A primal finite element scheme of the Hodge Laplace problem

In this paper, a unified family, for any $n⩾2$ and $1⩽k⩽n-1$, of nonconforming finite element schemes are presented for the primal weak formulation of the $n$-dimensional Hodge-Laplace equation on $H{\mathrm{\Lambda }}^k\cap H_0^{\ast }{\mathrm{\Lambda }}^k$ and on the simplicial subdivisions of the domain. The finite element scheme possesses an $\mathcal{O}(h)$-order convergence rate for sufficiently regular data, and an $\mathcal{O}(h^s)$-order rate on any $s$-regular domain, $0<s⩽1$, no matter what topology the domain has.