Nonconforming finite element spaces for $H\Lambda^k$ in $\mathbb{R}^n$

Abstract

This paper constructs a unified family of nonconforming finite element spaces for $H\Lambda^k$ in $\mathbb{R}^n$ ($0\leqslant k\leqslant n$, $n\geqslant 1$). The spaces employ piecewise Whitney forms as shape functions, and include the lowest-degree Crouzeix-Raviart element space for $H\Lambda^0$. Optimal approximations and uniform discrete Poincar\'e inequalities are presented. Based on the newly constructed finite element spaces, discrete de Rham complexes with commutative diagrams, and the discrete Helmholtz decomposition and Hodge decomposition for piecewise constant spaces are established. All discrete operators involved are local, acting cell-wise. A framework of nonconforming finite element exterior calculus is then established, and is naturally connected to the classical conforming one. The cooperation of conforming and nonconforming finite element spaces leads to new discretization schemes of the Hodge Laplace problem. The new finite element spaces are constructed by a novel approach that seeks to mimic the dual connections between adjoint operators; novel construction methods and basic estimations are presented. Although the new spaces do not fit Ciarlet's finite element definition, they admit locally supported basis functions each spanning at most two adjacent cells, which makes the computation of the local stiffness matrices and the assembling of the global stiffness matrix implementable by following the standard procedure. Some numerical experiments are given to show the implementability and the performance of the new kind of spaces.