Nonconforming Finite Element Spaces for $2m$-th Order Partial Differential Equations on $\mathbb{R}^n$ Simplicial Grids When $m=n+1$
Provides a theoretical extension of existing finite element methods to a previously unsolved case, enabling numerical solutions for higher-order PDEs in higher dimensions.
This paper extends nonconforming finite elements for 2m-th order PDEs to the case m=n+1 on simplicial grids, proving unisolvency, convergence, and quasi-optimal error estimates; numerical tests for 2D tri-harmonic problems validate the theory.
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.