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The Hm-nonconforming virtual elements of any order k on any shape of polytope in Rn with constraints m > n and k ≥ m are constructed in a universal way. A generalized Green’s identity for Hm inner product m > n is derived, which is essential to devise the Hm-nonconforming virtual elements. The dimension of the Hm-nonconforming virtual elements can reduced by the Serendipity approach. By means of the local Hm projection and a stabilization term using the boundary degrees of freedom, the Hm-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 Hm 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. Finally, the optimal error estimates for the Hmnonconforming virtual element methods are achieved from an estimate of the weak continuity and the error estimate of the canonical interpolation.

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