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arxiv: 2605.19037 · v1 · pith:OB7CUJ3Rnew · submitted 2026-05-18 · 🧮 math.NA · cs.NA

DG = FEM + flat elements, Part I: Diffusion

Ji\v{r}\'i Szotkowski , V\'aclav Ku\v{c}era , Chi-Wang Shu , Antoine Quiriny , Jonathan Lambrechts , Nicolas Mo\"es , Jean-Fran\c{c}ois Remacle This is my paper

Pith reviewed 2026-05-20 07:42 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords finite element methoddiscontinuous Galerkindummy elementsdiffusionconvergenceerror estimatesPoisson problem
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The pith

Adding vanishing-thickness dummy elements with scaled diffusion turns FEM into Babuška-Zlámal DG for Poisson problems.

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

The paper shows how to convert a standard continuous finite element method into a discontinuous Galerkin method for diffusion by inserting a thin layer of dummy elements along cell boundaries. These dummy elements have their diffusion coefficient adjusted to be proportional to their vanishing thickness. As the thickness approaches zero, the FEM solution converges to the solution of the Babuška-Zlámal DG scheme using trapezoidal quadrature. This connection allows implementing DG features inside existing FEM software with minimal changes, such as a mesh modification and a Jacobian threshold. Optimal error estimates in H1 and L2 norms are proved, and the approach supports adaptive switching between FEM and DG on an element basis.

Core claim

The central discovery is that by inserting a vanishing-thickness layer of dummy elements along cell interfaces and setting the diffusion coefficient on these elements proportional to their thickness, the standard FEM formulation converges exactly to the Babuška-Zlámal discontinuous Galerkin method with trapezoidal edge quadrature as the thickness tends to zero, yielding optimal H¹ and L² error estimates.

What carries the argument

Vanishing-thickness dummy elements with diffusion coefficient proportional to thickness, handled via tempered finite element (TFEM) framework for degenerate Jacobians.

Load-bearing premise

The mathematical limit of dummy-element thickness to zero, with the chosen trapezoidal quadrature and tempered treatment of degenerate Jacobians, recovers the target DG scheme without introducing consistency or stability errors.

What would settle it

Numerical tests in which the dummy element thickness is successively reduced and the difference between the FEM solution and a reference DG solution fails to approach zero at the expected rate.

Figures

Figures reproduced from arXiv: 2605.19037 by Antoine Quiriny, Chi-Wang Shu, Jean-Fran\c{c}ois Remacle, Ji\v{r}\'i Szotkowski, Jonathan Lambrechts, Nicolas Mo\"es, V\'aclav Ku\v{c}era.

Figure 1
Figure 1. Figure 1: FEM solution with auxiliary nodes yi inserted in the mesh (left) and limiting DG solution (right). where we consider two types of intervals: Ii = [xi , yi+1] and Ji = [yi , xi ]. The idea is that we let each of the intervals Ji degenerate to the single point xi , cf [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: TFEM-DG solution of the problem −u ′′ = 4 on [0, 1] with D = h −3 . Coarse mesh with 10 elements. degenerate elements along every edge of Th, cf [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: Triangulation Th for DG discretization. Center: Intermediate ‘mesh’ with shrunken elements. Right: Final mesh T β h for equivalent FEM discretization; ET – vertical hatch (green), EE – horizontal hatch (red), EV – cross-hatch (blue). for all v ∈ V the following equation holds: Z Ω ∇u · ∇v dx = Z Ω fv dx, (18) where f ∈ L 2 (Ω) is a given function. This problem is a weak formulation of Poisson’s probl… view at source ↗
Figure 4
Figure 4. Figure 4: Mesh with vertex elements replaced by ‘holes’ with Neumann boundaries. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Moreover, denote the foot of the altitude from vertex [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Edge element KE which degenerates to an edge Γ. Since e∥ is orthogonal to e⊥, we get (omitting the index β for clarity): Z KE βD∇φj · ∇φk dx = Z KE βD∇∥φj · ∇∥φk dx + Z KE βD∇⊥φj · ∇⊥φk dx = (A) + (B). The integral (A) vanishes as β → 0+: |(A)| ≤ Z KE βD [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence of TFEM-DG in 2D for various [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence for Jmin lower than h 4.0 . Optimal convergence is preserved, as the theory states. However, some problems arise when Jmin nears machine precision. The results from [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: zoomed image of fine unstructured mesh of [0 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of how three artificial interface elements (tetrahedra) can be inserted between two [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: As predicted by the theory, we get optimal convergence rates in both [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 10
Figure 10. Figure 10: TFEM-DG solution for a problem with the exact solution [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Convergence behavior of TFEM-DG in 3D for various values of [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Convergence behavior for values of Jmin that are lower than h 5.0 . The order is preserved, as the theory states. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: TFEM-DG solution on a series of meshes with a moving circular front which switches individual [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
read the original abstract

We establish a simple, rigorous, and easy to implement connection between the classical continuous finite element method (FEM) and the discontinuous Galerkin (DG) method for Poisson's problem. The key idea is to insert a vanishing-thickness layer of "dummy" elements along cell interfaces. By modifying the diffusion coefficient on these elements to be proportional to their thickness, we prove the FEM formulation converges to Babu\v{s}ka-Zl\'amal DG with trapezoidal edge quadrature. The scheme is trivial to implement by (i) a mesh edit that introduces degenerate interface elements and (ii) a single Jacobian threshold in an otherwise unmodified FEM code to handle the degenerate elements via the tempered finite element (TFEM) framework. We provide a rigorous derivation of the resulting TFEM-DG scheme, prove optimal $H^1$ and $L^2$ error estimates, and present numerical experiments in 2D and 3D. The method allows for simple implementation of DG in a FEM code and even adaptive element-by-element switching between FEM and DG with minimal coding effort. The framework is readily extensible, as we will demonstrate in a companion paper dedicated to evolutionary nonlinear first-order hyperbolic systems.

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 / 2 minor

Summary. The manuscript establishes a connection between continuous FEM and DG methods for the Poisson problem by inserting a vanishing-thickness layer of dummy flat elements along cell interfaces and scaling the diffusion coefficient proportionally to element thickness. This modification causes the standard FEM variational form to converge to the Babuška-Zlámal DG scheme with trapezoidal edge quadrature. Implementation requires only a mesh edit to add degenerate interface elements and a Jacobian threshold within the tempered finite element (TFEM) framework. The authors provide a rigorous derivation of the resulting TFEM-DG scheme, prove optimal H¹ and L² error estimates, and include supporting numerical experiments in 2D and 3D. The approach enables straightforward DG implementation in FEM codes and element-wise adaptive switching between the methods.

Significance. If the central limit argument holds, the result is significant for providing a practical and mathematically grounded bridge between FEM and DG discretizations. It allows DG to be realized with minimal modifications to existing FEM software and supports adaptive FEM-DG switching. The rigorous derivation, optimal error estimates, and 2D/3D experiments strengthen the contribution; the promised extensibility to nonlinear hyperbolic systems in a companion paper could increase its broader utility in numerical PDEs.

major comments (2)
  1. [Derivation of TFEM-DG scheme and limit process] The convergence argument for the vanishing-thickness limit (described in the derivation of the TFEM-DG scheme): the tempered Jacobian thresholding must be shown to commute with ε → 0 without producing a non-vanishing O(1) perturbation in the interface integrals. An explicit estimate bounding any threshold-induced remainder term is needed to confirm exact recovery of the target DG formulation and to support the subsequent consistency analysis.
  2. [Proof of optimal error estimates] The optimal H¹ and L² error estimates: these implicitly assume that the Jacobian threshold introduces no additional consistency or stability error that survives the limit. Without a uniform bound on the threshold remainder, the estimates may not hold at the claimed rates; a concrete test or additional analysis isolating this effect would be required.
minor comments (2)
  1. [Implementation section] Clarify the precise definition and application of the Jacobian threshold (e.g., the cutoff value and its dependence on ε) with a short algorithmic description or pseudocode to aid reproducibility.
  2. [Throughout manuscript] Check notation consistency for the scaled diffusion coefficient and the thickness parameter ε across the derivation and numerical sections.

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 of the major comments below. We agree that the analysis of the Jacobian threshold in the vanishing limit requires additional rigor, and we will revise the manuscript accordingly to include the requested estimates.

read point-by-point responses

  1. Referee: The convergence argument for the vanishing-thickness limit (described in the derivation of the TFEM-DG scheme): the tempered Jacobian thresholding must be shown to commute with ε → 0 without producing a non-vanishing O(1) perturbation in the interface integrals. An explicit estimate bounding any threshold-induced remainder term is needed to confirm exact recovery of the target DG formulation and to support the subsequent consistency analysis.

    Authors: We thank the referee for pointing this out. Upon re-examination, our current derivation shows that the dummy element contributions vanish as ε → 0, but we acknowledge that an explicit bound on the remainder due to the Jacobian threshold is not provided. In the revised version, we will add a lemma establishing that the threshold-induced perturbation in the interface integrals is bounded by Cε, which vanishes in the limit. This will confirm that the TFEM-DG scheme exactly recovers the Babuška-Zlámal formulation without O(1) terms, supporting the consistency analysis. revision: yes

  2. Referee: The optimal H¹ and L² error estimates: these implicitly assume that the Jacobian threshold introduces no additional consistency or stability error that survives the limit. Without a uniform bound on the threshold remainder, the estimates may not hold at the claimed rates; a concrete test or additional analysis isolating this effect would be required.

    Authors: We agree with this assessment. The error estimates in the manuscript are derived assuming the limit process is exact. By incorporating the explicit estimate from the first comment, we will show that any additional error from the threshold is of higher order and does not affect the optimal convergence rates. We will include this in the revised proof of the error estimates and, if appropriate, add a numerical test isolating the threshold effect in the experiments section. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds by explicit limit and independent proof

full rationale

The paper derives the DG scheme from FEM by inserting a vanishing-thickness dummy-element layer, scaling the diffusion coefficient proportionally to thickness, and taking the mathematical limit of the resulting variational form while applying a Jacobian threshold for degenerate elements. It then proves optimal H¹ and L² estimates directly from this construction. No step reduces by definition to its own inputs, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose validity is presupposed by the present work; the central claim rests on an explicit convergence argument that is self-contained once the limit and quadrature choices are stated.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the limit behavior of degenerate elements and standard Sobolev-space theory for FEM and DG; the dummy elements are the primary modeling device introduced by the paper.

axioms (1)
  • domain assumption The limit of the modified FEM variational form as dummy thickness tends to zero equals the Babuška-Zlámal DG formulation with trapezoidal quadrature.
    This limit is the load-bearing step that converts the FEM statement into the target DG scheme.
invented entities (1)
  • dummy flat elements no independent evidence
    purpose: Create a vanishing interface layer whose scaled diffusion coefficient produces the DG jump terms in the limit.
    These elements are introduced as a mathematical device to unify the two methods; no independent physical evidence is claimed.

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