Primal finite element scheme of the Hodge-Laplace problem

Abstract

In this paper, we construct nonconforming finite element spaces $\boldsymbol{V}^{\mathbf{d}\cap\mathring{\boldsymbol{\delta}}}_h\Lambda^k$ for the approximation of $H\Lambda^k\cap H^*_0\Lambda^k$ 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 $\boldsymbol{\delta}_{k,h}$ denoting respectively the piecewise action of differential and codifferential operators, and $\boldsymbol{\mathfrak{H}}_h\Lambda^k$ being the discrete harmonic forms in the FEEC sense, that $\boldsymbol{\mathfrak{H}}_h\Lambda^k=\{\boldsymbol{\mu}_h\in \boldsymbol{V}^{\mathbf{d}\cap\mathring{\boldsymbol{\delta}}}_h\Lambda^k:\mathbf{d}^k_h\boldsymbol{\mu}_h=0,\ \boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h=0\}$, 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.