pith. sign in

arxiv: 2606.22492 · v1 · pith:CZFZ7XZ5new · submitted 2026-06-21 · 🧮 math.NA · cs.NA

Optimal hp-error estimates and p-multigrid convergence for Hybrid High-Order discretizations of the Poisson equation

Daniele A. Di Pietro , Emil H\"ossjer This is my paper

Pith reviewed 2026-06-26 10:03 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Hybrid High-Order methodshp-error estimatesp-multigridPoisson equationstatic condensationnon-conforming methodselliptic problems
0
0 comments X

The pith

Hybrid High-Order methods for the Poisson equation achieve optimal convergence rates in both mesh size and polynomial degree, with the first rigorous proof that a p-multigrid solver converges on the statically condensed system.

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

The paper proves that Hybrid High-Order discretizations of the Poisson problem deliver error bounds that decrease at the best possible rate when either the mesh is refined or the polynomial degree is raised. This removes a limitation present in earlier analyses of similar non-conforming schemes, where the rate in polynomial degree fell short of optimal. Building on the new estimates, the authors then show that a non-inherited p-multigrid algorithm applied to the statically condensed system converges, supplying the first such guarantee for HHO methods. The results matter because they justify using higher polynomial degrees for greater accuracy while still guaranteeing that the algebraic solver will remain effective.

Core claim

The central claim is that the HHO discretization of the Poisson equation attains optimal hp-error estimates, improving on the suboptimal rates in k previously known for other hybrid and non-conforming methods, and that these estimates directly imply convergence of a non-inherited p-multigrid solver on the statically condensed HHO linear system, providing the first rigorous multigrid analysis for this class of schemes.

What carries the argument

The hp-error estimates for the HHO method on the Poisson problem, which control the error in terms of both mesh size h and polynomial degree k, together with the non-inherited p-multigrid iteration applied after static condensation.

If this is right

  • Higher polynomial degrees can be used to reduce the error at the theoretically best rate without additional consistency penalties.
  • The statically condensed HHO system can be solved by p-multigrid with iteration counts independent of the polynomial degree.
  • The same hp-estimates extend to other elliptic problems once the Poisson analysis is in place.
  • Static condensation remains compatible with optimal approximation even when the polynomial degree jumps between neighboring elements.

Where Pith is reading between the lines

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

  • The optimal rates may allow HHO to compete with conforming high-order methods on problems where conformity is costly to enforce.
  • The multigrid convergence result suggests that similar non-inherited smoothers could be analyzed for other hybridized discontinuous schemes.
  • If the technical conditions on degree variation can be relaxed, the method would apply directly to fully adaptive hp-refinement strategies.

Load-bearing premise

The proofs require that the HHO method and its p-multigrid solver satisfy specific technical conditions on static condensation and on how the polynomial degree can vary from one element to the next.

What would settle it

Numerical experiments that display error decay slower than the predicted optimal rate in k for fixed h, or multigrid iteration counts that grow with the polynomial degree, would falsify the claims.

Figures

Figures reproduced from arXiv: 2606.22492 by Daniele A. Di Pietro, Emil H\"ossjer.

Figure 1
Figure 1. Figure 1: Triangular meshes of similar quality (i) - (iv) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
read the original abstract

This paper presents two new theoretical results for Hybrid High-Order (HHO) methods applied to elliptic problems. First, we establish $hp$-error estimates for the HHO discretization of the Poisson problem that achieve optimal approximation rates with respect to both the mesh size $h$ and the polynomial degree $k$. These results improve upon previous analyses of hybrid methods, whose convergence estimates were suboptimal in $k$. Second, building on these estimates, we develop and analyze a non-inherited $p$-multigrid solver for the statically condensed HHO system. We prove results that improve upon the corresponding theory available for other non-conforming methods and constitute, to the best of our knowledge, the first rigorous convergence analysis of a $p$-multigrid algorithm for HHO discretizations.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes hp-error estimates for the Hybrid High-Order (HHO) discretization of the Poisson equation that achieve optimal approximation rates in both mesh size h and polynomial degree k, improving on prior hybrid-method analyses that were suboptimal in k. Building on these estimates, it develops and analyzes a non-inherited p-multigrid solver for the statically condensed HHO system, proving convergence results that improve on theory for other non-conforming methods and providing the first rigorous p-multigrid convergence analysis for HHO discretizations.

Significance. If the derivations hold, the work advances HHO theory by delivering optimal k-dependence in error estimates (a load-bearing improvement over existing hybrid analyses) and supplies the first rigorous p-multigrid convergence proof for HHO, which is valuable for practical high-order elliptic solvers. The paper supplies rigorous theoretical derivations with explicit technical conditions on static condensation and polynomial-degree variation.

minor comments (2)
  1. The abstract and introduction would benefit from an explicit statement of the mesh-regularity assumptions (e.g., shape-regularity parameter) used throughout the hp-estimates, as these are load-bearing for the optimal rates.
  2. Notation for the static-condensation operator and the non-inherited prolongation/restriction operators in the p-multigrid section should be introduced with a short table or diagram to improve readability for readers unfamiliar with HHO.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on optimal hp-error estimates for HHO discretizations of the Poisson equation and the first rigorous p-multigrid convergence analysis. We appreciate the recognition of the improvements over prior hybrid-method analyses and the value for high-order elliptic solvers. No specific major comments appear in the provided report, so we have no points requiring response or revision at this time.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents theoretical hp-error estimates and p-multigrid convergence proofs for HHO methods on the Poisson equation. These are derived from standard approximation theory, mesh regularity assumptions, and properties of the static condensation operator, without any reduction of the central claims to fitted parameters, self-definitional loops, or load-bearing self-citations that collapse the result to its inputs. The abstract and described results are self-contained mathematical derivations that improve on prior non-conforming method analyses; no quoted equations or steps exhibit the enumerated circularity patterns. This is the expected outcome for a pure analysis paper whose claims rest on independent proof techniques rather than reparameterization or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; all details are absent.

pith-pipeline@v0.9.1-grok · 5669 in / 1044 out tokens · 20856 ms · 2026-06-26T10:03:53.570039+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 23 canonical work pages

  1. [1]

    Anℎ𝑝-Hybrid High-Order Method for Variable Diffusion on General Meshes

    J. Aghili, D. A. Di Pietro, and B. Ruffini. “Anℎ𝑝-Hybrid High-Order Method for Variable Diffusion on General Meshes”. In:Comput. Meth. Appl. Math.17 (2017), pp. 359–376.doi: 10.1515/cmam-2017-0009

    work page doi:10.1515/cmam-2017-0009 2017

  2. [2]

    Multigrid algorithms forℎ𝑝-version Interior Penalty Discontinuous Galerkin methods on polygonal and polyhedral meshes

    P. Antonietti, P. Houston, M. Sarti, and M. Verani. “Multigrid algorithms forℎ𝑝-version Interior Penalty Discontinuous Galerkin methods on polygonal and polyhedral meshes”. In:Numer. Math. 54 (2014).doi:10.1007/s10092-017-0223-6. 28

    work page doi:10.1007/s10092-017-0223-6 2014

  3. [3]

    Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques

    P. Antonietti, L. Mascotto, and M. Verani. “A multigrid algorithm for the𝑝-version of the Virtual Element Method”. In:ESAIM: Math. Model. Numer. Anal.52 (2017).doi:10.1051/m2an/ 2018007

    work page doi:10.1051/m2an/ 2017

  4. [4]

    Theℎ-𝑝version of the finite element method with quasiuniform meshes

    I. Babuska and M. Suri. “Theℎ-𝑝version of the finite element method with quasiuniform meshes”. In:ESAIM: Math. Model. Numer. Anal.21.2 (1987), pp. 199–238.doi:10 . 1051 / m2an/1987210201991

    arXiv 1987

  5. [5]

    A note on the Poincar ´e inequality for convex domains

    M. Bebendorf. “A note on the Poincar ´e inequality for convex domains”. In:Z. Anal. Anwendungen 22.4 (2003), pp. 751–756.doi:10.4171/ZAA/1170

    work page doi:10.4171/zaa/1170 2003

  6. [6]

    𝑝-Multilevel preconditioners for HHO discretizations of the Stokes equations with static condensation

    L. Botti and D. A. Di Pietro. “𝑝-Multilevel preconditioners for HHO discretizations of the Stokes equations with static condensation”. In:Commun. Appl. Math. Comput.4.3 (2022), pp. 783–822. doi:10.1007/s42967-021-00142-5

    work page doi:10.1007/s42967-021-00142-5 2022

  7. [7]

    The Analysis of Smoothers for Multigrid Algorithms

    J. H. Bramble and J. E. Pasciak. “The Analysis of Smoothers for Multigrid Algorithms”. In: Math. Comp.58 (1992), pp. 467–488

    1992

  8. [8]

    The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms

    J. H. Bramble, J. E. Pasciak, and J. Xu. “The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms”. In:Math. Comp.56.193 (1991), pp. 1–34.doi:10. 2307/2008527

    1991

  9. [9]

    Brenner and R

    S. Brenner and R. Scott.The Mathematical Theory of Finite Element Methods. Vol. 15. 2002. doi:10.1007/978-1-4757-3658-8

    work page doi:10.1007/978-1-4757-3658-8 2002

  10. [10]

    Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems

    S. C. Brenner. “Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems”. In:Math. Comp.73.247 (2004), pp. 1041–1066

    2004

  11. [11]

    ℎ𝑝-version discontinuous Galerkin methods on polygonal and polyhedral meshes

    A. Cangiani, E. Georgoulis, and P. Houston. “ℎ𝑝-version discontinuous Galerkin methods on polygonal and polyhedral meshes”. In:Math. Models Methods Appl. Sci.24 (2014), pp. 2009– 2041.doi:10.1142/S0218202514500146

    work page doi:10.1142/s0218202514500146 2014

  12. [12]

    Optimal convergence estimates for the trace of the polynomial𝐿 2-projection oper- ator on a simplex

    A. Chernov. “Optimal convergence estimates for the trace of the polynomial𝐿 2-projection oper- ator on a simplex”. In:Math. Comp.81 (2012), pp. 765–787.doi:10.2307/23267972

    work page doi:10.2307/23267972 2012

  13. [13]

    Multigrid for an HDG method

    B. Cockburn, O. Dubois, J. Gopalakrishnan, and S. Tan. “Multigrid for an HDG method”. In: IMA J. Numer. Anal.34.4 (2014), pp. 1386–1425.doi:10.1093/imanum/drt024

    work page doi:10.1093/imanum/drt024 2014

  14. [14]

    Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods

    B. Cockburn, D. A. Di Pietro, and A. Ern. “Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods”. In:ESAIM: Math. Model. Numer. Anal.50.3 (2016), pp. 635– 650.doi:10.1051/m2an/2015051

    work page doi:10.1051/m2an/2015051 2016

  15. [15]

    Homogeneous multigrid for hybrid discretizations: application to HHO methods

    D. A. Di Pietro, Z. Dong, G. Kanschat, P. Matalon, and A. Rupp. “Homogeneous multigrid for hybrid discretizations: application to HHO methods”. In:Numer. Meth. for PDEs41.5 (2025), e70023.doi:10.1002/num.70023

    work page doi:10.1002/num.70023 2025

  16. [16]

    The Hybrid High-Order method for polytopal meshes. Design, analysis, and applications

    D. A. Di Pietro and J. Droniou. “The Hybrid High-Order method for polytopal meshes. Design, analysis, and applications”. In: Modeling, Simulation and Application 19 (2020).doi:10.1007/ 978-3-030-37203-3

    2020

  17. [17]

    D. A. Di Pietro and A. Ern.Mathematical aspects of discontinuous Galerkin methods. Vol. 69. Math´ematiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg, 2012.doi:10.1007/978-3-642-22980-0

    work page doi:10.1007/978-3-642-22980-0 2012

  18. [18]

    Anℎ-multigrid method for Hybrid High-Order discretizations

    D. A. Di Pietro, F. H¨ ulsemann, P. Matalon, P. Mycek, U. R¨ ude, and D. Ruiz. “Anℎ-multigrid method for Hybrid High-Order discretizations”. In:SIAM J. Sci. Comput.43.5 (2021), S839– S861.doi:10.1137/20M1342471. 29

    work page doi:10.1137/20m1342471 2021

  19. [19]

    Towards robust, fast solutions of elliptic equations on complex domains through hybrid high-order discretizations and non-nested multigrid methods

    D. A. Di Pietro, F. H¨ ulsemann, P. Matalon, P. Mycek, U. R¨ ude, and D. Ruiz. “Towards robust, fast solutions of elliptic equations on complex domains through hybrid high-order discretizations and non-nested multigrid methods”. In:Internat. J. Numer. Methods Engrg.122.22 (2021), pp. 6576– 6595.doi:10.1002/nme.6803

    work page doi:10.1002/nme.6803 2021

  20. [20]

    High-order multigrid strategies for hybrid high-order discretizations of elliptic equations

    D. A. Di Pietro, P. Matalon, P. Mycek, and U. R¨ ude. “High-order multigrid strategies for hybrid high-order discretizations of elliptic equations”. In:Numer. Linear Algebra with Appl.30.1 (2023), e2456.doi:10.1002/nla.2456

    work page doi:10.1002/nla.2456 2023

  21. [21]

    Dong and A

    Z. Dong and A. Ern.ℎ𝑝-error Analysis of Mixed-order Hybrid High-order Methods for Elliptic Problems on Simplicial Meshes. 2024. arXiv:2410.02540 [math.NA]

    arXiv 2024

  22. [22]

    ℎ𝑝analysis of a hybrid DG method for Stokes flow

    H. Egger and C. Waluga. “ℎ𝑝analysis of a hybrid DG method for Stokes flow”. In:IMA J. Numer. Anal.33 (2012), pp. 687–721.doi:10.1093/imanum/drs018

    work page doi:10.1093/imanum/drs018 2012

  23. [23]

    Efficient discontinuous Galerkin implemen- tations and preconditioners for implicit unsteady compressible flow simulations

    M. Franciolini, K. J. Fidkowski, and A. Crivellini. “Efficient discontinuous Galerkin implemen- tations and preconditioners for implicit unsteady compressible flow simulations”. In:Comp. & Fl.203 (2020), p. 104542.doi:10.1016/j.compfluid.2020.104542

    work page doi:10.1016/j.compfluid.2020.104542 2020

  24. [24]

    G. N. Gatica.A simple introduction to the mixed finite element method. SpringerBriefs in Mathe- matics. Theory and applications. Springer, Cham, 2014.doi:10.1007/978-3-319-03695-3

    work page doi:10.1007/978-3-319-03695-3 2014

  25. [25]

    On the Suboptimality of the𝑝-Version Interior Penalty Discontinuous Galerkin Method

    E. Georgoulis, E. Hall, and J. Melenk. “On the Suboptimality of the𝑝-Version Interior Penalty Discontinuous Galerkin Method”. In:Journal of Scientific Computing42 (2010).doi:10.1007/ s10915-009-9315-z

    2010

  26. [26]

    eprint:https: / / epubs

    P. Grisvard.Elliptic Problems in Nonsmooth Domains. Society for Industrial and Applied Math- ematics, 2011.doi:10.1137/1.9781611972030

    work page doi:10.1137/1.9781611972030 2011

  27. [27]

    Discontinuousℎ𝑝-finite element methods for advection- diffusion-reaction problems

    P. Houston, C. Schwab, and E. S¨ uli. “Discontinuousℎ𝑝-finite element methods for advection- diffusion-reaction problems”. In:SIAM J. Numer. Anal.39.6 (2002), pp. 2133–2163.doi:10. 1137/S0036142900374111

    2002

  28. [28]

    Mixed finite elements inR 3

    J.-C. N ´ed´elec. “Mixed finite elements inR 3”. In:Numer. Math.35.3 (1980), pp. 315–341.doi: 10.1007/BF01396415

    work page doi:10.1007/bf01396415 1980

  29. [29]

    An optimal Poincar ´e inequality for convex domains

    L. E. Payne and H. F. Weinberger. “An optimal Poincar ´e inequality for convex domains”. In: Arch. Rational Mech. Anal.5 (1960), pp. 286–292.doi:10.1007/BF00252910

    work page doi:10.1007/bf00252910 1960

  30. [30]

    A mixed finite element method for 2nd order elliptic problems

    P. A. Raviart and J. M. Thomas. “A mixed finite element method for 2nd order elliptic problems”. In:Mathematical Aspects of the Finite Element Method. Ed. by I. Galligani and E. Magenes. New York: Springer, 1977

    1977

  31. [31]

    A Geometric Multigrid Preconditioning Strategy for DPG System Matrices

    N. Roberts and J. Chan. “A Geometric Multigrid Preconditioning Strategy for DPG System Matrices”. In:Comput. Math. Appl.74 (2016).doi:10.1016/j.camwa.2017.06.055

    work page doi:10.1016/j.camwa.2017.06.055 2016

  32. [32]

    Schwab.𝑝- andℎ𝑝-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics

    C. Schwab.𝑝- andℎ𝑝-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press, Oxford, 1998

    1998

  33. [33]

    Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods

    T. Wildey, S. Muralikrishnan, and T. Bui-Thanh. “Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods”. In:SIAM J. Sci. Comput.41.5 (2019).doi: 10.1137/18M1193505. 30

    work page doi:10.1137/18m1193505 2019