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In this paper, we propose a family of nonconforming finite elements for$2m$-th order partial differential equations in $\mathbb{R}^n$ on simplicialgrids when $m=n+1$. This family of nonconforming elements naturally extends theelements 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 thedimensions using the similarity properties of both shape function spaces anddegrees of freedom. The proposed elements have approximability, pass thegeneralized patch test and hence converge. We also establish quasi-optimalerror estimates in the broken $H^3$ norm for the 2D nonconforming element. Inaddition, we propose an $H^3$ nonconforming finite element that is robust forthe sixth order singularly perturbed problems in 2D. These theoretical resultsare further validated by the numerical tests for the 2D tri-harmonic problem.

Nonconforming Finite Element Spaces for $2m$-th Order Partial Differential Equations on $\mathbb{R}^n$ Simplicial Grids When $m=n+1$