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

TL;DR

This paper introduces a spectral element method for efficiently approximating the nonlocal Cahn-Hilliard equation, enabling high-resolution simulations of phase separation with singular kernels.

Contribution

It develops a pseudospectral multishape numerical framework that accurately handles the singular convolution kernel in the nonlocal Cahn-Hilliard model.

Findings

Achieves accurate numerical solutions with limited computational resources.

Handles singular kernels effectively in a spectral element context.

Provides high-resolution simulations of phase separation phenomena.

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.