The choice of stabilization term is a critical component of the virtual element method (VEM). However, the theory of VEM provides only asymptotic guidance for selecting the stabilization term, which ensures convergence as the mesh size approaches zero, but does not provide a unique prescription for its exact form. Thus, the selection of a suitable stabilization term is often guided by numerical experimentation and analysis of the resulting solution, including factors such as stability, accuracy, and efficiency. In this paper, we establish a new link between VEM and generalized barycentric coordinates, in particular isoparametric finite elements as a specific case. This connection enables the interpretation of the stability as the energy of a particular function in the discrete space, commonly known as the `hourglass mode.' Through this approach, this study sheds light on how the virtual element solution depends on the stabilization term, providing insights into the behavior of the method in more general scenarios.
翻译:选择稳定项是虚拟元素法 (VEM) 的关键组成部分。然而,VEM 的理论仅提供了指导选择稳定项的渐近指导,确保网格大小趋近于零时的收敛性,但并没有为其精确形式提供唯一处方。因此,选择合适的稳定项通常是由数值实验和分析其结果的解决方案所指导的,包括稳定性、精度和效率等因素。本文建立了虚拟元素法和广义重心坐标之间的新联系,特别是同位参数有限元作为特定情况。这种联系使得稳定性能够被解释为离散空间中特定函数的能量,通常被称为“钟表模式”。通过这种方法,本研究阐明了虚拟元素解法如何取决于稳定项,为在更一般的情况下研究该方法的行为提供了有益的见解。