  In this paper, we propose a family of nonconforming finite elements for
$2m$-th order partial differential equations in $\mathbb{R}^n$ on simplicial
grids when $m=n+1$. This family of nonconforming elements naturally extends the
elements proposed by Wang and Xu [Math. Comp. 82(2013), pp. 25-43] , where $m
\leq n$ is required. We prove the unisolvent property by induction on the
dimensions using the similarity properties of both shape function spaces and
degrees of freedom. The proposed elements have approximability, pass the
generalized patch test and hence converge. We also establish quasi-optimal
error estimates in the broken $H^3$ norm for the 2D nonconforming element. In
addition, we propose an $H^3$ nonconforming finite element that is robust for
the sixth order singularly perturbed problems in 2D. These theoretical results
are further validated by the numerical tests for the 2D tri-harmonic problem.
