Generalized Gaffney inequality and discrete compactness for discrete differential forms

TL;DR

This paper establishes generalized Gaffney inequalities and discrete compactness for finite element differential forms on various domains, extending previous electromagnetism results to broader settings with new analytical tools.

Contribution

It introduces a new Hodge mapping and its approximation properties to generalize key inequalities and compactness results for discrete differential forms on Lipschitz domains.

Findings

Proves generalized Gaffney inequalities for finite element differential forms.

Establishes discrete compactness for these forms on Lipschitz domains.

Provides $L^{p}$ estimates for finite element solutions of Laplacian problems.

Abstract

We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on $s$-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have been established for edge elements with weakly imposed divergence-free conditions and used in the analysis of nonlinear and eigenvalue problems. In this paper, we generalize these results to discrete differential forms, not necessarily with strongly or weakly imposed constraints. The analysis relies on a new Hodge mapping and its approximation property. As an application, we show $L^{p}$ estimates for several finite element approximations of the scalar and vector Laplacian problems.