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arXiv · 2604.19521

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

Andrés Miniguano-Trujillo·Andrea Poiatti·Maurizio Grasselli·Benjamin Goddard·John Pearson
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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.

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Andrés Miniguano-Trujillo, Andrea Poiatti, Maurizio Grasselli, Benjamin Goddard, John Pearson. 2026-04-21. Singularities in phase separation models: a spectral element approach for the nonlocal Cahn-Hilliard equation. https://arxiv.org/abs/2604.19521

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