Nonconforming Virtual Element Method for $2m$-th Order Partial Differential Equations in $\mathbb R^n$

A unified construction of the $H^{m$-nonconforming} virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^{m$-nonconforming} virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal error estimates are derived for the $H^{m$-nonconforming} virtual element methods.