A nonconforming framework for finite element exterior calculus

Abstract This paper presents a theory of nonconforming finite element exterior calculus based on a unified family of nonconforming finite element spaces for $H\varLambda ^{k}$ in ${\mathbb{R}^{\boldsymbol{{n}}}}$ ($0\leqslant k\leqslant n$, $n\geqslant 1$), which are constructed in this paper by a novel approach that seeks to mimic the dual connections between adjoint operators. The family each employs piecewise Whitney forms as shape functions, including the lowest-degree Crouzeix–Raviart element space for $H\varLambda ^{0}$, and optimal approximations and uniform discrete Poincaré inequalities are presented. Further, with these 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, based on which the Poincaré–Leftschetz duality can be reconstructed discretely as an equality. The consequent framework of nonconforming finite element exterior calculus is naturally connected to the classical conforming one, but significantly different. Notably, all discrete operators involved are local, namely acting cell by cell separately. The newly constructed finite element spaces do not fit Ciarlet’s finite element definition, though 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 matrices implementable by following the standard procedure. Some numerical experiments are given to show the implementability and the performance of the new kind of spaces. The cooperation of conforming and nonconforming finite element spaces leads to new discretization schemes of the Hodge–Laplace problem.