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：本文针对具有Cordes系数的非散度形式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。该方法基于适用于任意维度$d \ge 2$的层次化构造。分析过程依赖于凸区域$\Omega$上的连续Miranda-Talenti估计，具有基础性。我们在数据满足最小正则性条件下，证明了$H^2(\Omega)$空间中的稳定性与误差估计，并考虑了数值积分的影响。数值实验揭示了系数正则性与$H^2(\Omega)$收敛速率之间的内在关联。