### Abstract

The Cahn-Hilliard (CH) equation is a time-dependent fourth-order partial differential equation (PDE). When solving the CH equation via the finite element method (FEM), the domain is discretized by ^{C1}-continuous basis functions or the equation is split into a pair of second-order PDEs, and discretized via ^{C0}-continuous basis functions. In the current work, a quantitative comparison between ^{C1} Hermite and ^{C0} Lagrange elements is carried out using a continuous Galerkin FEM formulation. The different discretizations are evaluated using the method of manufactured solutions solved with Newton's method and Jacobian-Free Newton Krylov. It is found that the use of linear Lagrange elements provides the fastest computation time for a given number of elements, while the use of cubic Hermite elements provides the lowest error. The results offer a set of benchmarks to consider when choosing basis functions to solve the CH equation. In addition, an example of microstructure evolution demonstrates the different types of elements for a traditional phase-field model.

Original language | English (US) |
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Pages (from-to) | 74-80 |

Number of pages | 7 |

Journal | Journal of Computational Physics |

Volume | 236 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2013 |

### All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

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## Cite this

^{C0}and

^{C1}elements for solving the Cahn-Hilliard equation.

*Journal of Computational Physics*,

*236*(1), 74-80. https://doi.org/10.1016/j.jcp.2012.12.001