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

Abstract

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 $m\le n$ 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.

References in corpus (1)

• ${\mathcal{P}}_m$ Interior Penalty Nonconforming Finite Element Methods for $2m$-th Order PDEs in ${\mathbb{R}}^n$

Cited by in corpus (5)

• The Finite Neuron Method and Convergence Analysis
• On optimal finite element schemes for biharmonic equation
• A Unified Study of Continuous and Discontinuous Galerkin Methods
• Minimal consistent finite element space for the biharmonic equation on quadrilateral grids
• Optimal quadratic element on rectangular grids for $H^1$ problems