and clearly explained; someone wanting an entry point to these methods will get a good map of the area and a sensible discussion of the HDG rationale.

The paper is honest about being an overview, and the cautionary negative result in Section 3—where a fixed cubic representation of the geometry lets the adaptive process converge to the solution of a different problem—is a genuinely useful warning. That part deserves credit.

The soft spot is exactly the one the stress-test flags. The central claim in Section 3 is that with HDG-NEFEM the exact error decreases monotonically to the prescribed tolerance. But the manuscript never defines how the 'exact error' is computed. There is no analytic solution for the filleted-square electrostatic problem, and the text does not say whether the reference is an overkill HDG-NEFEM solution, a different discretization, or a manufactured solution. Without that definition, the monotonic-decrease claim and the comparison with the cubic-geometry case cannot be independently checked. Relatedly, the local error indicator (7) and the degree-update formula (9) are used for NEFEM curved elements without an effectivity check; the cited analyses are by the same group and mostly concern standard polynomial elements. Figure 3 (right) shows how badly the indicator can fail when the geometry is wrong. The same degree of scrutiny is not shown for the NEFEM run. I find the qualitative conclusion plausible, but as written this is one unreproducible numerical observation, not an established property.

The rest of the paper is less problematic. The HDG-Voigt and FCFV sections are standard demonstrations, though they too rely on prior papers for the convergence analysis and do not include error tables with computed versus exact errors. For an overview that is acceptable, but a few tabulated numbers would help.

The citation pattern is self-heavy by design and not a flaw here; the prior work is published in reputable journals. Still, an independent reader has to go elsewhere for verification.

Bottom line: this is a useful review-style paper that deserves a serious referee, but the referee should ask for the exact-error definition in Section 3, a short effectivity study for the NEFEM error indicator, and ideally an artifact or a reproducible description of the experiments. I would not accept it as-is, but I would send it to review with those requests.

Referee Report

3 major / 4 minor

Summary. The paper revisits hybridizable discontinuous Galerkin (HDG) methods combined with exact geometry representation via NURBS-enhanced finite elements (NEFEM), degree adaptivity, a Voigt-based symmetric-tensor HDG formulation for linear elasticity, and the lowest-order face-centered finite volume (FCFV) method for large-scale Stokes flows. The central numerical claim is in Section 3: on a filleted-square electrostatic problem, degree adaptivity with HDG-NEFEM reduces the exact error monotonically to a prescribed tolerance, whereas degree adaptivity on a fixed cubic polynomial geometry converges instead to the solution of a different problem. The other sections summarize the authors' prior HDG-Voigt and FCFV developments and show representative benchmark results, including Cook's membrane and a 39-particle red-blood-cell Stokes flow.

Significance. If the monotonic-decrease and reliability claims hold, the HDG-NEFEM degree-adaptive strategy would be a practically valuable alternative to isoparametric or fixed-polynomial-geometry adaptivity for problems requiring tolerance-driven p-refinement on curved domains. The paper is clearly written and the formulations are standard. However, the central new evidence is a single numerical example whose reference ('exact') error is never defined, and the local error indicator's reliability for NEFEM elements is not independently verified. The manuscript also does not ship code or machine-checked proofs, so the numerical claims cannot be reproduced from the text alone. The contribution is best assessed as a well-structured survey plus one promising but under-validated numerical demonstration.

major comments (3)

[Section 3, Figure 3] The exact-error curves in Figure 3 are not defined. The filleted-square problem has no known analytic solution, and the text does not state whether the reference solution is an overkill HDG-NEFEM computation, a different discretization, or a manufactured solution. Without this definition, the central claim that the HDG-NEFEM exact error decreases monotonically to the tolerance, and the contrast with the fixed-cubic-geometry case, cannot be independently checked. Please specify the reference solution, the discretization parameters used, and how the exact error is computed element-wise.
[Section 3, Eqs. (8)-(9) and Figure 3] The degree-update formula (9) presumes that the local indicator E_u^e defined in Eq. (7) is a faithful proxy for the true local error and that the a priori estimate (8), with the constant C estimated by Richardson extrapolation from two polynomial degrees, correctly predicts the error reduction when the degree is increased. No effectivity indices or per-iteration estimated-versus-exact error tables are given for the NEFEM run. This matters because Figure 3 (right) demonstrates that the same assumption can fail badly when the geometry is represented by fixed cubic polynomials. Please provide effectivity indices for both the NEFEM and cubic-geometry runs, and justify the validity of the exponent k+1+nsd/2 and the Richardson extrapolation over the degree range actually used.
[Sections 4-5, Figures 4, 6-7] The quantitative support for the HDG-Voigt and FCFV contributions is thin: the optimal-convergence claim for HDG-Voigt is illustrated only by the value of the displacement at one point of Cook's membrane, and the FCFV claim of first-order accuracy is not accompanied by any convergence table on a problem with a known solution. If these sections are intended as original numerical contributions rather than as summaries of prior published work, please add computed-versus-exact error tables and measured convergence rates. At minimum, state clearly that these are review sections and provide pointers to the original papers containing the full convergence studies.

minor comments (4)

[Section 5, Eq. (26)] The incompressibility constraint is written as ∫_{∂Ω_e\Γ_D} υ̂·n dΓ = -∫_{∂Ω_e∩Γ_D} u_D·n dΓ = 0, but the final “= 0” is not true in general because the Dirichlet data u_D need not vanish on Γ_D. The equality should be just ∫_{∂Ω_e\Γ_D} υ̂·n dΓ = -∫_{∂Ω_e∩Γ_D} u_D·n dΓ.
[Section 5, large-scale simulation paragraph] The text says “the evaluation of the element-by-element solution in all 61 millions elements,” but the mesh consists of about 8.97 million tetrahedral elements; 61.5 million is the number of unknowns in the global system. Please correct this wording.
[Section 3, Eq. (9)] In the degree-update formula, log(h_e) is negative when the nondimensional element size h_e is less than 1, so the sign convention in the ceiling expression should be explained explicitly, together with the nondimensionalization used to define h_e.
[General] The figures (Figures 1-7) are referenced in the captions but the images are not present in the manuscript text provided; please ensure the final version contains all figures with clearly readable axis labels and legends.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper is an overview with self-citations to prior methods, but its claims are not forced by definition or by the cited results.

full rationale

The paper is a review-style account of previously developed HDG, HDG-NEFEM, HDG-Voigt and FCFV methods, and it frequently cites the authors' own prior papers ([57], [82], [83], [106], [107], [118]). Self-citation alone is not circular, and here it is not the load-bearing mechanism: the formulations are restated in the manuscript and the numerical demonstrations are anchored to external benchmarks, e.g. the Cook's membrane reference value taken from [120]. The Section 3 degree-adaptivity procedure does use the error indicator (7) and the degree update (9), and the 'estimated error' curves reaching the tolerance are, in part, a consequence of the stopping criterion; however, the paper does not present that as a prediction, and the central comparison is made with the separate 'exact error' curves. The fixed-cubic-geometry failure shown in Figure 3 (right) is a numerical observation about a particular subparametric strategy, not an implication of Eq. (9) by construction. No equation in the derivation is equivalent by definition to its input, no fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' earlier work to force a choice. The unspecified reference solution for the 'exact error' curves is a reproducibility weakness rather than a circular step. Overall, the derivation chain is self-contained enough that no specific circular reduction can be exhibited.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central numerical demonstrations rest on user-chosen stabilization parameters, an a priori error estimate with a fitted constant, and convergence/stability results from the authors' own prior publications. No new physical entities are introduced.

free parameters (2)

Stabilization parameter τ = Unspecified for examples
Appears in numerical flux definitions (Eqs. (2), (15), (23)) and affects accuracy and convergence; its value is not reported for the numerical tests, so the results are not fully reproducible from the text.
Per-element constant C in error estimate = Estimated per element by Richardson extrapolation from two polynomial degrees
Used to compute the degree update (Eq. (9)) in the degree adaptive procedure. Figure 3 shows the estimated and exact errors diverge for the fixed-geometry case, indicating the fitted C can mislead adaptivity.

assumptions (3)

domain assumption Local a priori error estimate ε_e ≤ C h_e^(k_e+1+n_sd/2) (Eq. (8)) holds elementwise.
Adopted from [115] and used to derive the adaptive degree update; assumes a uniform hidden constant C and local error behavior.
domain assumption HDG postprocessing u* superconverges with order k+2 and provides a reliable error indicator (Eq. (7)).
Taken from [111] and the authors' previous work [82,112,113]; underpins the error estimator that drives degree adaptivity.
domain assumption FCFV inherits LBB stability and first-order convergence from the lowest-order HDG framework.
Stated in Section 5 and cited to [106,107]; not re-derived or independently verified in this paper.

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Cite this review

Pith. "Pith review of Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity." pith.science (2026). https://pith.science/paper/N3NHTPB4

@misc{pith2026190806604,
  author       = {Pith},
  title        = {Pith review of: Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N3NHTPB4}},
  note         = {Machine review of arXiv:1908.06604}
}

read the original abstract

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).

Figures

Figures reproduced from arXiv: 1908.06604 by the authors.

**Figure 2.** Left: intensity of the electric field computed with the proposed HDG-NEFEM [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Evolution of the exact and estimated errors as a function of the number of itera [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Convergence of the displacement of the mid-point of the right end of Cook’s [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Error indicator based on the displacement field (left) and the strain rate tensor [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Pressure field on 39 particles modelling RBCs immersed in an incompressible [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: Magnitude of the velocity field of an incompressible Stokes flow in a channel [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

Discussion (0). Continue with ORCID to comment.

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