The $H^m$-conforming virtual elements of any degree $k$ on any shape of polytope in $\mathbb R^n$ with $m, n\geq1$ and $k\geq m$ are recursively constructed by gluing conforming virtual elements on faces in a universal way. For the lowest degree case $k=m$, the set of degrees of freedom only involves function values and derivatives up to order $m-1$ at the vertices of the polytope. The inverse inequality and several norm equivalences for the $H^m$-conforming virtual elements are rigorously proved. The $H^m$-conforming virtual elements are then applied to discretize a polyharmonic equation with a lower order term. With the help of the interpolation error estimate and norm equivalences, the optimal error estimates are derived for the $H^m$-conforming virtual element method.

翻译：本文递归构造了 $\mathbb{R}^n$ 空间中任意多面体上任意阶数 $k$ 的 $H^m$ 协调虚拟元（其中 $m, n \geq 1$ 且 $k \geq m$），其核心思想是采用统一方式通过粘合面上的协调虚拟元实现构造。对于最低阶情形 $k=m$，自由度的集合仅包含多面体顶点处的函数值及前 $m-1$ 阶导数。严格证明了 $H^m$ 协调虚拟元的逆不等式及若干范数等价性。进一步将 $H^m$ 协调虚拟元应用于含低阶项的多调和方程的离散化。基于插值误差估计与范数等价性，推导了 $H^m$ 协调虚拟元方法的最优误差估计。