A residual-type a posteriori error estimation is developed for a $C^1$-conforming virtual element method (VEM) to solve a Kirchhoff plate bending problem. To derive the reliability and efficiency of the a posteriori error bound, the inverse inequalities and norm equivalence are developed over the underlying $C^1$-conforming virtual element space, and a weak interpolation operator together with its error estimates is given as well. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results on various benchmark tests confirm the robustness of the proposed error estimator and show the efficiency of the resulting adaptive VEM.

为一个残差型后验误差估计$C^1$-符合虚拟元法（VEM）来解决基尔霍夫板弯曲问题。为了推导出后验误差界的可靠性和效率，逆不等式和范数等价在基础上发展$C^1$-符合虚拟元素空间，并给出了弱插值算子及其误差估计。作为误差估计器的结果，通过具有单挂节点规则的网格细化策略引入了自适应 VEM。各种基准测试的数值结果证实了所提出的误差估计器的稳健性，并显示了生成的自适应 VEM 的效率。