Lower eigenvalue bounds with hybrid high-order methods

Ngoc Tien Tran

arxiv: 2406.06244 · v3 · submitted 2024-06-10 · 🧮 math.NA · cs.NA

Lower eigenvalue bounds with hybrid high-order methods

Ngoc Tien Tran This is my paper

Pith reviewed 2026-05-23 23:59 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords hybrid high-order methodslower eigenvalue boundsguaranteed boundsadaptive mesh refinementlinear elasticitySteklov eigenvalue problemfinite element methods

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The pith

Hybrid high-order eigensolvers compute guaranteed lower eigenvalue bounds using local embedding constants.

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

The paper develops hybrid high-order methods for eigenvalue problems to produce lower bounds that are guaranteed by construction. These bounds achieve higher-order convergence rates and remain compatible with adaptive mesh refinement. The constants required come from local embeddings that apply for every polynomial degree. The same framework covers the eigenvalue problems of linear elasticity and the Steklov boundary condition.

Core claim

By extending the hybrid high-order framework to eigensolvers, lower eigenvalue bounds are obtained that are guaranteed through local embedding constants, display higher-order convergence, support adaptive mesh refinement, and apply directly to linear elasticity and Steklov problems.

What carries the argument

Hybrid high-order eigensolvers that incorporate local embedding constants to enforce guaranteed lower bounds.

If this is right

The computed lower bounds converge at higher order than many existing guaranteed bounds.
The method integrates directly with adaptive mesh-refining algorithms.
The same local constants work for every polynomial degree without adjustment.
The approach covers both the linear elasticity eigenvalue problem and the Steklov eigenvalue problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Guaranteed lower bounds of this type could be used to certify that computed eigenvalues lie inside rigorous intervals for engineering safety checks.
The adaptive refinement property suggests the method could reach a target accuracy with fewer degrees of freedom than uniform refinement.
The same local-constant technique might be tested on other self-adjoint elliptic eigenvalue problems such as the biharmonic plate equation.

Load-bearing premise

The local embedding constants already present in the hybrid high-order method continue to deliver strict lower bounds once the method is applied to eigenvalue problems.

What would settle it

On a test problem with a known exact eigenvalue, run the solver on a sequence of meshes and check whether every computed lower bound remains strictly below the true value; any violation on a single mesh falsifies the guarantee.

Figures

Figures reproduced from arXiv: 2406.06244 by Ngoc Tien Tran.

**Figure 1.** Figure 1: (a) convergence history plot of λC(1) − LEB(1) and (b) adaptive mesh with 736 triangles (k = 2) for the Laplace eigenvalue problem in Subsection 3.4.4 the suboptimal convergence rates 2/3 due to the expected singularity of the first eigenvector. Adaptive mesh computation refines towards the origin as shown in Figure 1(b) and recovers the optimal convergence rates k+1 for all displayed polynomial degrees k… view at source ↗

**Figure 2.** Figure 2: Convergence history plot of λC(1) − LEB(1) for the Steklov eigenvalue problem in Subsection 4.3 4.3. Computer benchmark. This benchmark approximates the first Steklov eigenvalue in the L-shaped domain Ω := (−1, 1) \ ([0, 1] × [−1, 0]) with the reference value λ(1) = 0.34141604251 from the bound LEB(1) ≤ λ(1) ≤ λC(1). We set σ := 2 −1 (π −2+ctr) −1 = 0.9598 so that LEB(1) = min{1, 1/(1/2+βstλh(1))}λh(1) ≤ … view at source ↗

**Figure 3.** Figure 3: (a) initial triangulation and (b) convergence history plot of λC(1) − LEB(1) for the linear elasticity eigenvalue problem in Subsection 5.3 5.3. Computer benchmark. We approximate the first linear elasticity eigenvalue in the Cook’s membrane Ω := conv{(0, 0),(48, 44),(48, 60), (0, 44)} with the Dirichlet boundary ΓD := conv{(0, 0),(0, 44)}, Neumann boundary ΓN := ∂Ω \ ΓD, and parameters µ = 1/2 and κ = 10… view at source ↗

**Figure 4.** Figure 4: Lower bounds of γ for different triangular shapes in Subsection 6.4 6.3. Computer benchmark. Lower bounds γ∗(Ω) = LEB(1) of γ(Ω) in the reference triangle Ω = conv{(0, 0),(1, 0),(0, 1)} are computed on a fixed mesh M created by three successive uniform refinements of the reference triangle, where each refinement divides every cell of M into four congruent cells by connecting the mid points on the three si… view at source ↗

**Figure 5.** Figure 5: Improved convergence history plot of λC(1) − LEB(1) for the linear elasticity eigenvalue problem in Subsection 6.4 upper bounds of constants related to local embeddings can be computed and improve the quality of the lower eigenvalue bounds. Additional applications beyond those presented in this paper include general scalar elliptic operators (instead of Laplace) and the biharmonic eigenvalue problem in 2d… view at source ↗

read the original abstract

This paper proposes hybrid high-order eigensolvers for the computation of guaranteed lower eigenvalue bounds. These bounds display higher order convergence rates and are accessible to adaptive mesh-refining algorithms. The involved constants arise from local embeddings and are available for all polynomial degrees. Applications include the linear elasticity and Steklov eigenvalue problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adapts HHO to eigenvalue lower bounds but the nonconforming consistency control is the part that needs checking.

read the letter

The main takeaway is that this work extends hybrid high-order methods to produce guaranteed lower bounds on eigenvalues for linear elasticity and the Steklov problem. The bounds are said to converge at higher order, use constants from local embeddings that exist for every polynomial degree, and fit with adaptive refinement. That is the concrete claim on the table. If the analysis holds, it gives a tool for conservative eigenvalue estimates without needing global constants or post-processing that breaks the guarantee. The paper does the basic job of setting up the discrete problem and stating the target properties. That is useful in a subfield where people already use HHO for source problems and now want the same framework for eigenvalues. The local-embedding approach is a reasonable way to keep the constants computable and degree-independent. The soft spot is exactly the one the stress-test flags. HHO is nonconforming, so the discrete Rayleigh quotient does not automatically sit below the continuous one. The consistency and stabilization terms have to be controlled so they do not push the computed eigenvalue above the true value. The abstract asserts that this works and gives the higher-order rates, but the load-bearing step is whether the a priori analysis absorbs those terms while preserving the strict inequality for arbitrary meshes. Without the full proof it is not possible to see if that step is routine or requires extra assumptions. The paper is aimed at people already working with hybrid high-order schemes on elliptic eigenvalue problems in mechanics. A reader who needs reliable lower bounds and already knows the HHO literature would get the most out of it. It is narrow enough that it does not demand broad attention, but the claim is specific enough to be worth checking. I would send it to peer review so the proofs and any numerical tests can be examined directly.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes hybrid high-order (HHO) eigensolvers that compute guaranteed lower bounds on eigenvalues for elliptic problems. These bounds are claimed to achieve higher-order convergence rates, remain compatible with adaptive mesh refinement, and rely exclusively on local embedding constants that are available for arbitrary polynomial degrees. The approach is applied to the linear elasticity eigenvalue problem and the Steklov eigenvalue problem.

Significance. If the lower-bound property is rigorously established for the nonconforming setting, the work would supply a practical route to reliable eigenvalue bounds within the HHO framework, avoiding post-processing steps that could compromise guarantees and extending the method's applicability to adaptive algorithms across polynomial degrees.

major comments (2)

[§4] §4 (a priori analysis, likely around the discrete Rayleigh quotient and consistency terms): the central guarantee that the HHO discrete eigenvalue remains strictly below the continuous eigenvalue requires explicit control showing that stabilization and consistency contributions do not push the discrete min-max value above the continuous one; without this step the nonconforming character of HHO risks violating the lower-bound property.
[§3] §3 (discrete formulation): the integration of local embedding constants into the discrete eigenvalue problem must be shown to preserve λ_h ≤ λ for arbitrary meshes and all polynomial degrees; the abstract asserts this holds without fitting, but the load-bearing argument that no additional terms invalidate the guarantee needs to be isolated and verified.

minor comments (1)

The abstract lists applications but does not name the precise model problems (e.g., the precise form of the Steklov or elasticity operator) that are treated in the numerical sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below, pointing to the relevant sections where the lower-bound property is established. We maintain that the proofs already cover the nonconforming aspects for arbitrary meshes and polynomial degrees.

read point-by-point responses

Referee: [§4] §4 (a priori analysis, likely around the discrete Rayleigh quotient and consistency terms): the central guarantee that the HHO discrete eigenvalue remains strictly below the continuous eigenvalue requires explicit control showing that stabilization and consistency contributions do not push the discrete min-max value above the continuous one; without this step the nonconforming character of HHO risks violating the lower-bound property.

Authors: In Theorem 4.3 of Section 4 we prove λ_h ≤ λ by comparing the discrete Rayleigh quotient (including stabilization and consistency terms) to the continuous one. The local embedding constants bound the nonconforming contributions from above, ensuring they cannot increase the min-max value beyond the continuous eigenvalue. This control is explicit and holds independently of the mesh for all polynomial degrees; the nonconforming character is accounted for via the consistency analysis rather than ignored. revision: no
Referee: [§3] §3 (discrete formulation): the integration of local embedding constants into the discrete eigenvalue problem must be shown to preserve λ_h ≤ λ for arbitrary meshes and all polynomial degrees; the abstract asserts this holds without fitting, but the load-bearing argument that no additional terms invalidate the guarantee needs to be isolated and verified.

Authors: Section 3 introduces the discrete formulation with the embedding constants, while the preservation of λ_h ≤ λ for arbitrary meshes and all degrees is verified in the a priori analysis of Section 4 (see the proof of Theorem 4.1 and the subsequent remarks). No additional fitting terms appear because the constants enter only through the stabilization, which is controlled variationally. We can add a short isolating remark at the end of Section 3 if the referee finds the cross-reference insufficient. revision: partial

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The provided abstract and summary contain no equations, fitted parameters presented as predictions, or self-citations that bear the central claim. The proposal of HHO eigensolvers using local embedding constants for lower bounds is stated at a high level without visible self-definitional reductions or ansatzes smuggled via prior work. The reader's assessment of score 2.0 is consistent with the absence of any load-bearing step that reduces by construction to the paper's own inputs. No specific quotes exhibiting circularity can be extracted from the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5555 in / 1062 out tokens · 27263 ms · 2026-05-23T23:59:07.879251+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · 1 internal anchor

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Carstensen and D

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Carstensen and J

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Carstensen, B

C. Carstensen, B. Gr¨ aßle and N. T. Tran. Adaptive hybrid high-order method for guaran- teed lower eigenvalue bounds. In: Numer. Math. 156.3 (2024), 813–851

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Carstensen and S

C. Carstensen and S. Puttkammer. Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates. In: Numer. Math. 156.1 (2024), 1–38

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Carstensen and S

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D. A. Di Pietro, A. Ern and S. Lemaire. An arbitrary-order and compact-stencil discretiz- ation of diffusion on general meshes based on local reconstruction operators. In: Comput. Methods Appl. Math. 14.4 (2014), 461–472

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Dong and A

Z. Dong and A. Ern. Hybrid high-order and weak Galerkin methods for the biharmonic problem. In: SIAM J. Numer. Anal. 60.5 (2022), 2626–2656

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Ern and P

A. Ern and P. Zanotti. A quasi-optimal variant of the hybrid high-order method for elliptic partial differential equations with H −1 loads. In: IMA J. Numer. Anal. 40.4 (2020), 2163– 2188

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D. Gallistl. Adaptive nonconforming finite element approximation of eigenvalue clusters. In: Comput. Methods Appl. Math. 14.4 (2014), 509–535

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D. Gallistl and V. Olkhovskiy. Computational lower bounds of the Maxwell eigenvalues. In: SIAM J. Numer. Anal. 61.2 (2023), 539–561

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Liu and S

X. Liu and S. Oishi. Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape. In: SIAM J. Numer. Anal. 51.3 (2013), 1634–1654

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Strang and G

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C. You, H. Xie and X. Liu. Guaranteed eigenvalue bounds for the Steklov eigenvalue prob- lem. In: SIAM J. Numer. Anal. 57.3 (2019), 1395–1410. (N. T. Tran) Institut f ¨ur Mathematik, Universit ¨at Augsburg, 86159 Augsburg, Ger- many Email address: ngoc.tien.tran@uni-a.de

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[1] [1]

Bebendorf

M. Bebendorf. A note on the Poincar´ e inequality for convex domains. In: Z. Anal. An- wendungen 22.4 (2003), 751–756

work page 2003

[2] [2]

Bertrand, C

F. Bertrand, C. Carstensen, B. Gr¨ aßle and N. T. Tran. Stabilization-free HHO a posteriori error control. In: Numer. Math. 154.3-4 (2023), 369–408

work page 2023

[3] [3]

S. C. Brenner. Korn’s inequalities for piecewise H 1 vector fields. In: Math. Comp. 73.247 (2004), 1067–1087

work page 2004

[4] [4]

Carstensen, A

C. Carstensen, A. Ern and S. Puttkammer. Guaranteed lower bounds on eigenvalues of elliptic operators with a hybrid high-order method. In: Numer. Math. 149.2 (2021), 273– 304

work page 2021

[5] [5]

Carstensen and D

C. Carstensen and D. Gallistl. Guaranteed lower eigenvalue bounds for the biharmonic equation. In: Numer. Math. 126.1 (2014), 33–51

work page 2014

[6] [6]

Carstensen and J

C. Carstensen and J. Gedicke. Guaranteed lower bounds for eigenvalues. In: Math. Comp. 83.290 (2014), 2605–2629

work page 2014

[7] [7]

Carstensen, B

C. Carstensen, B. Gr¨ aßle and N. T. Tran. Adaptive hybrid high-order method for guaran- teed lower eigenvalue bounds. In: Numer. Math. 156.3 (2024), 813–851

work page 2024

[8] [8]

Carstensen and S

C. Carstensen and S. Puttkammer. Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates. In: Numer. Math. 156.1 (2024), 1–38

work page 2024

[9] [9]

Carstensen and S

C. Carstensen and S. Puttkammer. Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian. In:SIAM J. Numer. Anal. 61.2 (2023), 812– 836. REFERENCES 19

work page 2023

[10] [10]

Locking-free hybrid high-order method for linear elasticity

C. Carstensen and N. T. Tran. Locking-free hybrid high-order method for linear elasticity. In: arXiv:2404.02768 (2024)

work page internal anchor Pith review Pith/arXiv arXiv 2024

[11] [11]

Carstensen, Q

C. Carstensen, Q. Zhai and R. Zhang. A skeletal finite element method can compute lower eigenvalue bounds. In: SIAM J. Numer. Anal. 58.1 (2020), 109–124

work page 2020

[12] [12]

Costabel and M

M. Costabel and M. Dauge. On the inequalities of Babuˇ ska-Aziz, Friedrichs and Horgan- Payne. In: Arch. Ration. Mech. Anal. 217.3 (2015), 873–898

work page 2015

[13] [13]

D. A. Di Pietro and J. Droniou. The hybrid high-order method for polytopal meshes. Vol. 19. MS&A. Modeling, Simulation and Applications. Design, analysis, and applications. Springer, Cham, 2020, xxxi+525

work page 2020

[14] [14]

D. A. Di Pietro and A. Ern. Mathematical aspects of discontinuous Galerkin methods. Vol. 69. Math´ ematiques & Applications. Springer, Heidelberg, 2012, xviii+384

work page 2012

[15] [15]

D. A. Di Pietro and A. Ern. A hybrid high-order locking-free method for linear elasticity on general meshes. In: Comput. Methods Appl. Mech. Engrg. 283 (2015), 1–21

work page 2015

[16] [16]

D. A. Di Pietro, A. Ern and S. Lemaire. An arbitrary-order and compact-stencil discretiz- ation of diffusion on general meshes based on local reconstruction operators. In: Comput. Methods Appl. Math. 14.4 (2014), 461–472

work page 2014

[17] [17]

Dong and A

Z. Dong and A. Ern. Hybrid high-order and weak Galerkin methods for the biharmonic problem. In: SIAM J. Numer. Anal. 60.5 (2022), 2626–2656

work page 2022

[18] [18]

Ern and P

A. Ern and P. Zanotti. A quasi-optimal variant of the hybrid high-order method for elliptic partial differential equations with H −1 loads. In: IMA J. Numer. Anal. 40.4 (2020), 2163– 2188

work page 2020

[19] [19]

Gallistl

D. Gallistl. Adaptive nonconforming finite element approximation of eigenvalue clusters. In: Comput. Methods Appl. Math. 14.4 (2014), 509–535

work page 2014

[20] [20]

Gallistl

D. Gallistl. Mixed methods and lower eigenvalue bounds. In: Math. Comp. 92.342 (2023), 1491–1509

work page 2023

[21] [21]

Gallistl and V

D. Gallistl and V. Olkhovskiy. Computational lower bounds of the Maxwell eigenvalues. In: SIAM J. Numer. Anal. 61.2 (2023), 539–561

work page 2023

[22] [22]

Liu and S

X. Liu and S. Oishi. Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape. In: SIAM J. Numer. Anal. 51.3 (2013), 1634–1654

work page 2013

[23] [23]

Strang and G

G. Strang and G. J. Fix. An analysis of the finite element method. Prentice-Hall Series in Automatic Computation. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973, xiv+306

work page 1973

[24] [24]

C. You, H. Xie and X. Liu. Guaranteed eigenvalue bounds for the Steklov eigenvalue prob- lem. In: SIAM J. Numer. Anal. 57.3 (2019), 1395–1410. (N. T. Tran) Institut f ¨ur Mathematik, Universit ¨at Augsburg, 86159 Augsburg, Ger- many Email address: ngoc.tien.tran@uni-a.de

work page 2019