We propose and analyze an $H^2$-conforming Virtual Element Method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension $d \ge 2$. The analysis relies on the continuous Miranda-Talenti estimate for convex domains $\Omega$ and is rather elementary. We prove stability and error estimates in $H^2(\Omega)$, including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in $H^2(\Omega)$.

翻译：我们针对具有Cordes系数的最简非散度型线性椭圆偏微分方程，提出并分析了一种$H^2$-协调虚拟元方法（VEM）。该VEM基于对任意维度$d \ge 2$均有效的分层构造。分析过程依赖于凸区域$\Omega$上的连续Miranda-Talenti估计，且方法较为基础。在数据正则性最小的条件下，我们证明了$H^2(\Omega)$中的稳定性与误差估计（包含求积效应）。数值实验阐述了系数正则性与$H^2(\Omega)$收敛速率之间的相互作用。