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

TL;DR

This work develops a unified framework of nonconforming finite element spaces for $H\Lambda^k$ on $\mathbb{R}^n$, extending the Crouzeix--Raviart construction to differential forms via piecewise Whitney forms. By exploiting a discrete adjoint viewpoint, the authors build locally supported bases and provably stable, optimally-approximating interpolators, enabling cell-wise assembly and leading to nonconforming de Rham complexes with commuting diagrams. They establish uniform discrete Poincaré inequalities, discrete Helmholtz and Hodge decompositions for piecewise-constant forms, and a suite of nonconforming FE schemes for the Hodge Laplace problem, including novel mixed formulations and dualities with conforming spaces. The framework yields a practical discretization pathway with local operators, connecting conforming and nonconforming FEEC and enabling efficient numerical treatment of Hodge-type problems. Overall, the paper provides both theoretical advances in nonconforming exterior calculus and concrete discretization strategies with demonstrated implementability and robustness across test problems.

Abstract

This paper constructs a unified family of nonconforming finite element spaces for $HΛ^k$ 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 $HΛ^0$. Optimal approximations and uniform discrete Poincaré 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.

Figures (9)

• Figure 1: Illustration of the three basis functions of $\mathbb{RT}(T)$ on a cell $T$. We pay particular attention to the sign of the outward normal component at every edge.
• Figure 2: Profile of a global basis functions in $\mathcal{B}_M$, supported on two adjacent cells.
• Figure 3: Profiles of linearly independent basis functions of $\mathcal{B}_M$, $M\in\mathcal{X}_h^i$.
• Figure 4: Illustration of global basis functions of $\mathbb{RT}^{\rm nc}_{h0}$ based on a boundary vertex $M$
• Figure 5: The local basis functions associated with a boundary vertex $M$ can work as global basis functions of $\mathbb{RT}^{\rm nc}_h$.

Theorems & Definitions (58)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• Remark 2.5
• Theorem 2.6
• Remark 2.7
• Lemma 2.8