A primal finite element scheme of the Hodge Laplace problem

TL;DR

This paper introduces a family of nonconforming finite element schemes for solving the Hodge-Laplace problem in any dimension, achieving optimal convergence rates regardless of domain topology.

Contribution

It presents a unified, nonconforming finite element approach for the primal Hodge-Laplace problem applicable to arbitrary dimensions and domain topologies.

Findings

Achieves $ ext{O}(h)$ convergence rate for regular data.

Achieves $ ext{O}(h^s)$ convergence on $s$-regular domains.

Applicable to any domain topology without restrictions.

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 $H\Lambda^k\cap H^*_0\Lambda^k$ 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, $0<s\leqslant 1$, no matter what topology the domain has.