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.
Zhang, L., Tonks, M., Millett P., “A Quantitative Comparison Between C0 and C1 Elements for Solving the Cahn–Hilliard Equation”, Journal of Computational Physics, 236, 74-80, 2013.