From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab$\check{\mathrm{u}}$ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is $h^{2k-2}$ for $k\ge 2$. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming $H^1\left(\mathrm{\Omega }\right)\times H^1\left(\mathrm{\Omega }\right)$ formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order $h^2$ for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order.

从特征值问题理论，我们看到I. Bab中混合方法得到的双调和特征值的收敛速度$\check{\mathrm{你}}$ska 和 J. Osborn，[特征值问题，数值分析手册，卷。II，荷兰北部，阿姆斯特丹，1991 年。] 是$H^{2克-2}$ 为了 $克\ge 2$. 在本文中，我们给出了近似基尔霍夫板振动问题的最低阶虚元方法。这种离散方案基于一个符合$H^1\left(\mathrm{\Omega }\right)\times H^1\left(\mathrm{\Omega }\right)$公式，遵循 Ciarlet-Raviart 方法的变分公式，这使我们能够利用更简单和规则性较低的虚拟元素空间。通过在泛函分析中使用经典的谱逼近理论，我们证明了谱逼近和最优收敛阶$H^2$为特征值。最后，给出了一些数值实验，表明所提出的数值方案可以达到最优收敛阶次。