Primal finite element scheme of the Hodge-Laplace problem

Abstract

In this paper, we construct nonconforming finite element spaces $\boldsymbol{V}^{\mathbf{d}\cap\mathring{\boldsymbolδ}}_hΛ^k$ for the approximation of $HΛ^k\cap H^*_0Λ^k$ on simplicial meshes, for $n\ge 2$ and $1\le k\le n-1$, by enforcing adjoint continuity against piecewise Whitney spaces rather than trace matching. It holds, with $\mathbf{d}^k_h$ and $\boldsymbolδ_{k,h}$ denoting respectively the piecewise action of differential and codifferential operators, and $\boldsymbol{\mathfrak{H}}_hΛ^k$ being the discrete harmonic forms in the FEEC sense, that $\boldsymbol{\mathfrak{H}}_hΛ^k=\{\boldsymbolμ_h\in \boldsymbol{V}^{\mathbf{d}\cap\mathring{\boldsymbolδ}}_hΛ^k:\mathbf{d}^k_h\boldsymbolμ_h=0,\ \boldsymbolδ_{k,h}\boldsymbolμ_h=0\}$, which mirrors the continuous Hodge--Laplace kernel on domains with nontrivial topology. The space is not a classical Ciarlet-type finite element space; though, a uniform discrete Poincare inequality and locally supported basis functions (supported on at most two cells) are guaranteed. The resulting primal scheme yields an $O(h)$ error bound for smooth data and $O(h^s)$ on $s$-regular domains ($0<s\le 1$), nontrivial topology admitted. Two- and three-dimensional eigenvalue tests agree with the mixed method on perforated domains, which are given to verify the validity of the scheme.