We establish a general framework to study the conforming and nonconforming virtual element methods (VEMs) for solving a Kirchhoff plate contact problem with friction, which is a fourth-order elliptic variational inequality (VI) of the second kind. This VI contains a non-differentiable term due to the frictional contact. This theoretical framework applies to the existing virtual elements such as the conforming element, the |$C^0$|-continuous nonconforming element and the fully nonconforming Morley-type element. In the unified framework we derive a priori error estimates for these virtual elements and show that they achieve optimal convergence order for the lowest-order case. For demonstrating the performance of the VEMs we present some numerical results that confirm the theoretical prediction of the convergence order.

中文翻译：

---

Kirchhoff板接触问题的一致和不一致虚拟元素方法

我们建立了一个通用的框架来研究用于解决带有摩擦的Kirchhoff板接触问题的合格和不合格虚拟元素方法（VEM），这是第二种四阶椭圆变分不等式（VI）。由于摩擦接触，该VI包含一个不可微分的项。该理论框架适用于现有的虚拟元素，例如一致性元素|| $ C ^ 0 $ ||。-连续不合格元素和完全不合格的Morley型元素。在统一框架中，我们得出先验这些虚拟元素的误差估计，并表明它们在最低阶情况下达到了最佳收敛阶。为了证明VEM的性能，我们提供了一些数值结果，这些结果证实了收敛阶的理论预测。