High-order temporal parametric finite element methods for simulating solid-state dewetting

We propose a class of temporally high-order parametric finite element methods for simulating solid-state dewetting of thin films in two dimensions using a sharp-interface model. The process is governed by surface diffusion and contact point migration, along with appropriate boundary conditions. By incorporating the predictor-corrector strategy and the backward differentiation formula for time discretization into the energy-stable parametric finite element method developed by Zhao et al. (2021), we successfully construct temporally high-order schemes. The resulting numerical scheme is semi-implicit, requiring the solution of a linear system at each time step. The well-posedness of the fully discretized system is established. Moreover, the method maintains the long-term mesh equidistribution property. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, measured by the manifold distance, while maintaining good mesh quality throughout the evolution.