In this work, we aim to develop energy-stable parametric finite element approximations for a sharp-interface model with strong surface energy anisotropy, which is derived from the first variation of an energy functional composed of film/vapor interfacial energy, substrate energy, and regularized Willmore energy. By introducing two geometric relations, we innovatively establish an equivalent regularized sharp-interface model and further construct an energy-stable parametric finite element algorithm for this equivalent model. We provide a detailed proof of the energy stability of the numerical scheme, addressing a gap in the relevant theory. Additionally, we develop another structure-preserving parametric finite element scheme that can preserve both area conservation and energy stability. Finally, we present several numerical simulations to show accuracy and efficiency as well as some structure-preserving properties of the proposed numerical methods. More importantly, extensive numerical simulations reveal that our schemes provide better mesh quality and are more suitable for long-term computations.

翻译：本文旨在为具有强表面能各向异性的尖锐界面模型开发能量稳定的参数有限元近似方法。该模型源自能量泛函的一阶变分，该泛函由薄膜/气相界面能、基底能量和正则化Willmore能量构成。通过引入两个几何关系，我们创新性地建立了一个等价的正则化尖锐界面模型，并进一步为该等价模型构建了能量稳定的参数有限元算法。我们详细证明了该数值格式的能量稳定性，弥补了相关理论的空白。此外，我们开发了另一种结构保持型参数有限元格式，该格式能够同时保持面积守恒和能量稳定性。最后，我们通过若干数值模拟展示了所提数值方法的精度、效率以及一些结构保持特性。更重要的是，大量的数值模拟结果表明，我们的格式提供了更好的网格质量，并且更适用于长期计算。