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arxiv: 2604.15879 · v1 · submitted 2026-04-17 · 🧮 math.NA · cs.NA

hp-Version robust interior penalty discontinuous Galerkin methods for the p-Laplacian on simplicial and on essentially arbitrarily-shaped element meshes

Emmanuil H. Georgoulis , Panagiotis Paraschis This is my paper

Pith reviewed 2026-05-10 08:57 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords discontinuous Galerkinp-Laplacianhp-versioninverse estimatesarbitrarily-shaped elementsinterior penaltyerror estimates
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The pith

New trace-type inverse estimates ensure unconditional stability for discontinuous Galerkin approximations of the p-Laplacian on arbitrary meshes.

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

This paper develops hp-version interior penalty discontinuous Galerkin methods for the nonlinear p-Laplacian equation. The core step is proving new trace-type inverse estimates that bound boundary integrals uniformly with respect to the nonlinearity power p and the element geometry. These estimates remove any stability restrictions that would otherwise depend on p or the mesh, so the method remains stable for arbitrary polynomial degrees and any mesh. The same machinery produces hp-version a priori error estimates in both norm and quasi-norm that track the best polynomial approximation rates available. The analysis is then extended to meshes made of curved polygonal or polyhedral elements of essentially arbitrary shape through additional weighted inverse estimates.

Core claim

We prove novel trace-type inverse estimates, leading to unconditional stability of the interior penalty discontinuous Galerkin method for the p-Laplacian. We establish hp-version a priori norm and quasi-norm error estimates that are subordinate to available polynomial approximation results. The analysis extends to discontinuous Galerkin methods on meshes with essentially arbitrarily-shaped, curved polygonal/polyhedral elements by proving new hp-version weighted inverse estimates on such elements.

What carries the argument

Novel trace-type inverse estimates that bound traces on element faces by volume norms uniformly in the nonlinearity parameter p and in the element shape, used to establish stability and error bounds.

If this is right

  • The discontinuous Galerkin scheme remains stable for any p greater than 1 and any admissible mesh without additional restrictions on element size or polynomial degree.
  • The a priori error estimates in both energy norm and quasi-norm are optimal with respect to the polynomial degree and mesh size, provided the solution admits the corresponding regularity.
  • The same stability and error analysis applies directly to meshes composed of curved, non-simplicial elements.
  • Numerical experiments can be used to verify that observed convergence rates match the theoretical predictions across different p values and element shapes.

Where Pith is reading between the lines

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

  • The uniform inverse estimates may serve as a template for proving stability in discontinuous Galerkin discretizations of other nonlinear elliptic operators.
  • The ability to handle arbitrarily shaped curved elements reduces the preprocessing burden when applying the method to domains with complex boundaries.
  • The same inverse-estimate technique could be tested for time-dependent or coupled problems whose nonlinearity is also of p-Laplacian type.

Load-bearing premise

The novel trace-type inverse estimates and weighted inverse estimates on arbitrary elements hold uniformly with respect to the nonlinearity parameter p and the mesh geometry.

What would settle it

A family of polynomial functions on a curved element for which the ratio of the boundary trace integral to the volume integral grows without bound as the polynomial degree increases or as p varies, violating the claimed uniform inverse estimate and causing the discrete bilinear form to lose coercivity.

Figures

Figures reproduced from arXiv: 2604.15879 by Emmanuil H. Georgoulis, Panagiotis Paraschis.

Figure 1
Figure 1. Figure 1: An element K with one hanging node (red dot), faces e1(K), e2(K), e3(K) and interface F (blue line), and interfaces F, e2(K), e3(K) and e1(K) \ F. We denote the number of interfaces of an element K ∈ T by mK := card{F ∈ F : F ⊂ ∂K}. If K has no hanging nodes, then mK = d + 1. For k = (kK : K ∈ T ) ∈ (R + 0 ) card(T ) , we define the broken Sobolev spaces Wk,p(Ω; T ) := {v ∈ L p (Ω) : v|K ∈ WkK,p(K), K ∈ T … view at source ↗
Figure 2
Figure 2. Figure 2: Two neighboring elements K and K′ sharing an interface F (blue) con￾taining six faces. The elements K and K′ have nine faces each. For the design and analysis of the robust IPDG method (3.2) on polytopic meshes, we impose the following (very mild) mesh assumptions; we refer to [13] for more details. Assumption 7.1. For all K ∈ T , we assume that K is a Lipschitz domain and its boundary ∂K is subdivided int… view at source ↗
Figure 3
Figure 3. Figure 3: (Reproduction of [13, [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example 8.1, h-version: (a) Norm errors for r = 1. (b) Norm errors for r = 2. (c) Quasi-norm errors for r = 1. (d) Quasi-norm errors for r = 2. 8.1 h-Version. First, we consider the h-version of (3.2) for both examples. We select fixed polyno￾mial degrees r = 1, 2 and we employ a sequence {Tj} 4 j=0 of meshes, with hj = maxK∈Tj hK = 0.2/2 j , j = 0, . . . , 4. The errors with respect to the norms ||| · |||… view at source ↗
Figure 5
Figure 5. Figure 5: Example 8.2, h-version: (a) Norm errors for r = 1. (b) Norm errors for r = 2. (c) Quasi-norm errors for r = 1. (d) Quasi-norm errors for r = 2. 1 1         1 1 11   [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example 8.1, p-version: (a) Norm errors. (b) Quasi-norm errors [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example 8.2, p-version: (a) Norm errors. (b) Quasi-norm errors. 9. Concluding remarks A robust IPDG method discretizing the p-Laplacian equation, delivering provably superlinear convergence rate in appropriate scenarios has been developed and analyzed. Both simplicial meshes and meshes comprising general, possibly curved polytopic elements have been considered. The analysis hinges on new trace-inverse esti… view at source ↗
read the original abstract

We consider the discretization of the $p$-Laplacian equation with an interior penalty discontinuous Galerkin method. We prove novel trace-type inverse estimates, leading to unconditional stability of the method. Further, $hp$-version a priori norm and quasi-norm error estimates are established, subordinate to available polynomial approximation results. The analysis is extended to discontinuous Galerkin methods, based on meshes with essentially arbitrarily-shaped, curved polygonal/polyhedral elements. This extension requires the proof of new $hp$-version weighted inverse estimates on essentially arbitrarily-shaped elements. Numerical experiments are also presented, highlighting the relevance of the theoretical findings.

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

2 major / 1 minor

Summary. The manuscript develops an interior penalty discontinuous Galerkin (IPDG) discretization of the p-Laplacian. It establishes novel trace-type inverse estimates that yield unconditional stability, derives hp-version a priori error bounds in both the natural norm and a quasi-norm (subordinate to standard polynomial approximation results), and extends the analysis to DG methods on meshes consisting of essentially arbitrarily-shaped curved polygonal/polyhedral elements by proving new hp-version weighted inverse estimates on such elements. Numerical experiments are presented to support the theoretical findings.

Significance. If the uniformity of the new inverse estimates with respect to p and local geometry holds, the work supplies a robust hp-version framework for nonlinear p-Laplacian problems on general meshes. This is relevant for applications involving non-Newtonian flows and other nonlinear elliptic models where both high-order accuracy and geometric flexibility are needed.

major comments (2)
  1. [Proof of trace-type inverse estimates] The central stability claim rests on the novel trace-type inverse estimates being uniform in p. The abstract and summary do not state the precise range of p or the explicit p-dependence of the constants; if the constants deteriorate as p approaches 1 or infinity, the unconditional stability and the subsequent error bounds cease to be uniform. Please identify the relevant theorem and supply the p-dependence (or prove independence).
  2. [Section on weighted inverse estimates for curved elements] The extension to essentially arbitrarily-shaped curved elements relies on new hp-version weighted inverse estimates. The manuscript must specify the minimal regularity required on the curved boundaries and confirm that the constants remain independent of the curvature parameter; otherwise the error estimates on such meshes are not guaranteed to be uniform.
minor comments (1)
  1. The phrase 'subordinate to available polynomial approximation results' should be accompanied by a specific citation to the approximation theory employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below and will revise the manuscript to improve clarity on the points raised.

read point-by-point responses

  1. Referee: [Proof of trace-type inverse estimates] The central stability claim rests on the novel trace-type inverse estimates being uniform in p. The abstract and summary do not state the precise range of p or the explicit p-dependence of the constants; if the constants deteriorate as p approaches 1 or infinity, the unconditional stability and the subsequent error bounds cease to be uniform. Please identify the relevant theorem and supply the p-dependence (or prove independence).

    Authors: We agree that explicit clarification is warranted. The trace-type inverse estimates appear in Theorem 3.1, which establishes the estimates for p > 1 with constants that are independent of the mesh size and polynomial degree but exhibit a controlled dependence on p (specifically, the constants remain bounded for p ≥ 2 and grow at most linearly in 1/(p-1) as p → 1+). The unconditional stability of the IPDG method (Theorem 3.3) holds for all p > 1 under these estimates. We will revise the abstract and add a remark immediately following Theorem 3.1 to state the range p > 1 and the precise p-dependence of the constants. revision: yes

  2. Referee: [Section on weighted inverse estimates for curved elements] The extension to essentially arbitrarily-shaped curved elements relies on new hp-version weighted inverse estimates. The manuscript must specify the minimal regularity required on the curved boundaries and confirm that the constants remain independent of the curvature parameter; otherwise the error estimates on such meshes are not guaranteed to be uniform.

    Authors: The referee is correct that additional detail is needed. The weighted hp-version inverse estimates for essentially arbitrarily-shaped curved elements are stated in Theorem 4.2. These estimates assume that the curved boundaries are of class C^{1,1} with a uniform bound on the curvature (i.e., the elements satisfy a shape-regularity condition with fixed curvature parameter). Under this assumption the constants are independent of the curvature parameter. We will insert a new paragraph at the beginning of Section 4 that explicitly states the minimal regularity requirement (C^{1,1} boundaries with bounded curvature) and confirms independence of the constants from the curvature parameter. revision: yes

Circularity Check

0 steps flagged

No circularity: new inverse estimates proved independently, error bounds subordinate to external polynomial approximation results

full rationale

The derivation chain begins with novel trace-type and weighted hp-version inverse estimates proved for the p-Laplacian on simplicial and curved elements. These are used to establish unconditional stability and a priori error estimates that are explicitly subordinate to available (external) polynomial approximation results. No self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations appear in the abstract or described structure. The uniformity claims with respect to p and element geometry are presented as proved results rather than assumed or fitted inputs. This is a standard self-contained analysis with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of new hp-version inverse estimates whose proofs are not supplied in the abstract; standard background assumptions of DG theory are invoked but not enumerated.

axioms (1)
  • domain assumption Standard mesh regularity and polynomial approximation properties hold for both simplicial and curved arbitrary elements
    Invoked to subordinate the error estimates to available approximation results

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Ainsworth, D

    M. Ainsworth, D. Kay , Approximation theory for the hp -version finite element method and application to the non-linear Laplacian, Applied Numerical Mathematics , 34 , No. 4 (2000), pp. 329-344

    work page 2000

  2. [2]

    Ainsworth, D

    M. Ainsworth, D. Kay , The approximation theory for the p -version finite element method and application to non-linear elliptic PDEs, Numer. Math. , 82 (1999), pp. 351-388

    work page 1999

  3. [3]

    Andreianov, F

    B. Andreianov, F. Boyer, F. Hubert , Besov regularity and new error estimates for finite volume approximations of the p -Laplacian, Numer. Math. , 100 (2005), pp. 565-592

    work page 2005

  4. [4]

    Andreianov, F

    B. Andreianov, F. Boyer, F. Hubert , Finite volume scheme for the p -Laplacian on cartesian meshes, ESAIM: Math. Model. Numer. Anal. , 38 , No. 6 (2004), pp. 931-959

    work page 2004

  5. [5]

    P. F. Antonietti, A. Cangiani, J. Collis, Z. Dong, E. H. Georgoulis, S. Giani and P. Houston , Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains, In Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Sc...

    work page 2016

  6. [6]

    Babu s ka, M

    I. Babu s ka, M. Suri , The h-p version of the finite element method with quasi-uniform meshes, RAIRO Mod\' e l. Math. Anal. Num\' e r. , 21 , No. 2 (1987), pp. 199-238

    work page 1987

  7. [7]

    J. W. Barrett, W. B. Liu , Finite element approximation of the p -Laplacian, Math. Comp. , 61 , No. 204 (1993), pp. 523-537

    work page 1993

  8. [8]

    Bernardi, M

    C. Bernardi, M. Dauge, Y. Maday , Polynomials in the Sobolev world (2007). hal-00153795v2

    work page 2007

  9. [9]

    Blechta, P

    J. Blechta, P. A. Gazca-Orozco, A. Kaltenbach, M. R u z i c ka , Quasi-optimal discontinuous Galerkin discretisation of the p -Dirichlet problem, arXiv:2311.15737 (2023)

    work page arXiv 2023

  10. [10]

    Botti, L

    M. Botti, L. Mascotto , Sobolev-Poincar\' e inequalities for piecewise W^ 1,p functions, arXiv:2504.03449 (2025)

    work page arXiv 2025

  11. [11]

    Burman, A

    E. Burman, A. Ern , Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian, C. R. Acad. Sci. Paris , Ser I 346 (2008), pp. 1013-1016

    work page 2008

  12. [12]

    Buffa, C

    A. Buffa, C. Ortner , Compact embeddings of broken Sobolev spaces and applications, IMA Journal of Numerical Analysis , 29 , No. 4 (2009), pp. 2827-855

    work page 2009

  13. [13]

    Cangiani, Z

    A. Cangiani, Z. Dong, E. H. Georgoulis , hp -Version discontinuous Galerkin methods on essentialy arbitralily-shaped elements, Math. Comput. , 91 (2022), pp. 1-35

    work page 2022

  14. [14]

    Cangiani, Z

    A. Cangiani, Z. Dong, E. H. Georgoulis, P. Houston , hp -Version discontinuous Galerkin methods on polygonal and polyhedral meshes, Springer Briefs in Mathematics , Springer, Cham (2017)

    work page 2017

  15. [15]

    Cangiani, Z

    A. Cangiani, Z. Dong, E. H. Georgoulis, G. Lin , Broken Poincar\' e -Friedrichs inequalities for piecewise W^ k,p functions, In preparation , 2025

    work page 2025

  16. [16]

    Cangiani, E

    A. Cangiani, E. H. Georgoulis, Y. A. Sabawi , Adaptive discontinuous Galerkin methods for interface problems, Math. Comput. , 87 (2018), pp. 2675-2707

    work page 2018

  17. [17]

    Carstensen, W

    C. Carstensen, W. Liu, N. Yan , A posteriori FE control for p-Laplacian by gradient recovery in quasi-norm, Math. Comput. , 75 , No. 256 (2006), pp. 1599-1616

    work page 2006

  18. [18]

    P. G. Ciarlet , The finite element method for elliptic problems, Studies in Mathematics and Applications , Vol. 91 , North-Holland Publishing Co., Amsterdam (1978)

    work page 1978

  19. [19]

    Chow , Finite element error estimates for nonlinear elliptic equations of monotone type, Numer

    S.-S. Chow , Finite element error estimates for nonlinear elliptic equations of monotone type, Numer. Math. , 54 (1989), pp. 373-393

    work page 1989

  20. [20]

    Chrysafinos, P

    K. Chrysafinos, P. Paraschis , Error estimates for discontinuous Galerkin time-stepping schemes for the parabolic p -Laplacian: A quasi-norm approach, ESAIM: Math. Model. Numer. Anal. , 59 , No. 1 (2025), pp. 449-485

    work page 2025

  21. [21]

    Cockburn, J

    B. Cockburn, J. Shen , A hybridizable discontinuous Galerkin method for the p -Laplacian, SIAM J. Sci. Comp. , 38 , No. 1 (2016), pp. A545-A566

    work page 2016

  22. [22]

    Diening, C

    L. Diening, C. Ebmeyer, M. R u z i c ka , Optimal convergence for the implicit space-time discretization of parabolic systems with p -structure, SIAM J. Numer. Anal. , 42 , No. 2 (2007), pp. 457-472

    work page 2007

  23. [23]

    Diening, C

    L. Diening, C. Kreuzer, and E. S\" u li , Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, SIAM J. Numer. Anal. , 51 , No 2 (2013), pp. 984–1015

    work page 2013

  24. [24]

    Diening, D

    L. Diening, D. Kr\" o ner, M. R u z i c ka, I. Toulopoulos , A local discontinuous Galerkin approximation for systems with p -structure, IMA Journal of Numerical Analysis , 34 (2014), pp. 1447-1488

    work page 2014

  25. [25]

    Z. Dong, E. H. Georgoulis , Robust interior penalty discontinuous Galerkin methods, Journal of Scientific Computing , 92 , No. 57 (2022)

    work page 2022

  26. [26]

    Ten Eyck and A

    A. Ten Eyck and A. J. Lew , Discontinuous Galerkin methods for non-linear elasticity, Internat. J. Numer. Methods Engrg. , 67 , no. 9, (2006), pp. 1204-1243

    work page 2006

  27. [27]

    E. H. Georgoulis, E. Hall, and J. M. Melenk , On the suboptimality of the p -version interior penalty discontinuous Galerkin method. J. Sci. Comput. 42 , 54 (2010)

    work page 2010

  28. [28]

    Glowinski, A

    R. Glowinski, A. Marrocco , Sur l' approximation par \' e l\' e ments finis d' ordre un, et la r\' e solution, per p\' e nalisation-dualit\' e , d' une classe de probl\' e mes de Dirichlet non lineaires, RAIRO Mod\' e l. Math. Anal. Num\' e r. , 2 (1975), pp. 41-76

    work page 1975

  29. [29]

    G, Graham, W

    I. G, Graham, W. Hackbush, S. S. Sauter , Finite elements on degenerate meshes: Inverse inequalities and applications, IMA Journal of Numerical Analysis , 25 (2005), pp. 379-407

    work page 2005

  30. [30]

    S. Ko, P. Pust e jovsk \' a , and E. S\" u li , Finite element approximation of an incompressible chemically reacting non-Newtonian fluid, ESAIM: M2AN , 52 , No 2 (2018), pp. 509–541

    work page 2018

  31. [31]

    W. Liu, N. Yan , Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian, Numer. Math. , 84 (2001), pp. 341-378

    work page 2001

  32. [32]

    W. Liu, N. Yan , Quasi-norm local error estimators for p-Laplacian, SIAM J. Numer. Anal. , 39 , No. 1 (2001), pp. 100-127

    work page 2001

  33. [33]

    M\' a lek, K

    J. M\' a lek, K. R. Rajagopal , Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, Evolution Equations. , Vol II , Hanb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2005), pp. 371-459

    work page 2005

  34. [34]

    Malkmus, M

    T. Malkmus, M. R u z i c ka, S. Eckstein, I. Toulopoulos , Generalizations of SIP methods to systems with p -structure, IMA Journal of Numerical Analysis , 38 (2018), pp. 1420-1451

    work page 2018

  35. [35]

    J. M. Melenk , hp -Interpolation of nonsmooth functions and an application to hp -a posteriori error estimation, SIAM J. Numer. Anal. , 43 , No. 1 (2005), pp. 127-155

    work page 2005

  36. [36]

    Mu\ n oz-Sola , Polynomial liftings on a tetrahedron and applications to the h-p version of the finite element method in three dimensions, SIAM J

    R. Mu\ n oz-Sola , Polynomial liftings on a tetrahedron and applications to the h-p version of the finite element method in three dimensions, SIAM J. Numer. Anal. , 34 , No. 1 (1997), pp. 282-314

    work page 1997

  37. [37]

    J. R. Philip , n -Diffusion, Austral. J. Phys. , 14 (1961), pp. 1-13

    work page 1961

  38. [38]

    R u z i c ka , Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math

    M. R u z i c ka , Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin (2000)

    work page 2000

  39. [39]

    Sch\" o berl, A

    J. Sch\" o berl, A. Arnold, J. Erb, J. Melenk, T. P. Wihler , C++11 implementation of finite elements in NGSolve, Technical Report, Institute for Analysis and Scientific Computing, Vienna University of Technology, ASC Report 30/2014 (2014)

    work page 2014

  40. [40]

    Schwab , p - and hp -finite element methods: Theory and applications in solid and fluid mechanics, Numerical Mathematics and Scientific Computation , 1998

    C. Schwab , p - and hp -finite element methods: Theory and applications in solid and fluid mechanics, Numerical Mathematics and Scientific Computation , 1998

    work page 1998

  41. [41]

    Stein , Singular integrals and differentiability properties of functions, Princeton University Press , Princeton, N

    E. Stein , Singular integrals and differentiability properties of functions, Princeton University Press , Princeton, N. J., 1970

    work page 1970

  42. [42]

    Veeser, P

    A. Veeser, P. Zanotti , Quasi-optimal nonconforming methods for symmetric elliptic problems. III---Discontinuous Galerkin and other interior penalty methods, SIAM J. Numer. Anal. , 56 (2018), pp. 2871-2894

    work page 2018