A $$\theta $$ -L Approach for the Simulation of Solid-State Dewetting Problems with Strongly Anisotropic Surface Energies

This paper aims to develop an efficient numerical scheme for simulating solid-state dewetting with strongly anisotropic surface energies in two dimensions. The governing equation is a sixth-order, highly nonlinear geometric partial differential equation, which makes it quite challenging to design an efficient numerical scheme. To tackle this problem, we first introduce an appropriate tangent velocity of the interface curve which could help mesh points equally distribute along the curve, then we reformulate the governing equation in terms of the tangent angle \(\theta \) and the length L of the interface curve with a release of the stiffness brought by surface tension. To further reduce the numerical stability constraint from the high-order PDE, we propose a mixed finite element method for solving the reformulated \(\theta \)-L equations. Numerical results are provided to demonstrate that the \(\theta \)-L approach is not only efficient and accurate, but also has the mesh equidistribution property with an improved numerical stability.