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arxiv: 2604.19521 · v1 · submitted 2026-04-21 · 🧮 math.NA · cs.NA· math.AP

Recognition: unknown

Singularities in phase separation models: a spectral element approach for the nonlocal Cahn-Hilliard equation

Andr\'es Miniguano-Trujillo , Andrea Poiatti , Maurizio Grasselli , Benjamin Goddard , John Pearson
Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:57 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP
keywords nonlocal Cahn-Hilliard equationphase separationsingular convolution kernelpseudospectral methodnumerical approximationNewtonian interaction
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The pith

The nonlocal Cahn-Hilliard equation with singular long-range interactions can be simulated at high resolution using a pseudospectral multishape method that requires only limited computational resources.

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 technique to simulate how materials separate into different phases when particles interact over long distances through a kernel that blows up at short range. Standard discretizations create dense operators and struggle with the singularity in the kernel. Their pseudospectral multishape method approximates these convolution terms reliably while respecting no-flux boundaries. This matters because realistic phase separation models with long-range effects become solvable on ordinary computers instead of requiring massive resources. If successful, researchers gain practical access to detailed time evolution of such mixtures.

Core claim

The authors introduce an efficient and error-controlled numerical framework for the nonlocal Cahn-Hilliard system with constant mobility, logarithmic potential, Newtonian interaction kernel, and no-flux boundary conditions. Their approach rests on a pseudospectral multishape method that accurately approximates the action of singular convolution operators, and they show that high-resolution numerical solutions can be achieved with limited computational resources.

What carries the argument

The pseudospectral multishape method, which divides the computational domain to approximate the dense singular convolution operator without special post-processing.

If this is right

  • High-resolution numerical solutions of the nonlocal system become practical on standard hardware.
  • The singular Newtonian kernel can be treated accurately within the chosen discretization without post-hoc adjustments.
  • Constant-mobility logarithmic-potential cases with no-flux boundaries admit controlled error behavior at moderate cost.
  • The framework supports systematic exploration of long-range interaction effects in phase separation dynamics.

Where Pith is reading between the lines

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

  • The same discretization strategy could be tested on other singular kernels or boundary conditions to check broader applicability.
  • Practical high-resolution runs might allow direct comparison against existence and regularity theorems for the continuous model.
  • Reducing the computational barrier could enable parameter sweeps that reveal how interaction range affects coarsening rates.
  • The multishape idea may transfer to related nonlocal evolution equations that appear in fluid or biological models.

Load-bearing premise

The multishape pseudospectral discretization controls approximation errors near the kernel singularity and stays stable for the Newtonian interaction under no-flux boundary conditions without hidden instabilities or extra fixes.

What would settle it

Running the scheme on a benchmark problem with a known reference solution and observing either growing pointwise errors near the singularity or unphysical oscillations in the phase field would show the accuracy claim does not hold.

Figures

Figures reproduced from arXiv: 2604.19521 by Andrea Poiatti, Andr\'es Miniguano-Trujillo, Benjamin Goddard, John Pearson, Maurizio Grasselli.

Figure 1
Figure 1. Figure 1: Quadrilateral. Among the shapes available in MultiShape, we will rely on the Quadrilateral class to efficiently compute the Newtonian potential term I1 in (18). As its name suggests, this class allows us to build planar quadrilaterals that together compose a multishape; see [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Maximal quadrilateral partitions based on a neighbourhood around a collocation point. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Swarm plots of the relative errors eε for approximating the Newtonian potential under refinement for the maximal partition strategy. The horizontal labels are presented in the format N/α, corresponding to the grid resolution N ∈ {10, 20} and the scaling factor α. Additionally, two options for the neighbourhood radius are presented, specifically ε ∈ {10−2 , 10−5}. The results are presented in [PITH_FULL_IM… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution (in log-log scale) of the absolute error (vertical axes) as [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of ρ(x;t) from the initial condition (31) and different values of η. As η became increasingly negative, the system started to approximate the step function 2  1R≥0 (x1 − 1/2)  − 1. Note that this function does not depend on x2. Overall, we found that the system evolves faster towards a stationary state if we increase the absolute value of η. In [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Free energy dissipation and convergence towards equilibrium for different values of the scaling parameter [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of ρ(x;t) from the initial condition (33) and two choices of η. Running the system for T = 1 and η ∈ [−100, 500], we obtained a similar behaviour to what we observed in the previous two examples: Energies increase as η increases, there is a plateau for δ, mass is conserved, and stationary states are reached in finite time [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of ρ(x;t) from the initial condition (33) at times 3σ, T + 3σ, 1, and 10. The numerical values of ρ(x, 3σ) and ρ(x, T + 3σ) were computed using the DAE solver in Matlab with absolute and relative tolerances set to 10−11. We then set ρ(x, 3σ) as initial condition for the regularised problem and computed ρω(x, T). The numerical L 2 (Ω) error, [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of ρ(x;t) from the initial condition (33) and η ∈ {−5, −1, 2, 3, 5, 10} × 102 . We tested the effect of Lη for several values of η inside the interval [−500, 1 000], the initial condition of compact support (33), and maximum time T = 1. The resulting dynamics displayed many familiar shapes that we observed in previous sections, but additional structures were formed for some values of η. We collec… view at source ↗
Figure 9
Figure 9. Figure 9: (Continued) Evolution of ρ(x;t) from the initial condition (33) and η ∈ {−5, −1, 2, 3, 5, 10} × 102 . 5 Conclusions We have successfully found solutions to the nonlocal Cahn–Hilliard system with constant mobility, featuring logarithmic and Newtonian potentials. Our study began with a review of the system, presenting an extension of the existence and uniqueness theory to polygonal domains. The Newtonian pot… view at source ↗
Figure 10
Figure 10. Figure 10: Square to rectangle map. Ψ : Ω ∋ (x1, x2) 7−→ [b1 − a1]x1 + a1, [b2 − a2]x2 + a2  ∈ Θ, which is given by Ψ −1 : Θ ∋ (y1, y2) 7−→  y1 − a1 b1 − a1 , y2 − a2 b2 − a2  ∈ Ω. For convolutions, this transformation yields [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Bulged square map. Ψ : Ω ∋ (x1, x2) 7−→  (2x1 − 1) 1 + 4kx2 (1 − x2) [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Minimal quadrilateral partitions based on a neighbourhood around a collocation point. [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Swarm plots of the relative errors eε for approximating the Newtonian potential under refinement. The horizontal labels are presented in the format N/α, corresponding to the grid resolution N ∈ {10, 20} and the scaling factor α ∈ {4, 5, 8, 10}. Additionally, two options for the neighbourhood radius are presented, specifically ε ∈ {10−2 , 10−5}. The geometric construction in [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 14
Figure 14. Figure 14: Proposed partition method for a disc based on a neighbourhood around a collocation point. [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: New shapes added to MultiShape for disc partition. [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Evolution (in log-log scale) of the absolute error (vertical axes) as [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Final states of ρ(x;t) starting from the initial conditions (A–3.4) with η ∈ {10, 5, −1, −2} × 10. To approximate the quadrature values of the convolution, we introduce the regularised kernel Kσ : R 3 \ {0} ∋ x 7−→ − 1 4π max{σ, ∥x∥} ∈ R<0. A simple choice for σ is to select a value smaller than the minimum distance between any two collocation points. Here we recall that, for a line of N Chebyshev–Gauss–L… view at source ↗
Figure 18
Figure 18. Figure 18: Final states ρ(x; T) obtained from the adapted initial condition (A–3.4), for η ∈ {5, −1, −2, −3} × 10. sections, values close to −1 correspond to near-black shades, while values close to 1 appear as the brightest areas. Intermediate grayscale tones represent values around zero. We observe that for positive scalings the kernel diffuses, leading ρ(T) to become nearly constant, whereas for negative values o… view at source ↗
read the original abstract

The nonlocal Cahn-Hilliard equation provides a natural extension of the classical model for phase separation by incorporating long-range interactions through a singular convolution kernel. While this formulation admits a rich existence and regularity theory, its numerical approximation remains challenging: discretisation of the nonlocal term leads to dense operators, and the singularity of the kernel requires special treatment in collocation-based schemes. In this work, we develop an efficient and error-controlled numerical framework for the nonlocal Cahn-Hilliard system with constant mobility, logarithmic potential, Newtonian interaction kernel, and no-flux boundary conditions. Our approach is based on a pseudospectral multishape method that accurately approximates the action of singular convolution operators. We present high-resolution numerical solutions for this nonlocal system of equations that can be achieved with limited computational resources.

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

1 major / 0 minor

Summary. The manuscript develops a pseudospectral multishape discretization for the nonlocal Cahn-Hilliard equation with Newtonian interaction kernel, logarithmic potential, constant mobility, and no-flux boundary conditions. It claims this yields an efficient, error-controlled framework that produces high-resolution numerical solutions with limited computational resources.

Significance. If the error-control and accuracy claims for the singular convolution operator are substantiated, the work would provide a practical advance in the numerical treatment of nonlocal phase-field models, enabling reliable high-resolution simulations of long-range interactions that are otherwise hindered by dense operators and kernel singularities.

major comments (1)
  1. [Abstract] Abstract: The assertion of an 'error-controlled numerical framework' and 'accurate' approximation of singular convolution operators lacks any supporting a priori error bounds, convergence-rate analysis, benchmark comparisons, or quantitative tables (e.g., error versus polynomial degree or element count). This is load-bearing for the central claim, as the skeptic correctly identifies that the method's ability to control quadrature errors near the kernel origin and domain boundaries under no-flux conditions remains unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive criticism. We address the single major comment below and agree that revisions are needed to align the abstract claims more closely with the manuscript's content.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The assertion of an 'error-controlled numerical framework' and 'accurate' approximation of singular convolution operators lacks any supporting a priori error bounds, convergence-rate analysis, benchmark comparisons, or quantitative tables (e.g., error versus polynomial degree or element count). This is load-bearing for the central claim, as the skeptic correctly identifies that the method's ability to control quadrature errors near the kernel origin and domain boundaries under no-flux conditions remains unverified.

    Authors: We acknowledge the validity of this observation. The manuscript presents numerical experiments in Section 4 demonstrating the practical performance of the pseudospectral multishape method through high-resolution simulations and comparisons with reference solutions, including observed convergence behavior with respect to polynomial degree. However, no a priori error bounds or formal convergence-rate analysis are derived, and the abstract's phrasing overstates the level of theoretical control. We will revise the abstract by replacing 'error-controlled numerical framework' with 'efficient numerical framework' and 'accurate' with 'effective', while adding explicit references to the numerical validation studies. We will also insert a brief paragraph in the conclusions noting that rigorous a priori analysis of quadrature errors near singularities and boundaries is left for future work. These changes ensure the claims are supported by the presented evidence without misrepresentation. revision: yes

Circularity Check

0 steps flagged

No significant circularity: numerical discretization method is self-contained

full rationale

The paper proposes and demonstrates a pseudospectral multishape discretization for approximating singular convolution operators in the nonlocal Cahn-Hilliard equation. No derivation chain exists in which a claimed prediction or first-principles result reduces by construction to fitted inputs, self-citations, or renamed ansatzes. The central claims concern the efficiency and error control of the numerical scheme for the given kernel, potential, and boundary conditions, presented via high-resolution solutions rather than any self-referential mathematical reduction. The approach is a standard numerical framework development without load-bearing circular steps of the enumerated kinds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the established existence and regularity theory for the nonlocal Cahn-Hilliard equation and standard assumptions of spectral methods for PDEs. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The nonlocal Cahn-Hilliard equation with Newtonian kernel admits a rich existence and regularity theory
    Stated directly in the abstract as background for the numerical challenge.
  • domain assumption Pseudospectral methods can be extended to multishape domains to approximate singular convolution operators
    Implicit in the choice of discretization technique.

pith-pipeline@v0.9.0 · 5452 in / 1404 out tokens · 40242 ms · 2026-05-10T01:57:49.398674+00:00 · methodology

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