A primal finite element scheme of the Hodge Laplace problem

Abstract: In this paper, a unified family, for any $n\geqslant 2$ and $1\leqslant k\leqslant n-1$, of nonconforming finite element schemes are presented for the primal weak formulation of the $n$-dimensional Hodge-Laplace equation on $H\Lambda^k\cap H<sup>*_0\Lambda<sup>k$ 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 $\mathcal{O}(h<sup>s)$-order rate on any $s$-regular domain, $0&lt;s\leqslant 1$, no matter what topology the domain has.