Nonconforming finite element spaces for 2mth order partial differential equations on ℝn simplicial grids when m = n + 1

In this paper, we propose a family of nonconforming finite elements for 2mth order partial differential equations in ℝn 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 ≤ 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 H3 norm for the 2D nonconforming element. In addition, we propose an H3 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 triharmonic problem.