Singularities in phase separation models: a spectral element approach for the nonlocal Cahn-Hilliard equation
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918459706703872 |
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| author | Miniguano-Trujillo, Andrés Poiatti, Andrea Grasselli, Maurizio Goddard, Benjamin Pearson, John |
| author_facet | Miniguano-Trujillo, Andrés Poiatti, Andrea Grasselli, Maurizio Goddard, Benjamin Pearson, John |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_19521 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Singularities in phase separation models: a spectral element approach for the nonlocal Cahn-Hilliard equation Miniguano-Trujillo, Andrés Poiatti, Andrea Grasselli, Maurizio Goddard, Benjamin Pearson, John Numerical Analysis Analysis of PDEs 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. |
| title | Singularities in phase separation models: a spectral element approach for the nonlocal Cahn-Hilliard equation |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2604.19521 |