A quantitative comparison between ^C0 and ^C1 elements for solving the Cahn-Hilliard equation

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.