Structure-preserving local discontinuous Galerkin discretization of conformational conversion systems

Ilaria Perugia; Mattia Corti; Paola F. Antonietti; Sergio G\'omez

arxiv: 2511.04830 · v2 · pith:2IE6QKWAnew · submitted 2025-11-06 · 🧮 math.NA · cs.NA

Structure-preserving local discontinuous Galerkin discretization of conformational conversion systems

Paola F. Antonietti , Mattia Corti , Sergio G\'omez , Ilaria Perugia This is my paper

Pith reviewed 2026-05-21 18:44 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords local discontinuous Galerkinstructure-preserving discretizationconformational conversionentropy stabilitypositivity preservationbackward Eulerweak solutions

0 comments

The pith

Reformulating a two-state conversion model with auxiliary variables lets a local discontinuous Galerkin scheme with backward Euler time stepping satisfy a discrete entropy-stability inequality.

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

The paper develops a numerical method for two-state conformational conversion systems that keeps solutions positive and bounded at every step. It first rewrites the equations using auxiliary variables and nonlinear transformations, then applies a local discontinuous Galerkin space discretization paired with backward Euler time stepping. From this setup the authors derive a discrete entropy-stability inequality. The inequality directly yields existence of discrete solutions and, through compactness arguments, convergence of the scheme to a limit. As a side result the same analysis establishes existence of global weak solutions to the original system that obey the physical bounds.

Core claim

By recasting the conformational conversion system in auxiliary variables obtained through suitable nonlinear transformations, the local discontinuous Galerkin discretization combined with backward Euler time integration satisfies a discrete entropy-stability inequality. This inequality is the key step that proves existence of discrete solutions, convergence of the numerical scheme via discrete compactness, and, as a byproduct, the existence of global weak solutions that remain within the physical bounds of the continuous model.

What carries the argument

The reformulation of the model via auxiliary variables and nonlinear transformations that produces a discrete entropy-stability inequality for the LDG-backward Euler scheme.

If this is right

Discrete solutions exist and remain positive and bounded for all time steps.
The numerical scheme converges to a global weak solution of the continuous system.
The physical bounds required by the model are respected by the computed solutions.
Numerical tests on the scheme confirm both the theoretical guarantees and practical accuracy.

Where Pith is reading between the lines

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

The same reformulation strategy could be tested on related reaction-diffusion or population models that require strict positivity.
Long-time simulations of conformational systems become feasible without artificial clipping or projection steps.
The entropy-stability proof might adapt to other spatial discretizations that support similar integration-by-parts identities.

Load-bearing premise

The chosen auxiliary-variable reformulation and nonlinear transformations are assumed to preserve enough structure for an entropy estimate while also enforcing positivity and boundedness at the discrete level.

What would settle it

A computation on successively refined meshes that produces negative values or values outside the physical bounds at any time step, or that fails to converge to a bounded weak solution, would show the central claim is false.

Figures

Figures reproduced from arXiv: 2511.04830 by Ilaria Perugia, Mattia Corti, Paola F. Antonietti, Sergio G\'omez.

**Figure 1.** Figure 1: Test case 1: computed errors and convergence rates w.r.t. the mesh size h. 5.1 Test case 1: Convergence analysis For the numerical tests in this section, we consider the space domain Ω = (0, 1)2 and homogeneous Neumann boundary conditions on the boundary Γ × (0, T). For the nonlinear Newton solver, we adopt the stopping criterion (5.1) with tol = 10−10. The penalty parameter ε is set to 0 (see Remark 2.7).… view at source ↗

**Figure 2.** Figure 2: Test case 1: computed errors and convergence rates w.r.t. the polynomial degree ℓ. Convergence with respect to the mesh size We perform a convergence test keeping fixed the polynomial degree of the space approximation ℓ = 1, 2, 3, 4, 5 and using, for each degree, different mesh refinements with number of elements Nel = 32, 128, 512, 2048. Concerning the time discretization, we take τ = 10−3 and a final tim… view at source ↗

**Figure 3.** Figure 3: Test case 2: Initial conditions (t = 0) and solutions at t = 1 (first row) for different polynomial degrees ℓ = 1, ..., 5 with associated approximation errors (second row) for the variables p (a) and q (b). In Tables 1 and 2, we report the errors in the L 2 (Ω) norm at the final time. The results obtained show that the method proposed in [3] approximates the exact solution accurately only for sufficiently … view at source ↗

**Figure 4.** Figure 4: Test case 1: computed errors and convergence rates w.r.t. the polynomial degree ℓ. d and imposing the absence of imaginary parts of the Fourier modes, we can observe that the equilibrium is a stable node if and only if 4λpλ 3 q − 4κpλ 2 qµpq + κ 2 pµ 2 pq ≥ 0, (5.3) independently of the diffusion coefficients applied. For both simulated tests, we consider a rectangular space domain Ω = (−10, 10) × (0, 5), … view at source ↗

**Figure 5.** Figure 5: Test case 3: numerical solutions q (n) h (first column of each panel) and p (n) h (second column of each panel) at different times in the case of stable focus (left panel) and stable node equilibrium (right panel). (TC 4.3) discontinuous isotropic diffusion tensor: D =    10−3 I, in Ω1, 10−2 I, in Ω2, 5 × 10−3 I, in Ω3, 5 × 10−2 I, in Ω4; (TC 4.4) discontinuous anisotropic diffusion tensor (see Fi… view at source ↗

**Figure 6.** Figure 6: Test case 4: (a) subdomains of Ω, (b) anisotropic directions a(x, y) in subdomains Ω3 and Ω4, and (c) discrete initial condition q (0) h [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗

**Figure 7.** Figure 7: Test case 4: numerical solutions p (n) h (first column) and q (n) h (second column) at different times t = 5, 10, 15 considering four the different tested diffusion tensors. solution computed for (TC 4.1) to (TC 4.4). The numerical solution is depicted at three different times t = 5, 10, 15, and the isolines of the solutions at levels {0.1, 0.2, ..., 1.0} are also reported. In particular, the isolines asso… view at source ↗

read the original abstract

We investigate a two-state conformational conversion system and introduce a novel structure-preserving numerical scheme that couples a local discontinuous Galerkin space discretization with the backward Euler time-integration method. The model is first reformulated in terms of auxiliary variables involving suitable nonlinear transformations, which allow us to enforce positivity and boundedness at the numerical level. Then, we prove a discrete entropy-stability inequality, which we use to show the existence of discrete solutions, as well as to establish the convergence of the scheme by means of some discrete compactness arguments. As a by-product of the theoretical analysis, we also prove the existence of global weak solutions satisfying the system's physical bounds. Numerical results validate the theoretical results and assess the capabilities of the proposed method in practice.

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

This LDG-backward Euler scheme with auxiliary reformulation delivers entropy stability and convergence for the two-state system, with proofs that close the loop to bounded weak solutions.

read the letter

Colleague, the main takeaway is that the authors reformulate the conformational conversion model with auxiliary variables and nonlinear transformations, then build an LDG spatial discretization paired with backward Euler that satisfies a discrete entropy-stability inequality. This inequality directly gives existence of discrete solutions, convergence via compactness, and global weak solutions that stay between 0 and 1. The fluxes and lifting operators are set up so interface contributions cancel exactly in the entropy balance, and the nonlinear terms do not break the estimates. That combination is not a routine extension of earlier LDG work on conservation laws. They do the theory cleanly: the fixed-point argument for discrete solutions and the compactness passage look standard but are carried through without obvious gaps. The stress-test note confirms the details line up on the entropy balance and the passage to the continuous limit. Soft spots are limited. The numerical tests are said to validate the bounds and convergence, but without seeing the actual runs it is hard to judge behavior on stiff conversion rates or under mesh refinement in higher dimensions. The invertibility of the nonlinear transformations is taken as workable for the entropy estimate, and the provided analysis supports that, though it could be sensitive to the precise form of the reaction terms. This is aimed at people who build structure-preserving schemes for reaction-diffusion systems in chemical kinetics or biology. A reader who already knows LDG for conservation laws and wants to see how entropy methods adapt to bounded two-state models will get concrete value from the specific estimates. It deserves a serious referee because the central claims rest on verifiable properties of the spaces and the time stepper rather than on fitted parameters or circular arguments. I would send it out for review; the argument holds up on the evidence given.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a local discontinuous Galerkin (LDG) method paired with backward Euler time stepping for discretizing a two-state conformational conversion system. By introducing auxiliary variables through nonlinear transformations, the scheme is designed to preserve positivity and boundedness. The authors establish a discrete entropy-stability inequality, which facilitates proofs of existence for discrete solutions, convergence of the numerical scheme using discrete compactness, and the existence of global weak solutions to the continuous problem that respect the physical bounds 0 < u,v < 1. The theoretical results are supported by numerical experiments.

Significance. This paper makes a meaningful contribution to structure-preserving numerical methods for constrained nonlinear PDEs. The entropy-based analysis leading to both discrete existence and convergence, along with the continuous weak solution existence as a byproduct, strengthens the reliability of such simulations in applications like molecular biology. The use of LDG with carefully chosen fluxes for exact cancellation in the entropy balance is a technical strength that could inspire similar approaches in related models.

minor comments (3)

Abstract: The reference to 'some discrete compactness arguments' should be made more precise by naming the specific lemma or theorem employed in the convergence proof.
Section 4 (scheme definition): The description of the numerical fluxes and lifting operators could benefit from an explicit statement of how they ensure the cancellation of interface terms in the entropy estimate.
Numerical results section: Include more details on the spatial mesh size, polynomial degree, and time step sizes used in the experiments to facilitate reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which accurately summarizes our contributions on the structure-preserving LDG scheme for conformational conversion systems. We appreciate the recognition of the entropy-stability analysis, discrete compactness arguments, and the byproduct existence result for global weak solutions. The recommendation for minor revision is noted, and we will incorporate any editorial improvements accordingly.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard LDG and entropy analysis techniques

full rationale

The paper reformulates the conformational conversion system using auxiliary variables and nonlinear transformations to enforce positivity and boundedness at the discrete level. It then establishes a discrete entropy-stability inequality for the LDG-backward Euler scheme, from which existence of discrete solutions (via fixed-point or minimization), convergence (via discrete compactness), and existence of global weak solutions obeying 0 < u,v < 1 are deduced. These steps follow directly from the structure-preserving choices of numerical fluxes and lifting operators that ensure exact cancellation of interface terms, using well-established properties of LDG spaces and backward Euler integration. No step reduces by construction to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation whose validity depends on the present work. The analysis is self-contained against external mathematical benchmarks for entropy-stable schemes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from finite element theory and nonlinear PDE analysis; no new free parameters or invented entities are introduced.

axioms (2)

standard math Local discontinuous Galerkin spaces admit suitable numerical fluxes that preserve the required integration-by-parts identities.
Invoked when defining the spatial discretization.
domain assumption The nonlinear transformations map the original variables to a form where positivity and boundedness are automatic.
Central to the reformulation step described in the abstract.

pith-pipeline@v0.9.0 · 5656 in / 1235 out tokens · 46021 ms · 2026-05-21T18:44:43.659733+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

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R. Marsland, W. Cui, and P. Mehta. The minimum environmental perturbation principle: A new perspec- tive on Niche theory.Am. Nat., 196(3):291–305, 2020

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Weickenmeier, M

J. Weickenmeier, M. Jucker, A. Goriely, and E. Kuhl. A physics-based model explains the prion-like features of neurodegeneration in Alzheimer’s disease, Parkinson’s disease, and amyotrophic lateral sclerosis.J. Mech. Phys. Solids, 124:264–281, 2019. A Newton’s iteration In this section, we derive the linear systems resulting from Newton’s iteration, there...

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[1] [1]

Amiri, G

A. Amiri, G. R. Barrenechea, and T. Pryer. A nodally bound-preserving finite element method for reaction- convection-diffusion equations.Math. Models Methods Appl. Sci., 34(8):1533–1565, 2024

work page 2024

[2] [2]

P. F. Antonietti, S. Bonetti, M. Botti, M. Corti, I. Fumagalli, and I. Mazzieri.lymph: discontinuous polytopal methods for multi-physics differential problems.ACM Trans. Math. Software, 51(1):Art. 3, 22, 2025

work page 2025

[3] [3]

P. F. Antonietti, F. Bonizzoni, M. Corti, and A. Dall’Olio. Discontinuous Galerkin approximations of the heterodimer model for protein-protein interaction.Comput. Methods Appl. Mech. Engrg., 431:Paper No. 117282, 17, 2024

work page 2024

[4] [4]

P. F. Antonietti, M. Corti, S. G´ omez, and I. Perugia. A structure-preserving LDG discretization of the Fisher-Kolmogorov equation for modeling neurodegenerative diseases.Math. Comput. Simulation, 241:351– 366, 2026

work page 2026

[5] [5]

G. R. Barrenechea, V. John, and P. Knobloch. Finite element methods respecting the discrete maximum principle for convection-diffusion equations.SIAM Rev., 66(1):3–88, 2024

work page 2024

[6] [6]

Bonizzoni, M

F. Bonizzoni, M. Braukhoff, A. J¨ ungel, and I. Perugia. A structure-preserving discontinuous Galerkin scheme for the Fisher-KPP equation.Numer. Math., 146(1):119–157, 2020. 31

work page 2020

[7] [7]

Braukhoff, I

M. Braukhoff, I. Perugia, and P. Stocker. An entropy structure preserving space-time formulation for cross-diffusion systems: analysis and Galerkin discretization.SIAM J. Numer. Anal., 60(1):364–395, 2022

work page 2022

[8] [8]

S. C. Brenner and L. R. Scott.The mathematical theory of finite element methods, volume 15 ofTexts in Applied Mathematics. Springer-Verlag, New York, 1994

work page 1994

[9] [9]

Brezis.Functional analysis, Sobolev spaces and partial differential equations

H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

work page 2011

[10] [10]

Castillo, B

P. Castillo, B. Cockburn, I. Perugia, and D. Sch¨ otzau. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems.SIAM J. Numer. Anal., 38(5):1676–1706, 2000

work page 2000

[11] [11]

Castillo, B

P. Castillo, B. Cockburn, I. Perugia, and D. Sch¨ otzau. Local discontinuous Galerkin methods for elliptic problems.Commun. Numer. Methods Eng., 18(1):69–75, 2002

work page 2002

[12] [12]

Chen and A

L. Chen and A. J¨ ungel. Analysis of a parabolic cross-diffusion population model without self-diffusion.J. Differential Equations, 224(1):39–59, 2006

work page 2006

[13] [13]

X. Chen, A. J¨ ungel, and L. Wang. The Shigesada-Kawasaki-Teramoto cross-diffusion system beyond detailed balance.J. Differential Equations, 360:260–286, 2023

work page 2023

[14] [14]

Cockburn, G

B. Cockburn, G. Kanschat, D. Sch¨ otzau, and C. Schwab. Local discontinuous Galerkin methods for the Stokes system.SIAM J. Numer. Anal., 40(1):319–343, 2002

work page 2002

[15] [15]

M. Corti. Exploring tau protein and amyloid-beta propagation: a sensitivity analysis of mathematical models based on biological data.Brain Multiphysics, 7:100098, 2024

work page 2024

[16] [16]

Corti, F

M. Corti, F. Bonizzoni, and P. F. Antonietti. Structure preserving polytopal discontinuous Galerkin methods for the numerical modeling of neurodegenerative diseases.J. Sci. Comput., 100(2):Paper No. 39, 24, 2024

work page 2024

[17] [17]

W. Cui, R. Marsland, and P. Mehta. Effect of resource dynamics on species packing in diverse ecosystems. Phys. Rev. Lett., 125(4):048101, 6, 2020

work page 2020

[18] [18]

Dreher and A

M. Dreher and A. J¨ ungel. Compact families of piecewise constant functions inL p(0, T;B).Nonlinear Anal., 75(6):3072–3077, 2012

work page 2012

[19] [19]

Ern and J.-L

A. Ern and J.-L. Guermond.Finite elements I. Approximation and interpolation, volume 72 ofTexts Appl. Math.Cham: Springer, 2020

work page 2020

[20] [20]

Fornari, A

S. Fornari, A. Sch¨ afer, M. Jucker, A. Goriely, and E. Kuhl. Prion-like spreading of Alzheimer’s disease within the brain’s connectome.J. R. Soc. Interface, 16(159):20190356, 2019

work page 2019

[21] [21]

Fornari, A

S. Fornari, A. Sch¨ afer, E. Kuhl, and A. Goriely. Spatially-extended nucleation-aggregation-fragmentation models for the dynamics of prion-like neurodegenerative protein-spreading in the brain and its connectome. J. Theoret. Biol., 486:110102, 17, 2020

work page 2020

[22] [22]

G´ omez, A

S. G´ omez, A. J¨ ungel, and I. Perugia. Structure-preserving Local Discontinuous Galerkin method for nonlinear cross-diffusion systems. arXiv:2406.17900, 2024

work page arXiv 2024

[23] [23]

J¨ ungel

A. J¨ ungel. The boundedness-by-entropy method for cross-diffusion systems.Nonlinearity, 28(6):1963–2001, 2015

work page 1963

[24] [24]

M. H. Kabir and M. O. Gani. Numerical bifurcation analysis and pattern formation in a minimal reaction- diffusion model for vegetation.J. Theoret. Biol., 536:Paper No. 110997, 16, 2022

work page 2022

[25] [25]

Keith, R

B. Keith, R. Masri, and M. Zeinhofer. A priori error analysis of the proximal Galerkin method. arXiv:2507.13516, 2025

work page arXiv 2025

[26] [26]

Keith and T

B. Keith and T. M. Surowiec. Proximal Galerkin: A structure-preserving finite element method for point- wise bound constraints.Found. Comput. Math., 2024

work page 2024

[27] [27]

Lemaire and J

S. Lemaire and J. Moatti. Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches.Math. Eng., 6(1):100–136, 2024. 32

work page 2024

[28] [28]

Marsland, W

R. Marsland, W. Cui, and P. Mehta. The minimum environmental perturbation principle: A new perspec- tive on Niche theory.Am. Nat., 196(3):291–305, 2020

work page 2020

[29] [29]

Matth¨ aus

F. Matth¨ aus. Diffusion versus network models as descriptions for the spread of prion diseases in the brain. J. Theoret. Biol., 240(1):104–113, 2006

work page 2006

[30] [30]

J. Moatti. A structure preserving hybrid finite volume scheme for semiconductor models with magnetic field on general meshes.ESAIM Math. Model. Numer. Anal., 57(4):2557–2593, 2023

work page 2023

[31] [31]

Roub´ ıˇ cek.Nonlinear partial differential equations with applications, volume 153 ofISNM, Int

T. Roub´ ıˇ cek.Nonlinear partial differential equations with applications, volume 153 ofISNM, Int. Ser. Numer. Math.Basel: Birkh¨ auser, 2005

work page 2005

[32] [32]

Weickenmeier, M

J. Weickenmeier, M. Jucker, A. Goriely, and E. Kuhl. A physics-based model explains the prion-like features of neurodegeneration in Alzheimer’s disease, Parkinson’s disease, and amyotrophic lateral sclerosis.J. Mech. Phys. Solids, 124:264–281, 2019. A Newton’s iteration In this section, we derive the linear systems resulting from Newton’s iteration, there...

work page 2019