The \(H^m\)-nonconforming virtual elements of any order k on any shape of polytope in \({\mathbb {R}}^n\) with constraints \(m> n\) and \(k\ge m\) are constructed in 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 virtual elements. By means of the local \(H^m\) projection and a stabilization term using only the boundary degrees of freedom, the \(H^m\)-nonconforming virtual element methods are proposed to approximate solutions of the m-harmonic equation. The norm equivalence of the stabilization on the kernel of the local \(H^m\) projection is proved by using the bubble function technique, the Poincaré inquality and the trace inequality, which implies the well-posedness of the virtual element methods. The optimal error estimates for the \(H^m\)-nonconforming virtual element methods are achieved from an estimate of the weak continuity and the error estimate of the canonical interpolation. Finally, the implementation of the nonconforming virtual element method is discussed.

具有约束\ {m> n \）和\ {k \ ge的\（{\ mathbb {R}} ^ n \）中任何形状的多边形上的任意阶k的\（H ^ m \）-不合格虚拟元素m \）以通用方式构造。推导了具有\（m> n \）的\（H ^ m \）内积的广义格林身份，这对于设计\（H ^ m \）-非协调虚拟元素至关重要。通过局部\（H ^ m \）投影和仅使用边界自由度的稳定项，提出了\（H ^ m \）-非协调虚拟元素方法来近似求解m谐波方程。利用气泡函数技术，庞加莱不等式和迹线不等式证明了局部\（H ^ m \）投影的核的稳定范数等价，这表明虚拟元素方法的适定性。从弱连续性的估计和典范插值的误差估计中获得了（H ^ m \）不符合虚拟元素方法的最佳误差估计。最后，讨论了非一致性虚拟元素方法的实现。