We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a new weak formulation for the problem, in which the interface and its contact line are evolved simultaneously. By using piecewise linear elements in space and backward Euler in time, we then discretize the weak formulation to obtain a fully discretized parametric finite element approximation. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is extended for solving the sharp interface model of solid-state dewetting with anisotropic surface energies in the Riemmanian metric form. Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.

翻译：我们提出了一种精确且能量稳定的参数有限元方法，用于求解三维空间中固态去湿问题的尖锐界面连续模型。该模型描述了伴随接触线迁移的薄膜/蒸汽界面的运动，由表面扩散方程及接触线处的适当边界条件控制。我们针对该问题提出了一种新的弱形式，其中界面及其接触线同步演化。通过空间上的分片线性单元和时间上的后向欧拉格式，我们对该弱形式进行离散，得到完全离散的参数有限元近似。所提出的数值方法被证明是适定且无条件能量稳定的。此外，该方法被扩展用于求解黎曼度量形式下具有各向异性表面能的固态去湿尖锐界面模型。数值结果展示了所提数值方法的收敛性与效率，以及各向异性效应对固态去湿中薄膜形貌演化的影响。