FEM Solution of the Cahn–Hilliard Equation

This project implements the finite element method (FEM) to solve the Cahn–Hilliard equation, a nonlinear PDE used to model phase separation.

This work was completed as part of the second project in TMA4220 – Numerical Solution of Partial Differential Equations Using Finite Element Methods at NTNU.
The project was originally intended for groups of three students, but I completed it individually and received a grade of 98/100 (A).

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Overview

The Cahn–Hilliard equation describes phase separation dynamics:

$$ \partial_t \phi = \nabla \cdot \Big( M \nabla \big( \gamma (\phi^3 - \phi) - \epsilon \Delta \phi \big) \Big) \quad \text{in } \Omega \times [0,T]. $$

This notebook:

• Derives the weak formulation.
• Proves mass conservation and energy dissipation.
• Implements fully discrete FEM schemes:
  • Fully Explicit scheme
  • IMEX (Implicit–Explicit) scheme
• Uses NGSolve to solve and visualize the system in 2D.

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Project structure

• Project2.ipynb – main Jupyter Notebook with derivations, code, and results.
• README.md – project documentation (this file).

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Requirements

• Python 3.x
• NGSolve
• NumPy, Pandas, Matplotlib

Install requirements:

pip install numpy pandas matplotlib