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The $H^m$-nonconforming virtual elements of any order $k$ on any shape ofpolytope in $\mathbb R^n$ with constraints $m>n$ and $k\geq m$ are constructedin a universal way. A generalized Green's identity for $H^m$ inner product with$m>n$ is derived, which is essential to devise the $H^m$-nonconforming virtualelements. By means of the local $H^m$ projection and a stabilization term usingonly the boundary degrees of freedom, the $H^m$-nonconforming virtual elementmethods are proposed to approximate solutions of the $m$-harmonic equation. Thenorm equivalence of the stabilization on the kernel of the local $H^m$projection is proved by using the bubble function technique, the Poincar\'einquality and the trace inequality, which implies the well-posedness of thevirtual element methods. The optimal error estimates for the$H^m$-nonconforming virtual element methods are achieved from an estimate ofthe weak continuity and the error estimate of the canonical interpolation.Finally, the implementation of the nonconforming virtual element method isdiscussed.

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